tag:blogger.com,1999:blog-26793727472857054782024-03-07T19:19:59.504-08:00 cbse ncert class 12 maths solutionsdear students here i offers Free Online NCERT Solutions of class 12 math text book. i tried to solved maths problem stepwise in simple manner which can easily understood by students. i hopes my this work would help lot of students to understand the maths and also they will get good marks .
Anonymoushttp://www.blogger.com/profile/17104854705255350948noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-2679372747285705478.post-6191109921622212912013-01-18T16:31:00.000-08:002013-01-18T16:31:17.351-08:00 Miscellaneous relations and functions<div dir="ltr" style="text-align: left;" trbidi="on">
<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 1:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>be defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = 10<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>+ 7. Find the function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>o<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>o<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g<span class="Apple-converted-space"> </span></i>= 1<sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">R</b></sub>.</div>
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solutions =</div>
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<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is given that<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = 10<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>+ 7.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
One-one:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>), where<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>∈<b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒ 10<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>+ 7 = 10<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>+ 7</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>is a one-one function.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Onto:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>, let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>= 10<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>+ 7.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="41" name="Object204" src="http://img1.meritnation.com/img/curr/1/12/15/230/6113/Chapter%201_html_m6cba91ea.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="109" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, for any<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>, there exists<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="41" name="Object205" src="http://img1.meritnation.com/img/curr/1/12/15/230/6113/Chapter%201_html_m1ce549cf.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="91" />such that<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="45" name="Object206" src="http://img1.meritnation.com/img/curr/1/12/15/230/6113/Chapter%201_html_m3508d959.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="319" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>is onto.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>is one-one and onto.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is an invertible function.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let us define<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>as<img align="ABSMIDDLE" height="41" name="Object207" src="http://img1.meritnation.com/img/curr/1/12/15/230/6113/Chapter%201_html_m2ecebd19.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="92" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="139" name="Object208" src="http://img1.meritnation.com/img/curr/1/12/15/230/6113/Chapter%201_html_m3371d141.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="405" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, the required function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is defined as<img align="ABSMIDDLE" height="41" name="Object209" src="http://img1.meritnation.com/img/curr/1/12/15/230/6113/Chapter%201_html_15420e2c.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="88" />.</div>
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-webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; 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text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 2:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>: W → W be defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>− 1, if is odd and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>+ 1, if<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>is even. Show that<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i><span class="Apple-converted-space"> </span>is invertible. Find the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>. Here, W is the set of all whole numbers.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<br /></div>
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<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is given that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>: W → W is defined as<img align="ABSMIDDLE" height="48" name="Object210" src="http://img1.meritnation.com/img/curr/1/12/15/230/6117/Chapter%201_html_21949e0e.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="172" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
One-one:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m</i>).</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can be observed that if<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>is odd and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m</i><span class="Apple-converted-space"> </span>is even, then we will have<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>− 1 =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m</i><span class="Apple-converted-space"> </span>+ 1.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m</i><span class="Apple-converted-space"> </span>= 2</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
However, this is impossible.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Similarly, the possibility of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>being even and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m</i><span class="Apple-converted-space"> </span>being odd can also be ignored under a similar argument.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴Both<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m</i><span class="Apple-converted-space"> </span>must be either odd or even.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, if both<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m</i><span class="Apple-converted-space"> </span>are odd, then we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m</i>) ⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>− 1 =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m</i><span class="Apple-converted-space"> </span>− 1 ⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Again, if both<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m</i><span class="Apple-converted-space"> </span>are even, then we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m</i>) ⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>+ 1 =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m</i><span class="Apple-converted-space"> </span>+ 1 ⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is one-one.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is clear that<span class="Apple-converted-space"> </span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">any odd number 2</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">r<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">+ 1 in co-domain<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N<span class="Apple-converted-space"> </span></b></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">is the image of 2</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">r<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">in domain<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N<span class="Apple-converted-space"> </span></b></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">and any even number 2</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">r<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">in co-domain<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N<span class="Apple-converted-space"> </span></b></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">is the image of 2</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">r<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">+ 1 in domain<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">.</span></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is onto.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>is an invertible function.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let us define<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>: W → W as:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="48" name="Object211" src="http://img1.meritnation.com/img/curr/1/12/15/230/6117/Chapter%201_html_5cc1a749.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="181" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, when<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>is odd:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="27" name="Object212" src="http://img1.meritnation.com/img/curr/1/12/15/230/6117/Chapter%201_html_5f1e987a.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="280" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
And, when<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>is even:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="27" name="Object213" src="http://img1.meritnation.com/img/curr/1/12/15/230/6117/Chapter%201_html_7809955c.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="281" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Similarly, when<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m<span class="Apple-converted-space"> </span></i>is odd:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="27" name="Object214" src="http://img1.meritnation.com/img/curr/1/12/15/230/6117/Chapter%201_html_112e0d8.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="299" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
When<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m<span class="Apple-converted-space"> </span></i>is even:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="27" name="Object215" src="http://img1.meritnation.com/img/curr/1/12/15/230/6117/Chapter%201_html_74c1f870.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="300" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<img align="ABSMIDDLE" height="24" name="Object216" src="http://img1.meritnation.com/img/curr/1/12/15/230/6117/Chapter%201_html_m5d3ea30e.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="161" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is invertible and the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>is given by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">—1</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g,<span class="Apple-converted-space"> </span></i>which is the same as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>itself.</div>
</div>
</div>
</div>
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-webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; 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font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><img border="0" height="1" src="http://googleads.g.doubleclick.net/pagead/viewthroughconversion/1036711378/?random=1358554497115&cv=7&fst=1358554497111&num=2&fmt=1&label=ZLO9COT-6wMQ0uur7gM&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=15&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6113/relations-and-functions/let-f-r-rarr-r-be-defined-as-f-x-10x&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6117/relations-and-functions/let-f-w-rarr-w-be-defined-as-f-n-n-mi&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 3:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
If<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is defined by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>− 3<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x<span class="Apple-converted-space"> </span></i>+ 2, find<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>)).</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solution=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
It is given that<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>− 3<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>+ 2.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
</div>
<div style="background-color: white; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<img align="ABSMIDDLE" height="112" name="Object217" src="http://img1.meritnation.com/img/curr/1/12/15/230/6119/Chapter%201_html_me696a15.gif" style="border: medium none; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;" width="375" /> </div>
<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 4:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Show that function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→ {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>: −1 <<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>< 1} defined by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<img align="ABSMIDDLE" height="47" name="Object218" src="http://img1.meritnation.com/img/curr/1/12/15/230/6122/Chapter%201_html_m111a0b79.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="40" />,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x<span class="Apple-converted-space"> </span></i>∈<b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is one-one and onto function.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions= </div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is given that<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</em>:<span class="Apple-converted-space"> </span><strong style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</strong><span class="Apple-converted-space"> </span>→ {<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><strong style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</strong>: −1 <<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>< 1} is defined as<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</em>(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>) =<img align="absmiddle" alt="" height="47" name="Object219" src="http://img1.meritnation.com/img/curr/1/12/15/230/6122/Chapter%201_html_m111a0b79.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="40" />,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x<span class="Apple-converted-space"> </span></em>∈<strong style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</strong>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Suppose<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</em>(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>) =<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</em>(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>), where<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y<span class="Apple-converted-space"> </span></em>∈<span class="Apple-converted-space"> </span><strong style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</strong>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="absmiddle" alt="" height="47" name="Object220" src="http://img1.meritnation.com/img/curr/1/12/15/230/6122/Chapter%201_html_6e6bb21.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="111" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can be observed that if<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is positive and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>is negative, then we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="absmiddle" alt="" height="44" name="Object221" src="http://img1.meritnation.com/img/curr/1/12/15/230/6122/Chapter%201_html_m59e831a3.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="176" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Since<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x<span class="Apple-converted-space"> </span></em>is positive and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>is negative:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>><span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>> 0</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
But, 2<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">xy</em><span class="Apple-converted-space"> </span>is negative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Then,<span class="Apple-converted-space"> </span><img align="absmiddle" alt="" height="21" name="Object222" src="http://img1.meritnation.com/img/curr/1/12/15/230/6122/Chapter%201_html_27a854af.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="76" />.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the case of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>being positive and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>being negative can be ruled out.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Under a similar argument,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>being negative and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>being positive can also be ruled out</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="absmiddle" alt="" height="13" name="Object223" src="http://img1.meritnation.com/img/curr/1/12/15/230/6122/Chapter%201_html_4dd19828.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="15" /><span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>have to be either positive or negative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
When<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>are both positive, we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="absmiddle" alt="" height="44" name="Object224" src="http://img1.meritnation.com/img/curr/1/12/15/230/6122/Chapter%201_html_m191a8ff7.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="360" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
When<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>are both negative, we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="absmiddle" alt="" height="44" name="Object225" src="http://img1.meritnation.com/img/curr/1/12/15/230/6122/Chapter%201_html_7583119d.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="360" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</em><span class="Apple-converted-space"> </span>is one-one.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, let<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><strong style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</strong><span class="Apple-converted-space"> </span>such that −1 <<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y<span class="Apple-converted-space"> </span></em>< 1.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
If<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is negative, then there exists <img align="absmiddle" alt="" height="44" name="Object226" src="http://img1.meritnation.com/img/curr/1/12/15/230/6122/Chapter%201_html_m314f86d5.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="88" />such that</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="absmiddle" alt="" height="93" name="Object227" src="http://img1.meritnation.com/img/curr/1/12/15/230/6122/Chapter%201_html_3cc4d3b.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="371" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
If<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is positive, then there exists <img align="absmiddle" alt="" height="44" name="Object228" src="http://img1.meritnation.com/img/curr/1/12/15/230/6122/Chapter%201_html_m43ea6b8f.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="88" />such that</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="absmiddle" alt="" height="96" name="Object229" src="http://img1.meritnation.com/img/curr/1/12/15/230/6122/Chapter%201_html_m783e1055.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="371" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</em><span class="Apple-converted-space"> </span>is onto.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</em><span class="Apple-converted-space"> </span>is one-one and onto.</div>
</div>
</div>
</div>
<img border="0" height="1" src="http://www.googleadservices.com/pagead/conversion/1036711378/?random=1358554762757&cv=7&fst=1358554762757&num=1&fmt=3&value=0&label=ZDPOCNTSkwMQ0uur7gM&bg=ffffff&hl=en&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=17&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6119/relations-and-functions/if-f-r-rarr-r-is-defined-by-f-x-x2-mi&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6122/relations-and-functions/show-that-function-f-r-rarr-x-isin-r&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><img border="0" height="1" src="http://googleads.g.doubleclick.net/pagead/viewthroughconversion/1036711378/?random=1358554762760&cv=7&fst=1358554762757&num=2&fmt=1&label=ZLO9COT-6wMQ0uur7gM&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=17&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6119/relations-and-functions/if-f-r-rarr-r-is-defined-by-f-x-x2-mi&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6122/relations-and-functions/show-that-function-f-r-rarr-x-isin-r&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 5:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Show that the function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>given by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>is injective.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions = </div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is given as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Suppose<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>), where<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup> … (1)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, we need to show that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Suppose<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>≠<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>, their cubes will also not be equal.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="16" name="Object230" src="http://img1.meritnation.com/img/curr/1/12/15/230/6126/Chapter%201_html_74849e29.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="20" /><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>≠<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
However, this will be a contradiction to (1).</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is injective.</div>
</div>
</div>
</div>
<img border="0" height="1" src="http://www.googleadservices.com/pagead/conversion/1036711378/?random=1358554806128&cv=7&fst=1358554806128&num=1&fmt=3&value=0&label=ZDPOCNTSkwMQ0uur7gM&bg=ffffff&hl=en&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=18&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6122/relations-and-functions/show-that-function-f-r-rarr-x-isin-r&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6126/relations-and-functions/show-that-the-function-f-r-rarr-r-given-b&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><img border="0" height="1" src="http://googleads.g.doubleclick.net/pagead/viewthroughconversion/1036711378/?random=1358554806136&cv=7&fst=1358554806128&num=2&fmt=1&label=ZLO9COT-6wMQ0uur7gM&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=18&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6122/relations-and-functions/show-that-function-f-r-rarr-x-isin-r&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6126/relations-and-functions/show-that-the-function-f-r-rarr-r-given-b&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<br /></div>
<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 6:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
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Give examples of two functions<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>o<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is injective but<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>is not injective.</div>
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(Hint: Consider<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<img align="ABSMIDDLE" height="27" name="Object231" src="http://img1.meritnation.com/img/curr/1/12/15/230/6127/Chapter%201_html_3549d168.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="19" />)</div>
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solution=</div>
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<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
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Define<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<img align="ABSMIDDLE" height="27" name="Object232" src="http://img1.meritnation.com/img/curr/1/12/15/230/6127/Chapter%201_html_3549d168.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="19" />.</div>
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We first show that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>is not injective.</div>
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It can be observed that:</div>
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>(−1) =<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="27" name="Object233" src="http://img1.meritnation.com/img/curr/1/12/15/230/6127/Chapter%201_html_m6c1f588b.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="47" /></div>
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>(1) =<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="27" name="Object234" src="http://img1.meritnation.com/img/curr/1/12/15/230/6127/Chapter%201_html_6cb1bc37.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="36" /></div>
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∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>(−1) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>(1), but −1 ≠ 1.</div>
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∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>is not injective.</div>
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Now,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>is defined as<img align="ABSMIDDLE" height="27" name="Object235" src="http://img1.meritnation.com/img/curr/1/12/15/230/6127/Chapter%201_html_m65077c5b.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="204" />.</div>
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Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>such that go<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>).</div>
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⇒<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="27" name="Object236" src="http://img1.meritnation.com/img/curr/1/12/15/230/6127/Chapter%201_html_m7ba6cfcf.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="49" /></div>
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Since<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N,</b><span class="Apple-converted-space"> </span>both are positive.</div>
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<img align="ABSMIDDLE" height="27" name="Object237" src="http://img1.meritnation.com/img/curr/1/12/15/230/6127/Chapter%201_html_244b96fa.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="117" /></div>
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Hence,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is injective</div>
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<img border="0" height="1" src="http://www.googleadservices.com/pagead/conversion/1036711378/?random=1358554862451&cv=7&fst=1358554862451&num=1&fmt=3&value=0&label=ZDPOCNTSkwMQ0uur7gM&bg=ffffff&hl=en&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=19&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6126/relations-and-functions/show-that-the-function-f-r-rarr-r-given-b&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6127/relations-and-functions/give-examples-of-two-functions-f-n-rarr-z&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><span style="-webkit-text-size-adjust: auto; 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color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; 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Question 7:</div>
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Given examples of two functions<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is onto but<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>is not onto.</div>
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(Hint: Consider<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>+ 1 and</div>
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<img align="ABSMIDDLE" height="48" name="Object238" src="http://img1.meritnation.com/img/curr/1/12/15/230/6129/Chapter%201_html_m72219cf.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="137" /> solution = </div>
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<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Define<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>by,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>x<span class="Apple-converted-space"> </span></i>+ 1</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
And,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>by,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
We first show that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>is not onto.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For this, consider element 1 in co-domain<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>. It is clear that this element is not an image of any of the elements in domain<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not onto.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>is defined by,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="48" name="Object240" src="http://img1.meritnation.com/img/curr/1/12/15/230/6129/Chapter%201_html_32937bc5.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="433" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Then, it is clear that for<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>, there exists<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is onto.</div>
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Question 8:</div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Given a non empty set<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>, consider P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) which is the set of all subsets of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Define the relation R in P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) as follows:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For subsets<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>in P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>),<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>R<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>if and only if<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>⊂<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i>. Is R an equivalence relation on P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>)? Justify you answer:</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<br /></div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Since every set is a subset of itself,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>R<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>for all<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>∈ P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>).</div>
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∴R is reflexive.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>R<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>⊂<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B.</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
This cannot be implied to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>⊂<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A.</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For instance, if<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A<span class="Apple-converted-space"> </span></i>= {1, 2} and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>= {1, 2, 3}, then it cannot be implied that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>is related to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>.</div>
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∴ R is not symmetric.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Further, if<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>R<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B<span class="Apple-converted-space"> </span></i>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i>R<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">C</i>, then<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>⊂<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B<span class="Apple-converted-space"> </span></i>⊂<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">C</i>.</div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>⊂<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">C</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>R<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">C</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴ R is transitive.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, R is not an equivalence relation since it is not symmetric.</div>
</div>
</div>
</div>
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color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; 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<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
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Question 9:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Given a non-empty set<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>, consider the binary operation *: P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) × P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) → P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) given by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>∩<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>in P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) is the power set of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>. Show that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X<span class="Apple-converted-space"> </span></i>is the identity element for this operation and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X<span class="Apple-converted-space"> </span></i>is the only invertible element in P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) with respect to the operation*.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is given that<img align="ABSMIDDLE" height="27" name="Object241" src="http://img1.meritnation.com/img/curr/1/12/15/230/6135/Chapter%201_html_m40d9865a.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="439" />.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
We know that<img align="ABSMIDDLE" height="27" name="Object242" src="http://img1.meritnation.com/img/curr/1/12/15/230/6135/Chapter%201_html_m72326739.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="213" />.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="27" name="Object243" src="http://img1.meritnation.com/img/curr/1/12/15/230/6135/Chapter%201_html_m4badca8f.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="221" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i><span class="Apple-converted-space"> </span>is the identity element for the given binary operation *.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, an element<img align="ABSMIDDLE" height="27" name="Object244" src="http://img1.meritnation.com/img/curr/1/12/15/230/6135/Chapter%201_html_mf05791b.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="68" />is invertible if there exists<img align="ABSMIDDLE" height="27" name="Object245" src="http://img1.meritnation.com/img/curr/1/12/15/230/6135/Chapter%201_html_m4bb5aa36.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="68" />such that</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="69" name="Object246" src="http://img1.meritnation.com/img/curr/1/12/15/230/6135/Chapter%201_html_m36354f52.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="391" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
This case is possible only when<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i><span class="Apple-converted-space"> </span>is the only invertible element in P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) with respect to the given operation*.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, the given result is proved.</div>
</div>
</div>
</div>
<img border="0" height="1" src="http://www.googleadservices.com/pagead/conversion/1036711378/?random=1358555051134&cv=7&fst=1358555051134&num=1&fmt=3&value=0&label=ZDPOCNTSkwMQ0uur7gM&bg=ffffff&hl=en&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=22&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6131/relations-and-functions/given-a-non-empty-set-x-consider-p-x-which&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6135/relations-and-functions/given-a-non-empty-set-x-consider-the-binary&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><img border="0" height="1" src="http://googleads.g.doubleclick.net/pagead/viewthroughconversion/1036711378/?random=1358555051137&cv=7&fst=1358555051134&num=2&fmt=1&label=ZLO9COT-6wMQ0uur7gM&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=22&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6131/relations-and-functions/given-a-non-empty-set-x-consider-p-x-which&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6135/relations-and-functions/given-a-non-empty-set-x-consider-the-binary&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 10:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Find the number of all onto functions from the set {1, 2, 3, … ,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i>) to itself.</div>
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solution= </div>
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Onto functions from the set {1, 2, 3, … ,<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i>} to itself is simply a permutation on<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>symbols 1, 2, …,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i>.</div>
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Thus, the total number of onto maps from {1, 2, … ,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i>} to itself is the same as the total number of permutations on<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>symbols 1, 2, …,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i>, which is<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i>.</div>
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Question 11:</div>
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Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">S</i><span class="Apple-converted-space"> </span>= {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>} and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><span class="Apple-converted-space"> </span>= {1, 2, 3}. Find F<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>of the following functions F from<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">S</i><span class="Apple-converted-space"> </span>to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i>, if it exists.</div>
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<span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">(i)</span> <span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">F</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>= {(</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">, 3), (</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">, 2), (</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">, 1)} (ii) </span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">F</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>= {(</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">, 2), (</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">, 1), (</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">, 1)}</span></div>
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<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;">solution= </span></div>
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<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">S</i><span class="Apple-converted-space"> </span>= {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>},<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><span class="Apple-converted-space"> </span>= {1, 2, 3}</div>
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(i) F:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">S</i><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><span class="Apple-converted-space"> </span>is defined as:</div>
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F = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>, 3), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>, 2), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>, 1)}</div>
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⇒ F (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) = 3, F (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) = 2, F(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) = 1</div>
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Therefore, F<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup>:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">S</i><span class="Apple-converted-space"> </span>is given by</div>
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F<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>= {(3,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>), (2,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>), (1,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>)}.</div>
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(ii) F:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">S</i><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><span class="Apple-converted-space"> </span>is defined as:</div>
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F = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>, 2), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>, 1), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>, 1)}</div>
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Since F (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) = F (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) = 1, F is not one-one.</div>
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Hence, F is not invertible i.e., F<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>does not exist.</div>
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Question 12:</div>
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Consider the binary operations*:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>×<b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>→ and o:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>×<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>defined as<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="27" name="Object247" src="http://img1.meritnation.com/img/curr/1/12/15/230/6142/Chapter%201_html_m4474dc4c.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="84" /><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>o<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>. Show that * is commutative but not associative, o is associative but not commutative. Further, show that<span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>*(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>o<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) = (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) o (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.</div>
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solution= </div>
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<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is given that *:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>×<b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>→ and o:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>×<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>isdefined as</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="27" name="Object248" src="http://img1.meritnation.com/img/curr/1/12/15/230/6142/Chapter%201_html_m4474dc4c.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="84" /><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>o<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>, we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="27" name="Object249" src="http://img1.meritnation.com/img/curr/1/12/15/230/6142/Chapter%201_html_m4474dc4c.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="84" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="29" name="Object250" src="http://img1.meritnation.com/img/curr/1/12/15/230/6142/Chapter%201_html_32b9b83e.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="211" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴ The operation * is commutative.</div>
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It can be observed that,</div>
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<img align="ABSMIDDLE" height="85" name="Object251" src="http://img1.meritnation.com/img/curr/1/12/15/230/6142/Chapter%201_html_m1d19d376.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="277" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴The operation * is not associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, consider the operation o:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can be observed that 1 o 2 = 1 and 2 o 1 = 2.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴1 o 2 ≠ 2 o 1 (where 1, 2 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴The operation o is not commutative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>. Then, we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>o<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) o<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>c<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>o (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>o<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>o<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>o<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) o c =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>o (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>o<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴ The operation o is associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>, then we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>* (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>o<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) = a *<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>=<img align="ABSMIDDLE" height="27" name="Object252" src="http://img1.meritnation.com/img/curr/1/12/15/230/6142/Chapter%201_html_m5a4ace0b.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="40" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) o (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) =<img align="ABSMIDDLE" height="29" name="Object253" src="http://img1.meritnation.com/img/curr/1/12/15/230/6142/Chapter%201_html_m6174c190.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="163" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>* (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>o<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) = (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) o (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>).</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
1 o (2 * 3) =<img align="ABSMIDDLE" height="29" name="Object254" src="http://img1.meritnation.com/img/curr/1/12/15/230/6142/Chapter%201_html_4872cc38.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="136" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(1 o 2) * (1 o 3) = 1 * 1 =<img align="ABSMIDDLE" height="27" name="Object255" src="http://img1.meritnation.com/img/curr/1/12/15/230/6142/Chapter%201_html_2f5f27b.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="57" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴1 o (2 * 3) ≠ (1 o 2) * (1 o 3) (where 1, 2, 3 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="13" name="Object256" src="http://img1.meritnation.com/img/curr/1/12/15/230/6142/Chapter%201_html_4dd19828.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="15" />The operation o does not distribute over *.</div>
</div>
</div>
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Question 13:</div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Given a non-empty set<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>, let *: P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) × P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) → P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) be defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>= (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i>) ∪ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>),<span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>∈ P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>). Show that the empty set<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Φ</i><span class="Apple-converted-space"> </span>is the identity for the operation * and all the elements<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>of P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) are invertible with<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>. (Hint: (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Φ</i>) ∪ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Φ</i><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>and (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>) ∪ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Φ</i>).</div>
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solution= </div>
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<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
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It is given that *: P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) × P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) → P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) is defined as</div>
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>= (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i>) ∪ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>)<span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>∈ P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>).</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A<span class="Apple-converted-space"> </span></i>∈ P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>). Then, we have:</div>
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Φ</i><span class="Apple-converted-space"> </span>= (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Φ</i>) ∪ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Φ</i><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>∪<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Φ</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i></div>
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Φ *<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Φ</i><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>) ∪ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Φ</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Φ</i><span class="Apple-converted-space"> </span>∪<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i></div>
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∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Φ</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Φ</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>.<span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>∈ P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Φ</i><span class="Apple-converted-space"> </span>is the identity element for the given operation*.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, an element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>∈ P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) will be invertible if there exists<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>∈ P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) such that</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Φ</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>. (As Φ is the identity element)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, we observed that<img align="ABSMIDDLE" height="27" name="Object257" src="http://img1.meritnation.com/img/curr/1/12/15/230/6144/Chapter%201_html_m3df80437.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="319" />.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, all the elements<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>of P(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>) are invertible with<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>.</div>
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color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; 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Question 14:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Define a binary operation *on the set {0, 1, 2, 3, 4, 5} as</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="48" name="Object258" src="http://img1.meritnation.com/img/curr/1/12/15/230/6145/Chapter%201_html_23b339d7.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="193" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Show that zero is the identity for this operation and each element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>≠ 0 of the set is invertible with 6 −<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>being the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solution= </div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i><span class="Apple-converted-space"> </span>= {0, 1, 2, 3, 4, 5}.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
The operation * on X is defined as:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="48" name="Object259" src="http://img1.meritnation.com/img/curr/1/12/15/230/6145/Chapter%201_html_m44f621db.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="221" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
An element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">e</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i><span class="Apple-converted-space"> </span>is the identity element for the operation *, if<img align="ABSMIDDLE" height="19" name="Object260" src="http://img1.meritnation.com/img/curr/1/12/15/230/6145/Chapter%201_html_m7e4668e9.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="156" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="96" name="Object261" src="http://img1.meritnation.com/img/curr/1/12/15/230/6145/Chapter%201_html_74ee7c50.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="275" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, 0 is the identity element for the given operation *.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
An element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i><span class="Apple-converted-space"> </span>is invertible if there exists<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>∈<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>= 0 =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="48" name="Object262" src="http://img1.meritnation.com/img/curr/1/12/15/230/6145/Chapter%201_html_3865723c.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="259" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
i.e.,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>= −<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>or<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>= 6 −<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
But,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i><span class="Apple-converted-space"> </span>= {0, 1, 2, 3, 4, 5} and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X.</i><span class="Apple-converted-space"> </span>Then,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>≠ −<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>= 6 −<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>is the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, the inverse of an element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>∈<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>≠ 0 is 6 −<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>i.e.,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>= 6 −<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>.</div>
</div>
</div>
</div>
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-webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; 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Question 15:</div>
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Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= {−1, 0, 1, 2},<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>= {−4, −2, 0, 2} and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>be functions defined by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>∈ A and<img align="ABSMIDDLE" height="45" name="Object263" src="http://img1.meritnation.com/img/curr/1/12/15/230/6146/Chapter%201_html_1989f306.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="164" />. Are<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>equal?</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Justify your answer. (Hint: One may note that two function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>and g:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>→ B such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) = g(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>)<span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>∈<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>, are called equal functions).</div>
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solution=</div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is given that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= {−1, 0, 1, 2},<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>= {−4, −2, 0, 2}.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Also, it is given that<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>are defined by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>and<img align="ABSMIDDLE" height="45" name="Object264" src="http://img1.meritnation.com/img/curr/1/12/15/230/6146/Chapter%201_html_1989f306.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="164" />.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is observed that:</div>
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<img align="ABSMIDDLE" height="441" name="Object265" src="http://img1.meritnation.com/img/curr/1/12/15/230/6146/Chapter%201_html_m2d9450ff.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="284" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, the functions<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>are equal.</div>
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Question 16:</div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(A) 1 (B) 2 (C) 3 (D) 4</div>
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solution=</div>
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<br /></div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
The given set is A = {1, 2, 3}.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
The smallest relation containing (1, 2) and (1, 3) which is reflexive and symmetric, but not transitive is given by:</div>
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R = {(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 1), (3, 1)}</div>
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This is because relation R is reflexive as (1, 1), (2, 2), (3, 3) ∈ R.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Relation R is symmetric since (1, 2), (2, 1) ∈R and (1, 3), (3, 1) ∈R.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
But relation R is not transitive as (3, 1), (1, 2) ∈ R, but (3, 2) ∉ R.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, if we add any two pairs (3, 2) and (2, 3) (or both) to relation R, then relation R will become transitive.</div>
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Hence, the total number of desired relations is one.</div>
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The correct answer is A.</div>
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color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; 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Question 17:</div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= {1, 2, 3}. Then number of equivalence relations containing (1, 2) is</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(A) 1 (B) 2 (C) 3 (D) 4</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solution=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
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<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is given that<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>A</i><span class="Apple-converted-space"> </span>= {1, 2, 3}.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
The smallest equivalence relation containing (1, 2) is given by,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
R<sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>= {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, we are left with only four pairs i.e., (2, 3), (3, 2), (1, 3), and (3, 1).</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
If we odd any one pair [say (2, 3)] to R<sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>, then for symmetry we must add (3, 2). Also, for transitivity we are required to add (1, 3) and (3, 1).</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, the only equivalence relation (bigger than R<sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>) is the universal relation.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
This shows that the total number of equivalence relations containing (1, 2) is two.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
The correct answer is B.</div>
</div>
</div>
</div>
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color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; 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Question 18:</div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>be the Signum Function defined as</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="75" name="Object266" src="http://img1.meritnation.com/img/curr/1/12/15/230/6149/Chapter%201_html_m37007912.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="123" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>be the Greatest Integer Function given by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = [<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>], where [<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>] is greatest integer less than or equal to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>. Then does<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>coincide in (0, 1]?</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutio</div>
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<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is given that,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is defined as<img align="ABSMIDDLE" height="75" name="Object267" src="http://img1.meritnation.com/img/curr/1/12/15/230/6149/Chapter%201_html_m37007912.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="123" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Also,<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>g</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = [<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>], where [<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>] is the greatest integer less than or equal to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>∈ (0, 1].</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Then, we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
[<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>] = 1 if<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>= 1 and [<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>] = 0 if 0 <<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>< 1.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="56" name="Object268" src="http://img1.meritnation.com/img/curr/1/12/15/230/6149/Chapter%201_html_m45f8aa1.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="480" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="83" name="Object269" src="http://img1.meritnation.com/img/curr/1/12/15/230/6149/Chapter%201_html_m1d030a9d.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="191" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, when<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>∈ (0, 1), we have<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = 0and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = 1.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>do not coincide in (0, 1].</div>
</div>
</div>
</div>
<img border="0" height="1" src="http://www.googleadservices.com/pagead/conversion/1036711378/?random=1358555371662&cv=7&fst=1358555371662&num=1&fmt=3&value=0&label=ZDPOCNTSkwMQ0uur7gM&bg=ffffff&hl=en&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=31&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6148/relations-and-functions/let-a-1-2-3-then-number-of-equivalenc&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6149/relations-and-functions/let-f-r-rarr-r-be-the-signum-function-def&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><img border="0" height="1" src="http://googleads.g.doubleclick.net/pagead/viewthroughconversion/1036711378/?random=1358555371665&cv=7&fst=1358555371662&num=2&fmt=1&label=ZLO9COT-6wMQ0uur7gM&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=31&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6148/relations-and-functions/let-a-1-2-3-then-number-of-equivalenc&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6149/relations-and-functions/let-f-r-rarr-r-be-the-signum-function-def&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 19:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Number of binary operations on the set {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>} are</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(A) 10 (B) 16 (C) 20 (D) 8</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solution=</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
A binary operation * on {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>} is a function from {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>} × {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>} → {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>}</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
i.e., * is a function from {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>)} → {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>}.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, the total number of binary operations on the set {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>} is 2<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">4</sup><span class="Apple-converted-space"> </span>i.e., 16.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
The correct answer is B.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
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<div style="background-color: white; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<span><span style="background-color: #444444;"></span></span></div>
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Anonymoushttp://www.blogger.com/profile/17104854705255350948noreply@blogger.com0tag:blogger.com,1999:blog-2679372747285705478.post-34698974743471750872013-01-18T15:57:00.000-08:002013-01-18T16:08:46.080-08:00excersie 1.4<div dir="ltr" style="text-align: left;" trbidi="on">
<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 1:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(i) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>, define * by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(ii) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>, define * by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ab</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iii) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>, define * by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ab</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iv) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>, define * by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>= |<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>|</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(v) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>, define * by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;">solutions =</i></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;"> </i></div>
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<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(i) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>, * is defined by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b = a − b</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is not a binary operation as the image of (1, 2) under * is 1 * 2 = 1 − 2<span class="Apple-converted-space"> </span><br style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" />= −1 ∉<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(ii) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>, * is defined by<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ab</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is seen that for each<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>, there is a unique element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ab</i><span class="Apple-converted-space"> </span>in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
This means that * carries each pair (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) to a unique element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ab<span class="Apple-converted-space"> </span></i>in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, * is a binary operation.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iii) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>, * is defined by<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ab</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is seen that for each<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>, there is a unique element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ab</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
This means that * carries each pair (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) to a unique element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ab</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2<span class="Apple-converted-space"> </span></sup>in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, * is a binary operation.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iv) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>, * is defined by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>= |<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a − b</i>|.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is seen that for each<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>, there is a unique element |<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a − b</i>| in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
This means that * carries each pair (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) to a unique element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><br style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" />|<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a − b</i>|<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><span class="Apple-converted-space"> </span></sup>in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, * is a binary operation.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(v) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>, * is defined by<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
* carries each pair (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) to a unique element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><span class="Apple-converted-space"> </span></sup>in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, * is a binary operation.</div>
</div>
</div>
</div>
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Question 2:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For each binary operation * defined below, determine whether * is commutative or associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(i) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b>, define<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(ii) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b>, define<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ab<span class="Apple-converted-space"> </span></i>+ 1</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iii) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b>, define<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i><img align="ABSMIDDLE" height="41" name="Object158" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_m1e843f5c.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="36" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iv) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>, define<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>= 2<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">ab</i></sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(v) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>, define<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">b</i></sup></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
(vi) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>− {−1}, define<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="41" name="Object159" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_m7356b404.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="79" /> solutions</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<br /></div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(i) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b>, * is defined by<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a − b</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can be observed that 1 * 2 = 1 − 2 = 1 and 2 * 1 = 2 − 1 = 1.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴1 * 2 ≠ 2 * 1; where 1, 2 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, the operation * is not commutative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Also we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(1 * 2) * 3 = (1 − 2) * 3 = −1 * 3 = −1 − 3 = −4</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
1 * (2 * 3) = 1 * (2 − 3) = 1 * −1 = 1 − (−1) = 2</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴(1 * 2) * 3 ≠ 1 * (2 * 3) ; where 1, 2, 3 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, the operation * is not associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(ii) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b>, * is defined by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ab</i><span class="Apple-converted-space"> </span>+ 1.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is known that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ab</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ba</i><span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ab</i><span class="Apple-converted-space"> </span>+ 1 =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ba<span class="Apple-converted-space"> </span></i>+ 1<span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, the operation * is commutative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can be observed that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(1 * 2) * 3 = (1 × 2 + 1) * 3 = 3 * 3 = 3 × 3 + 1 = 10</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
1 * (2 * 3) = 1 * (2 × 3 + 1) = 1 * 7 = 1 × 7 + 1 = 8</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴(1 * 2) * 3 ≠ 1 * (2 * 3) ; where 1, 2, 3 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, the operation * is not associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iii) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b>, * is defined by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="41" name="Object160" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_m5b900bab.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="40" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is known that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ab</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ba</i><span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="41" name="Object161" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_2fce9653.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="59" /><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a</i><span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, the operation * is commutative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For all<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i>,<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>c</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b>, we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="135" name="Object162" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_5b28e7e9.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="239" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<img align="ABSMIDDLE" height="27" name="Object163" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_1bd6e76f.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="138" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, the operation * is associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iv) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>, * is defined by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>= 2<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">ab</i></sup>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is known that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ab</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ba</i><span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒ 2<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">ab</i></sup><span class="Apple-converted-space"> </span>= 2<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">ba</i></sup><span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a</i><span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, the operation * is commutative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can be observed that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="28" name="Object164" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_m2c873722.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="241" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="27" name="Object165" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_m29ceb7e9.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="247" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴(1 * 2) * 3 ≠ 1 * (2 * 3) ; where 1, 2, 3 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, the operation * is not associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(v) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup>, * is defined by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">b</i></sup>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can be observed that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="20" name="Object166" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_1e5b10e4.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="79" /><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="20" name="Object167" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_2cf8b621.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="83" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴ 1 * 2 ≠ 2 * 1 ; where 1, 2 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, the operation * is not commutative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can also be observed that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="33" name="Object168" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_m4f475604.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="268" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="27" name="Object169" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_4f45b508.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="193" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴(2 * 3) * 4 ≠ 2 * (3 * 4) ; where 2, 3, 4 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, the operation * is not associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(vi) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>, * − {−1} is defined by<img align="ABSMIDDLE" height="41" name="Object170" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_m2b9c81a6.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="83" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can be observed that<img align="ABSMIDDLE" height="41" name="Object171" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_m7d6058cd.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="101" /><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="41" name="Object172" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_m6b55f783.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="125" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴1 * 2 ≠ 2 * 1 ; where 1, 2 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>− {−1}</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, the operation * is not commutative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can also be observed that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="60" name="Object173" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_531cf619.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="182" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="61" name="Object174" src="http://img1.meritnation.com/img/curr/1/12/15/230/6079/Chapter%201_html_66962d18.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="314" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴ (1 * 2) * 3 ≠ 1 * (2 * 3) ; where 1, 2, 3 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>− {−1}</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, the operation * is not associative.</div>
</div>
</div>
</div>
<img border="0" height="1" src="http://www.googleadservices.com/pagead/conversion/1036711378/?random=1358553547753&cv=7&fst=1358553547753&num=1&fmt=3&value=0&label=ZDPOCNTSkwMQ0uur7gM&bg=ffffff&hl=en&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=2&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6077/relations-and-functions/determine-whether-or-not-each-of-the-definit&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6079/relations-and-functions/for-each-binary-operation-defined-below&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; 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Question 3:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Consider the binary operation ∨ on the set {1, 2, 3, 4, 5} defined by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>∨<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>= min {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>}. Write the operation table of the operation∨.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions =</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
The binary operation ∨ on the set {1, 2, 3, 4, 5} is defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>∨<sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><span class="Apple-converted-space"> </span></sub><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>= min {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>}</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>∈ {1, 2, 3, 4, 5}.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation table for the given operation ∨ can be given as:</div>
<table border="1" cellpadding="7" cellspacing="0" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><tbody style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" valign="TOP"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
∨</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
5</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" valign="TOP"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" valign="TOP"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" valign="TOP"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" valign="TOP"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" valign="TOP"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
5</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
5</div>
</td></tr>
</tbody></table>
</div>
</div>
</div>
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Question 4:</div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(i) Compute (2 * 3) * 4 and 2 * (3 * 4)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(ii) Is * commutative?</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iii) Compute (2 * 3) * (4 * 5).</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(Hint: use the following table)</div>
<table border="1" cellpadding="7" cellspacing="0" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><tbody style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td height="7" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
*</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
5</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td height="8" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td height="8" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td height="8" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td height="8" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td height="7" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
5</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
5</div>
</td></tr>
</tbody></table>
</div>
solutions =</div>
<div class="html" style="background-color: white; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(i) (2 * 3) * 4 = 1 * 4 = 1</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
2 * (3 * 4) = 2 * 1 = 1</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(ii) For every<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i><span class="Apple-converted-space"> </span>∈{1, 2, 3, 4, 5}, we have<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b = b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a</i>. Therefore, the operation * is commutative.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(iii) (2 * 3) = 1 and (4 * 5) = 1</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴(2 * 3) * (4 * 5) = 1 * 1 = 1</div>
<div style="background-color: white; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<br /></div>
<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 5:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Let</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">*</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">′<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">be the binary operation on the set {1, 2, 3, 4, 5} defined by<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">*</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">′<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">= H.C.F. of<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">and<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">. Is the operation<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">*</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">′</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>same as the operation<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">*</span><span class="Apple-converted-space"> </span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">defined in Exercise 4 above? Justify your answer</span></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;">solutions=</span></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;"> </span></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
The binary operation *′ on the set {1, 2, 3 4, 5} is defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*′<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>= H.C.F of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
The operation table for the operation *′ can be given as:</div>
<table border="1" cellpadding="7" cellspacing="0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><tbody style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
*′</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
5</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
5</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
5</div>
</td></tr>
</tbody></table>
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We observe that the operation tables for the operations * and *′ are the same.</div>
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Thus, the operation *′ is same as the operation*.</div>
<div style="background-color: white; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
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Question 6:</div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let * be the binary operation on<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N<span class="Apple-converted-space"> </span></b>given by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>= L.C.M. of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>. Find</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(i) 5 * 7, 20 * 16 (ii) Is * commutative?</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iii) Is * associative? (iv) Find the identity of * in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(v) Which elements of<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N<span class="Apple-converted-space"> </span></b>are invertible for the operation *?</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
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<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
The binary operation * on<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>is defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>= L.C.M. of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(i) 5 * 7 = L.C.M. of 5 and 7 = 35</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
20 * 16 = L.C.M of 20 and 16 = 80</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(ii) It is known that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
L.C.M of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>= L.C.M of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation * is commutative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iii) For<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>, we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i>) *<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>c<span class="Apple-converted-space"> </span></i>= (L.C.M of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) *<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i><span class="Apple-converted-space"> </span>= LCM of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>, and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>* (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>* (LCM of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) = L.C.M of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>, and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i>) *<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>* (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>c</i>)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation * is associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iv) It is known that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
L.C.M. of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>and 1<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">= a</i><span class="Apple-converted-space"> </span>= L.C.M. 1 and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>* 1 =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>= 1 *<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, 1 is the identity of * in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(v) An element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>is invertible with respect to the operation * if there exists an element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>, such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b = e = b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Here,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">e</i><span class="Apple-converted-space"> </span>= 1</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
This means that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
L.C.M of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>= 1 = L.C.M of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
This case is possible only when<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>are equal to 1.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, 1 is the only invertible element of<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>with respect to the operation *.</div>
</div>
</div>
</div>
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Question 7:</div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Is<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">*</span><span class="Apple-converted-space"> </span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">defined on the set {1, 2, 3, 4, 5} by<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">*</span><span class="Apple-converted-space"> </span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">= L.C.M. of<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">and<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a binary operation? Justify your answer.</span></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;">solutions=</span></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;"> </span></div>
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<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
The operation * on the set A = {1, 2, 3, 4, 5} is defined as</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>= L.C.M. of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Then, the operation table for the given operation * can be given as:</div>
<table border="1" cellpadding="7" cellspacing="0" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><tbody style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
*</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
5</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
1</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
5</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
2</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
6</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
10</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
6</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
3</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
12</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
15</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
12</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
4</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
20</div>
</td></tr>
<tr style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
5</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
5</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
10</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
15</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
20</div>
</td><td style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><div align="CENTER" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px; margin-bottom: 0px; margin-top: 0px;">
5</div>
</td></tr>
</tbody></table>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can be observed from the obtained table that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
3 * 2 = 2 * 3 = 6 ∉ A, 5 * 2 = 2 * 5 = 10 ∉ A, 3 * 4 = 4 * 3 = 12 ∉ A</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
3 * 5 = 5 * 3 = 15 ∉ A, 4 * 5 = 5 * 4 = 20 ∉ A</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, the given operation * is not a binary operation.</div>
</div>
</div>
</div>
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Question 8:</div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Let<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">*</span><span class="Apple-converted-space"> </span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">be the binary operation on<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N<span class="Apple-converted-space"> </span></b></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">defined by<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">*</span><span class="Apple-converted-space"> </span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">= H.C.F. of<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">and<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">. Is<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">*</span><span class="Apple-converted-space"> </span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">commutative? Is<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">*</span><span class="Apple-converted-space"> </span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">associative? Does there exist identity for this binary operation on<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">?</span></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;">solutions=</span></div>
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<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
The binary operation * on<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>is defined as:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>= H.C.F. of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is known that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
H.C.F. of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>= H.C.F. of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span><span class="splChar" style="color: #333333; font-family: symbol; font-size: 17px;">"</span><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation * is commutative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>, we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i>)*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>c</i><span class="Apple-converted-space"> </span>= (H.C.F. of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) *<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i><span class="Apple-converted-space"> </span>= H.C.F. of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>, and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>c</i>)=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*(H.C.F. of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) = H.C.F. of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>, and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i>) *<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c = a</i><span class="Apple-converted-space"> </span>* (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>c</i>)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation * is associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, an element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">e<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>will be the identity for the operation * if<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>e</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>= e*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a</i><span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="17" name="Object175" src="http://img1.meritnation.com/img/curr/1/12/15/230/6085/Chapter%201_html_m24db17a.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="16" /><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
But this relation is not true for any<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation * does not have any identity in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>.</div>
</div>
</div>
</div>
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-webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><img border="0" height="1" src="http://googleads.g.doubleclick.net/pagead/viewthroughconversion/1036711378/?random=1358553878961&cv=7&fst=1358553878959&num=2&fmt=1&label=ZLO9COT-6wMQ0uur7gM&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=8&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6084/relations-and-functions/is-defined-on-the-set-1-2-3-4-5-by&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6085/relations-and-functions/let-be-the-binary-operation-on-n-defined&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<br /></div>
<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 9:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let * be a binary operation on the set<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q<span class="Apple-converted-space"> </span></b>of rational numbers as follows:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(i) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>(ii) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>+<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iii) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>+<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ab<span class="Apple-converted-space"> </span></i>(iv) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>= (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>)<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(v) <img align="ABSMIDDLE" height="41" name="Object176" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_66c978c7.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="68" />(vi) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">ab</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Find which of the binary operations are commutative and which are associative.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(i) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b>, the operation * is defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b = a − b</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can be observed that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="41" name="Object177" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_m17987fc5.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="161" />and<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="41" name="Object178" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_7ce79ff7.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="169" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<img align="ABSMIDDLE" height="41" name="Object179" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_m33d0397d.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="85" /><span class="Apple-converted-space"> </span>; where<img align="ABSMIDDLE" height="41" name="Object180" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_195a9be.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="67" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation * is not commutative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can also be observed that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="45" name="Object181" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_m337a2963.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="346" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="45" name="Object182" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_m58a3c011.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="355" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="45" name="Object183" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_m32386eb0.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="321" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation * is not associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(ii) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b>, the operation * is defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b = a</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>+ b</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b>, we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="21" name="Object184" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_m670d7202.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="196" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b = b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation * is commutative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can be observed that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="111" name="Object185" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_58ff96b4.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="371" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, ,the operation * is not associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iii) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b>, the operation * is defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b = a + ab</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can be observed that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="93" name="Object186" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_m565d07f9.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="195" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation * is not commutative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can also be observed that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="103" name="Object187" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_m1179ebba.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="333" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation * is not associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iv) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b>, the operation * is defined by<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>= (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a − b</i>)<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b>, we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>= (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a − b</i>)<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a</i><span class="Apple-converted-space"> </span>= (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b − a</i>)<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>= [− (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a − b</i>)]<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>= (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a − b</i>)<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b = b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation * is commutative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can be observed that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="111" name="Object188" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_m32b45009.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="387" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation * is not associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(v) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b>, the operation * is defined as<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="41" name="Object189" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_m3695afd3.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="72" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b>, we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="41" name="Object190" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_m1b52bbe8.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="146" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation * is commutative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b, c</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b>, we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="60" name="Object191" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_2951557c.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="214" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="60" name="Object192" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_m77554e0b.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="214" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i>) *<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>* (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>c</i>)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation * is associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(vi) On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b>, the operation * is defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b = ab</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can be observed that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="167" name="Object193" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_63ffb90a.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="215" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation * is not commutative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can also be observed that:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="184" name="Object194" src="http://img1.meritnation.com/img/curr/1/12/15/230/6093/Chapter%201_html_3d547ebd.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="383" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the operation * is not associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, the operations defined in (ii), (iv), (v) are commutative and the operation defined in (v) is associative.</div>
</div>
</div>
</div>
<img border="0" height="1" src="http://www.googleadservices.com/pagead/conversion/1036711378/?random=1358553915235&cv=7&fst=1358553915235&num=1&fmt=3&value=0&label=ZDPOCNTSkwMQ0uur7gM&bg=ffffff&hl=en&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=9&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6085/relations-and-functions/let-be-the-binary-operation-on-n-defined&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6093/relations-and-functions/let-be-a-binary-operation-on-the-set-q-of&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; 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line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><img border="0" height="1" src="http://googleads.g.doubleclick.net/pagead/viewthroughconversion/1036711378/?random=1358553915236&cv=7&fst=1358553915235&num=2&fmt=1&label=ZLO9COT-6wMQ0uur7gM&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=9&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6085/relations-and-functions/let-be-the-binary-operation-on-n-defined&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6093/relations-and-functions/let-be-a-binary-operation-on-the-set-q-of&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 10:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Find which of the operations given above has identity.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<br /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
An element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">e<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b><span class="Apple-converted-space"> </span>will be the identity element for the operation * if</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>e = a = e<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a</i>,<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="17" name="Object195" src="http://img1.meritnation.com/img/curr/1/12/15/230/6098/Chapter%201_html_m24db17a.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="16" /><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
However, there is no such element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">e<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Q</b><span class="Apple-converted-space"> </span>with respect to each of the six operations satisfying the above condition.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Thus, none of the six operations has identity.</div>
<div style="background-color: white; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<br /></div>
<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 11:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let A =<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N<span class="Apple-converted-space"> </span></b>×<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N<span class="Apple-converted-space"> </span></b>and * be the binary operation on A defined by</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) * (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">d</i>) = (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>+<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>+<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">d</i>)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Show that * is commutative and associative. Find the identity element for * on A, if any.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
A =<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>×<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
* is a binary operation on A and is defined by:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i>) * (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c, d</i>) = (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a + c, b + d</i>)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i>), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c, d</i>) ∈ A</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Then,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b, c, d</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
We have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i>) * (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c, d</i>) = (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a + c, b + d</i>)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c, d</i>) * (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i>) = (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c + a, d + b</i>) = (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a + c, b + d</i>)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
[Addition is commutative in the set of natural numbers]</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i>) * (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c, d</i>) = (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c, d</i>) * (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i>)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, the operation * is commutative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b</i>), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c, d</i>), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">e, f</i>) ∈A</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Then,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b, c, d, e</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
We have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="29" name="Object196" src="http://img1.meritnation.com/img/curr/1/12/15/230/6103/Chapter%201_html_ma309ea.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="441" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="82" name="Object197" src="http://img1.meritnation.com/img/curr/1/12/15/230/6103/Chapter%201_html_1f4f19c1.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="441" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, the operation * is associative.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
An element<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="27" name="Object198" src="http://img1.meritnation.com/img/curr/1/12/15/230/6103/Chapter%201_html_m107bb298.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="97" />will be an identity element for the operation * if</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="27" name="Object199" src="http://img1.meritnation.com/img/curr/1/12/15/230/6103/Chapter%201_html_m78ce6f74.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="216" />, i.e.,<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="27" name="Object200" src="http://img1.meritnation.com/img/curr/1/12/15/230/6103/Chapter%201_html_7e9327a.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="282" />which is not true for any element in A.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, the operation * does not have any identity element.</div>
</div>
</div>
</div>
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color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; 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Question 12:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
State whether the following statements are true or false. Justify.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(i) For an arbitrary binary operation * on a set<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="17" name="Object201" src="http://img1.meritnation.com/img/curr/1/12/15/230/6106/Chapter%201_html_m24db17a.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="16" /></i><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(ii) If * is a commutative binary operation on<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>, then<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>* (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) = (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) *<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;">solutions=</i></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;"> </i></div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(i) Define an operation * on<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>as:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a + b</i><span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="17" name="Object202" src="http://img1.meritnation.com/img/curr/1/12/15/230/6106/Chapter%201_html_m24db17a.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="16" /><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a, b<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Then, in particular, for<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>= 3, we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
3 * 3 = 3 + 3 = 6 ≠ 3</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, statement (i) is false.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(ii) R.H.S. = (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i>) *<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
= (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>c</i>) *<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a </i>[* is commutative]</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>* (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>c</i>) [Again, as * is commutative]</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
= L.H.S.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>* (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>c</i>) = (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i>) *<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, statement (ii) is true.</div>
</div>
</div>
</div>
<img border="0" height="1" src="http://www.googleadservices.com/pagead/conversion/1036711378/?random=1358554034893&cv=7&fst=1358554034893&num=1&fmt=3&value=0&label=ZDPOCNTSkwMQ0uur7gM&bg=ffffff&hl=en&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=12&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6103/relations-and-functions/let-a-n-times-n-and-be-the-binary-o&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6106/relations-and-functions/state-whether-the-following-statements-are-t&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><img border="0" height="1" src="http://googleads.g.doubleclick.net/pagead/viewthroughconversion/1036711378/?random=1358554034896&cv=7&fst=1358554034893&num=2&fmt=1&label=ZLO9COT-6wMQ0uur7gM&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=12&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6103/relations-and-functions/let-a-n-times-n-and-be-the-binary-o&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6106/relations-and-functions/state-whether-the-following-statements-are-t&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 13:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Consider a binary operation * on<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N<span class="Apple-converted-space"> </span></b>defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>+<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup>. Choose the correct answer.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(A) Is * both associative and commutative?</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(B) Is * commutative but not associative?</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(C) Is * associative but not commutative?</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(D) Is * neither commutative nor associative?</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
On<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>, the operation * is defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>+<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
For,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>, ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>, we have:</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>+<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>+<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>*<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>a</i> [Addition is commutative in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>]</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Therefore, the operation * is commutative.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
It can be observed that:</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<img align="ABSMIDDLE" height="63" name="Object203" src="http://img1.meritnation.com/img/curr/1/12/15/230/6109/Chapter%201_html_6455216b.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="541" /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴(1 * 2) * 3 ≠ 1 * (2 * 3) ; where 1, 2, 3 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Therefore, the operation * is not associative.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, the operation * is commutative, but not associative. Thus, the correct answer is B.</div>
<div style="background-color: white; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
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Anonymoushttp://www.blogger.com/profile/17104854705255350948noreply@blogger.com4Jodhpur, Rajasthan, India26.2389469 73.02430939999999326.0111109 72.7015859 26.4667829 73.347032899999988tag:blogger.com,1999:blog-2679372747285705478.post-67544927328161184192013-01-18T15:11:00.001-08:002013-01-18T15:11:48.168-08:00excersie 1.3 continued<div dir="ltr" style="text-align: left;" trbidi="on">
<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 6:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Show that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>: [−1, 1] →<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>, given by<img align="ABSMIDDLE" height="47" name="Object92" src="http://img1.meritnation.com/img/curr/1/12/15/230/6059/Chapter%201_html_m6a332d46.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="100" /> is one-one. Find the inverse of the function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>: [−1, 1] → Range<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(Hint: For<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>∈Range<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>=<img align="ABSMIDDLE" height="41" name="Object93" src="http://img1.meritnation.com/img/curr/1/12/15/230/6059/Chapter%201_html_m3252a6d5.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="88" />, for some<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>in [−1, 1], i.e.,<img align="ABSMIDDLE" height="47" name="Object94" src="http://img1.meritnation.com/img/curr/1/12/15/230/6059/Chapter%201_html_m51a30695.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="72" />)</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions =</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<br /></div>
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</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<br /></div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>: [−1, 1] → R is given as<img align="ABSMIDDLE" height="47" name="Object95" src="http://img1.meritnation.com/img/curr/1/12/15/230/6059/Chapter%201_html_735b2aef.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="104" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>).</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="117" name="Object96" src="http://img1.meritnation.com/img/curr/1/12/15/230/6059/Chapter%201_html_2b620184.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="133" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is a one-one function.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is clear that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>: [−1, 1] → Range<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is onto.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>: [−1, 1] → Range<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is one-one and onto and therefore, the inverse of the function:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>: [−1, 1] → Range<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>exists.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>: Range<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>→ [−1, 1] be the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>y</i><span class="Apple-converted-space"> </span>be an arbitrary element of range<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Since<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>: [−1, 1] → Range<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is onto, we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="165" name="Object97" src="http://img1.meritnation.com/img/curr/1/12/15/230/6059/Chapter%201_html_m6dada837.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="193" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, let us define<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>: Range<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i><span class="Apple-converted-space"> </span>→ [−1, 1] as</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="219" name="Object98" src="http://img1.meritnation.com/img/curr/1/12/15/230/6059/Chapter%201_html_7466b8c7.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="468" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>=<img align="ABSMIDDLE" height="27" name="Object99" src="http://img1.meritnation.com/img/curr/1/12/15/230/6059/Chapter%201_html_2819068d.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="36" /><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="25" name="Object100" src="http://img1.meritnation.com/img/curr/1/12/15/230/6059/Chapter%201_html_m386cf7c4.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="41" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="13" name="Object101" src="http://img1.meritnation.com/img/curr/1/12/15/230/6059/Chapter%201_html_4dd19828.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="15" /><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="44" name="Object102" src="http://img1.meritnation.com/img/curr/1/12/15/230/6059/Chapter%201_html_57eb3e9.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="136" /></div>
</div>
</div>
</div>
<img border="0" height="1" src="http://www.googleadservices.com/pagead/conversion/1036711378/?random=1358550166531&cv=7&fst=1358550166531&num=1&fmt=3&value=0&label=ZDPOCNTSkwMQ0uur7gM&bg=ffffff&hl=en&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=1&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/cbse-math-math-part-i/relations-and-functions/15/230/ncertsolutions.html&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6059/relations-and-functions/show-that-f-minus-1-1-rarr-r-given-b&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; 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text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 7:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Consider<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">:<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>given by<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">(</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">) = 4</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>+ 3. Show that<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>is invertible. Find the inverse of<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">.</span></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;">solutions=</span></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;"> </span></div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is given by,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = 4<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>+ 3</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
One-one:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>).</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="69" name="Object103" src="http://img1.meritnation.com/img/curr/1/12/15/230/6060/Chapter%201_html_m761b1178.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="116" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is a one-one function.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Onto:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>, let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>= 4<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>+ 3.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="41" name="Object104" src="http://img1.meritnation.com/img/curr/1/12/15/230/6060/Chapter%201_html_4195f8f1.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="108" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, for any<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>, there exists<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="41" name="Object105" src="http://img1.meritnation.com/img/curr/1/12/15/230/6060/Chapter%201_html_60a29e0a.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="89" /><span class="Apple-converted-space"> </span>such that</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="45" name="Object106" src="http://img1.meritnation.com/img/curr/1/12/15/230/6060/Chapter%201_html_m6d747ee9.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="240" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is onto.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is one-one and onto and therefore,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>exists.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let us define<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>by<img align="ABSMIDDLE" height="41" name="Object107" src="http://img1.meritnation.com/img/curr/1/12/15/230/6060/Chapter%201_html_1d7c51d1.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="87" />.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="91" name="Object108" src="http://img1.meritnation.com/img/curr/1/12/15/230/6060/Chapter%201_html_18a30fb0.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="411" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<img align="ABSMIDDLE" height="24" name="Object109" src="http://img1.meritnation.com/img/curr/1/12/15/230/6060/Chapter%201_html_m7e9141ec.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="100" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is invertible and the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is given by</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="41" name="Object110" src="http://img1.meritnation.com/img/curr/1/12/15/230/6060/Chapter%201_html_2a2d1bb7.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="153" /></div>
</div>
</div>
</div>
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color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; 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Question 8:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Consider<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">:<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b></span><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+<span class="Apple-converted-space"> </span></span></sub><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">→ [4,<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">∞</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">) given by<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">(</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">) =</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>x</i></span><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</span></sup><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>+ 4. Show that<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>is invertible with the inverse<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</span></sup><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>of given<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">by<img align="ABSMIDDLE" height="28" name="Object111" src="http://img1.meritnation.com/img/curr/1/12/15/230/6061/Chapter%201_html_2a333a87.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="111" />, where<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b></span><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</span></sub><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>is the set of all non-negative real numbers.</span></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;">solutions=</span></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;"> </span></div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
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<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sub><span class="Apple-converted-space"> </span>→ [4, ∞) is given as<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>+ 4.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
One-one:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>).</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="79" name="Object112" src="http://img1.meritnation.com/img/curr/1/12/15/230/6061/Chapter%201_html_m7cfa4fd1.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="197" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is a one-one function.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Onto:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>∈ [4, ∞), let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>+ 4.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="56" name="Object113" src="http://img1.meritnation.com/img/curr/1/12/15/230/6061/Chapter%201_html_506595d.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="209" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Therefore, for any<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>, there exists<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="27" name="Object114" src="http://img1.meritnation.com/img/curr/1/12/15/230/6061/Chapter%201_html_m4518fcda.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="100" /><span class="Apple-converted-space"> </span>such that</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="36" name="Object115" src="http://img1.meritnation.com/img/curr/1/12/15/230/6061/Chapter%201_html_1774a55e.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="319" />.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is onto.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is one-one and onto and therefore,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>exists.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let us define<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>: [4, ∞) →<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+<span class="Apple-converted-space"> </span></sub>by,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="28" name="Object116" src="http://img1.meritnation.com/img/curr/1/12/15/230/6061/Chapter%201_html_2c94539a.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="97" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="72" name="Object117" src="http://img1.meritnation.com/img/curr/1/12/15/230/6061/Chapter%201_html_m31641b49.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="453" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<img align="ABSMIDDLE" height="24" name="Object118" src="http://img1.meritnation.com/img/curr/1/12/15/230/6061/Chapter%201_html_m1c8142f3.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="109" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is invertible and the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is given by</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="28" name="Object119" src="http://img1.meritnation.com/img/curr/1/12/15/230/6061/Chapter%201_html_55a4df99.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="163" /></div>
</div>
</div>
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-webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; 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Question 9:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Consider<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">:<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b></span><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</span></sub><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>→ [−5,<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">∞</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">) given by<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">(</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">) = 9</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i></span><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</span></sup><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>+ 6</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>− 5. Show that<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>is invertible with<img align="ABSMIDDLE" height="67" name="Object120" src="http://img1.meritnation.com/img/curr/1/12/15/230/6065/Chapter%201_html_m6578c33e.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="160" />.</span></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;">solutions=</span></div>
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<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;"> </span></div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sub><span class="Apple-converted-space"> </span>→ [−5, ∞) is given as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = 9<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>+ 6<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>− 5.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>be an arbitrary element of [−5, ∞).</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>= 9<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>+ 6<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x<span class="Apple-converted-space"> </span></i>− 5.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="136" name="Object121" src="http://img1.meritnation.com/img/curr/1/12/15/230/6065/Chapter%201_html_m61e2d07d.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="295" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is onto, thereby range<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>= [−5, ∞).</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let us define<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>: [−5, ∞) →<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">+</sub><span class="Apple-converted-space"> </span>as<img align="ABSMIDDLE" height="45" name="Object122" src="http://img1.meritnation.com/img/curr/1/12/15/230/6065/Chapter%201_html_m52e3184b.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="124" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
We now have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="163" name="Object123" src="http://img1.meritnation.com/img/curr/1/12/15/230/6065/Chapter%201_html_md23d7aa.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="292" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="152" name="Object124" src="http://img1.meritnation.com/img/curr/1/12/15/230/6065/Chapter%201_html_46d51344.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="360" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<img align="ABSMIDDLE" height="25" name="Object125" src="http://img1.meritnation.com/img/curr/1/12/15/230/6065/Chapter%201_html_28d7b353.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="64" />and<img align="ABSMIDDLE" height="25" name="Object126" src="http://img1.meritnation.com/img/curr/1/12/15/230/6065/Chapter%201_html_m5eafb379.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="84" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is invertible and the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is given by</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="45" name="Object127" src="http://img1.meritnation.com/img/curr/1/12/15/230/6065/Chapter%201_html_m19f5c246.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="185" /></div>
</div>
</div>
</div>
<img border="0" height="1" src="http://www.googleadservices.com/pagead/conversion/1036711378/?random=1358550343138&cv=7&fst=1358550343138&num=1&fmt=3&value=0&label=ZDPOCNTSkwMQ0uur7gM&bg=ffffff&hl=en&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=4&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6061/relations-and-functions/consider-f-r-rarr-4-infin-given-b&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6065/relations-and-functions/consider-f-r-rarr-minus-5-infin&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><img border="0" height="1" src="http://googleads.g.doubleclick.net/pagead/viewthroughconversion/1036711378/?random=1358550343140&cv=7&fst=1358550343138&num=2&fmt=1&label=ZLO9COT-6wMQ0uur7gM&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=4&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6061/relations-and-functions/consider-f-r-rarr-4-infin-given-b&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6065/relations-and-functions/consider-f-r-rarr-minus-5-infin&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 10:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Y</i><span class="Apple-converted-space"> </span>be an invertible function. Show that<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f<span class="Apple-converted-space"> </span></i>has unique inverse.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(Hint: suppose<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub><span class="Apple-converted-space"> </span>are two inverses of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>. Then for all<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Y</i>,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>) = I<sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">Y</i></sub>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>). Use one-one ness of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>).</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Y</i><span class="Apple-converted-space"> </span>be an invertible function.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Also, suppose<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>has two inverses (say<sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><img align="BOTTOM" height="24" name="Object128" src="http://img1.meritnation.com/img/curr/1/12/15/230/6068/Chapter%201_html_7f5d1628.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;" width="64" />).</sub></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Then, for all<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>∈<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Y</i>, we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="109" name="Object129" src="http://img1.meritnation.com/img/curr/1/12/15/230/6068/Chapter%201_html_36b3004b.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="404" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence,<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i><span class="Apple-converted-space"> </span>has a unique inverse.</div>
</div>
</div>
</div>
<img border="0" height="1" src="http://www.googleadservices.com/pagead/conversion/1036711378/?random=1358550379175&cv=7&fst=1358550379175&num=1&fmt=3&value=0&label=ZDPOCNTSkwMQ0uur7gM&bg=ffffff&hl=en&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=5&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6065/relations-and-functions/consider-f-r-rarr-minus-5-infin&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6068/relations-and-functions/let-f-x-rarr-y-be-an-invertible-function&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; 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Question 11:</div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Consider<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">: {1, 2, 3} → {</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">,<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">,<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">} given by<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">(1) =<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">,<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">(2) =<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">(3) =<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">. Find<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</span></sup><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>and show that (</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</span></sup><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">)</span><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</span></sup><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>=</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">.</span></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;">solutions=</span></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;"> </span></div>
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<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>: {1, 2, 3} → {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>} is given by,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(1) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(2) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b,</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(3) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
If we define<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>: {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>} → {1, 2, 3} as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) = 1,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) = 2,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) = 3, then we have:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="199" name="Object130" src="http://img1.meritnation.com/img/curr/1/12/15/230/6072/Chapter%201_html_m6c2426fc.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="215" /></div>
<div align="JUSTIFY" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<img align="ABSMIDDLE" height="24" name="Object131" src="http://img1.meritnation.com/img/curr/1/12/15/230/6072/Chapter%201_html_m2d002022.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="60" />and<img align="ABSMIDDLE" height="24" name="Object132" src="http://img1.meritnation.com/img/curr/1/12/15/230/6072/Chapter%201_html_m2a3edc62.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="57" />, where<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i><span class="Apple-converted-space"> </span>= {1, 2, 3} and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Y</i>= {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>}.</div>
<div align="JUSTIFY" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>exists and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">=</b><span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>.</div>
<div align="JUSTIFY" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup>: {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>} → {1, 2, 3} is given by,</div>
<div align="JUSTIFY" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) = 1,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) = 2,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">-1</sup>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) = 3</div>
<div align="JUSTIFY" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let us now find the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>i.e., find the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>.</div>
<div align="JUSTIFY" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
If we define<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i>: {1, 2, 3} → {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>} as</div>
<div align="JUSTIFY" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i>(1) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i>(2) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i>(3) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>, then we have:</div>
<div align="JUSTIFY" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="199" name="Object133" src="http://img1.meritnation.com/img/curr/1/12/15/230/6072/Chapter%201_html_6a6a33b2.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="212" /></div>
<div align="JUSTIFY" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<img align="ABSMIDDLE" height="24" name="Object134" src="http://img1.meritnation.com/img/curr/1/12/15/230/6072/Chapter%201_html_3c64248.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="147" />, where<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i><span class="Apple-converted-space"> </span>= {1, 2, 3} and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Y</i><span class="Apple-converted-space"> </span>= {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>}.</div>
<div align="JUSTIFY" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g<span class="Apple-converted-space"> </span></i>exists and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i><span class="Apple-converted-space"> </span>⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup>)<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i>.</div>
<div align="JUSTIFY" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It can be noted that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>.</div>
<div align="JUSTIFY" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup>)<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>.</div>
</div>
</div>
</div>
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-webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; 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Question 12:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Y</i><span class="Apple-converted-space"> </span>be an invertible function. Show that the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>is<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>, i.e.,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup>)<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
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<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
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<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Y</i><span class="Apple-converted-space"> </span>be an invertible function.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Then, there exists a function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Y</i><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>= I<sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">X</i></sub>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g<span class="Apple-converted-space"> </span></i>= I<sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">Y</i></sub>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Here,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>= I<sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">X</i></sub>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g<span class="Apple-converted-space"> </span></i>= I<sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">Y</i></sub></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>= I<sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">X</i></sub>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup>= I<sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">Y</i></sub></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence,<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup>:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Y</i><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">X</i><span class="Apple-converted-space"> </span>is invertible and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
i.e., (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup>)<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>.</div>
</div>
</div>
</div>
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-webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; 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text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 13:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
If<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>be given by<img align="ABSMIDDLE" height="39" name="Object135" src="http://img1.meritnation.com/img/curr/1/12/15/230/6075/Chapter%201_html_m49ce42ca.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="108" />, then<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) is</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(A) <img align="ABSMIDDLE" height="41" name="Object136" src="http://img1.meritnation.com/img/curr/1/12/15/230/6075/Chapter%201_html_m3aeac40c.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="21" /> (B) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup> (C) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i> (D) (3 −<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup>)</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is given as<img align="ABSMIDDLE" height="39" name="Object137" src="http://img1.meritnation.com/img/curr/1/12/15/230/6075/Chapter%201_html_m49ce42ca.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="108" />.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<img align="ABSMIDDLE" height="176" name="Object138" src="http://img1.meritnation.com/img/curr/1/12/15/230/6075/Chapter%201_html_25362c5e.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="376" /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
The correct answer is C.</div>
<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 14:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Let</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><img align="ABSMIDDLE" height="45" name="Object139" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_3751b097.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="124" /> be a function defined as<img align="ABSMIDDLE" height="41" name="Object140" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_m354ab145.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="95" />. The inverse of<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>is map<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">: Range<img align="ABSMIDDLE" height="45" name="Object141" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_m4ca863d4.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="160" /></span></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(A) <img align="ABSMIDDLE" height="44" name="Object142" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_m3af6d2ec.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="96" /> (B) <img align="ABSMIDDLE" height="44" name="Object143" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_9e09296.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="96" /></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
(C) <img align="ABSMIDDLE" height="44" name="Object144" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_m1ab068e3.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="96" /> (D)<img align="ABSMIDDLE" height="44" name="Object145" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_29a6289f.gif" style="border: medium none; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;" width="96" /> solutions = </div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is given that<img align="ABSMIDDLE" height="45" name="Object146" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_341935bb.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="304" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>be an arbitrary element of Range<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Then, there exists<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>∈<img align="ABSMIDDLE" height="45" name="Object147" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_3e4b1502.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="69" />such that<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="27" name="Object148" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_m4786a736.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="68" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="LEFT" height="139" hspace="6" name="Object149" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_6a72f987.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="119" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let us define<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>: Range<img align="ABSMIDDLE" height="45" name="Object150" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_m7f2eeed8.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="103" />as<img align="ABSMIDDLE" height="44" name="Object151" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_m59b54d57.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="100" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now,<img align="ABSMIDDLE" height="45" name="Object152" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_3a8779b5.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="231" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="88" name="Object153" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_5420ebc8.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="279" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="144" name="Object154" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_m10d6f44e.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="383" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<img align="ABSMIDDLE" height="39" name="Object155" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_1f2105c2.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="195" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>is the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>i.e.,<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">−1</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is the map<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>: Range<img align="ABSMIDDLE" height="45" name="Object156" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_m7f2eeed8.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="103" />, which is given by</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="44" name="Object157" src="http://img1.meritnation.com/img/curr/1/12/15/230/6076/Chapter%201_html_m59b54d57.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="100" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
The correct answer is B.</div>
</div>
</div>
</div>
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-webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; 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Anonymoushttp://www.blogger.com/profile/17104854705255350948noreply@blogger.com0tag:blogger.com,1999:blog-2679372747285705478.post-48832461479408582852013-01-18T09:25:00.002-08:002013-01-18T09:25:47.988-08:00Exercise 1.3 <div dir="ltr" style="text-align: left;" trbidi="on">
<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 1:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
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<span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Let<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">: {1, 3, 4} → {1, 2, 5} and<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">: {1, 2, 5} → {1, 3} be given by<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">= {(1, 2), (3, 5), (4, 1)} and<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">= {(1, 3), (2, 3), (5, 1)}. Write down<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">o</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">.</span></div>
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<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;">SOLUTION=</span></div>
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<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;"> </span></div>
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The functions<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>: {1, 3, 4} → {1, 2, 5} and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>: {1, 2, 5} → {1, 3} are defined as</div>
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>= {(1, 2), (3, 5), (4, 1)} and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g<span class="Apple-converted-space"> </span></i>= {(1, 3), (2, 3), (5, 1)}.</div>
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<img align="ABSMIDDLE" height="117" name="Object75" src="http://img1.meritnation.com/img/curr/1/12/15/230/6045/Chapter%201_html_57f3b03d.gif" style="border: medium none; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;" width="449" /> </div>
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Question 2:</div>
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Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i><span class="Apple-converted-space"> </span>be functions from<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>to<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>. Show that</div>
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<img align="ABSMIDDLE" height="53" name="Object76" src="http://img1.meritnation.com/img/curr/1/12/15/230/6048/Chapter%201_html_m59307f71.gif" style="border: medium none; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;" width="161" /> </div>
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SOLUTIONS=</div>
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<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
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To prove:</div>
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<img align="ABSMIDDLE" height="27" name="Object81" src="http://img1.meritnation.com/img/curr/1/12/15/230/6048/Chapter%201_html_m377bec3b.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="217" /></div>
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Question 3:</div>
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Find<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>o<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>, if</div>
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(i) <img align="ABSMIDDLE" height="27" name="Object82" src="http://img1.meritnation.com/img/curr/1/12/15/230/6050/Chapter%201_html_5f375722.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="192" /></div>
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(ii)</div>
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<img align="ABSMIDDLE" height="37" name="Object83" src="http://img1.meritnation.com/img/curr/1/12/15/230/6050/Chapter%201_html_m2242302d.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="171" /> SOLUTIONS =</div>
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(i) <img align="ABSMIDDLE" height="27" name="Object84" src="http://img1.meritnation.com/img/curr/1/12/15/230/6050/Chapter%201_html_5f375722.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="192" /></div>
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<img align="ABSMIDDLE" height="59" name="Object85" src="http://img1.meritnation.com/img/curr/1/12/15/230/6050/Chapter%201_html_mc9dd7a9.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="345" /></div>
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(ii) <img align="ABSMIDDLE" height="37" name="Object86" src="http://img1.meritnation.com/img/curr/1/12/15/230/6050/Chapter%201_html_m2242302d.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="171" /></div>
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Question 4:</div>
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<span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">If</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><img align="ABSMIDDLE" height="49" name="Object88" src="http://img1.meritnation.com/img/curr/1/12/15/230/6054/Chapter%201_html_m2c54b729.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="153" />, show that</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f<span class="Apple-converted-space"> </span></i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">o<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">(</span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">) =<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">, for all<img align="ABSMIDDLE" height="41" name="Object89" src="http://img1.meritnation.com/img/curr/1/12/15/230/6054/Chapter%201_html_m39253282.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="40" />. What is the inverse of<span class="Apple-converted-space"> </span></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">?</span></div>
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<span style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;">SOLUTIONS=</span></div>
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It is given that<img align="ABSMIDDLE" height="49" name="Object90" src="http://img1.meritnation.com/img/curr/1/12/15/230/6054/Chapter%201_html_m2c54b729.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="153" />.</div>
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<img align="ABSMIDDLE" height="203" name="Object91" src="http://img1.meritnation.com/img/curr/1/12/15/230/6054/Chapter%201_html_m196322d8.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="371" /></div>
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Hence, the given function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is invertible and the inverse of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>itself.</div>
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Question 5:</div>
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State with reason whether following functions have inverse</div>
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(i) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>: {1, 2, 3, 4} → {10} with</div>
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>= {(1, 10), (2, 10), (3, 10), (4, 10)}</div>
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(ii) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>: {5, 6, 7, 8} → {1, 2, 3, 4} with</div>
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>= {(5, 4), (6, 3), (7, 4), (8, 2)}</div>
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(iii) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i>: {2, 3, 4, 5} → {7, 9, 11, 13} with</div>
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i><span class="Apple-converted-space"> </span>= {(2, 7), (3, 9), (4, 11), (5, 13)}</div>
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SOLUTIONS=</div>
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(i) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>: {1, 2, 3, 4} → {10}defined as:</div>
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>= {(1, 10), (2, 10), (3, 10), (4, 10)}</div>
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From the given definition of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>, we can see that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is a many one function as:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(1) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(2) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(3) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(4) = 10</div>
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∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>is not one-one.</div>
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Hence, function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>does not have an inverse.</div>
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(ii) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>: {5, 6, 7, 8} → {1, 2, 3, 4} defined as:</div>
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>= {(5, 4), (6, 3), (7, 4), (8, 2)}</div>
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From the given definition of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>, it is seen that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>is a many one function as:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>(5) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>(7) = 4.</div>
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∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g<span class="Apple-converted-space"> </span></i>is not one-one,</div>
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Hence, function g does not have an inverse.</div>
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(iii) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i>: {2, 3, 4, 5} → {7, 9, 11, 13} defined as:</div>
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i><span class="Apple-converted-space"> </span>= {(2, 7), (3, 9), (4, 11), (5, 13)}</div>
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It is seen that all distinct elements of the set {2, 3, 4, 5} have distinct images under<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i>.</div>
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∴Function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i><span class="Apple-converted-space"> </span>is one-one.</div>
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Also,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i><span class="Apple-converted-space"> </span>is onto since for every element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>of the set {7, 9, 11, 13}, there exists an element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>in the set {2, 3, 4, 5}such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>.</div>
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Thus,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i><span class="Apple-converted-space"> </span>is a one-one and onto function. Hence,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">h</i><span class="Apple-converted-space"> </span>has an inverse.</div>
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-webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><img border="0" height="1" src="http://googleads.g.doubleclick.net/pagead/viewthroughconversion/1036711378/?random=1358529884583&cv=7&fst=1358529884580&num=2&fmt=1&label=ZLO9COT-6wMQ0uur7gM&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=22&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6045/relations-and-functions/let-f-1-3-4-rarr-1-2-5-and-g-1&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6057/relations-and-functions/state-with-reason-whether-following-function&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
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Anonymoushttp://www.blogger.com/profile/17104854705255350948noreply@blogger.com0tag:blogger.com,1999:blog-2679372747285705478.post-80577297857666275762013-01-18T09:15:00.003-08:002013-01-18T09:15:52.275-08:00excersie 1.2 <div dir="ltr" style="text-align: left;" trbidi="on">
<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 1:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Show that the function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">*</sub><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">*</sub><span class="Apple-converted-space"> </span>defined by <img align="ABSMIDDLE" name="Object25" src="http://img1.meritnation.com/img/curr/1/12/15/230/6022/Chapter%201_html_m3c3e4298.gif" style="border: medium none; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px;" />is one-one and onto, where<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">*</sub><span class="Apple-converted-space"> </span>is the set of all non-zero real numbers. Is the result true, if the domain<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">*</sub><span class="Apple-converted-space"> </span>is replaced by<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>with co-domain being same as<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">*</sub>?</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions =</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
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<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<br /></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<br /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
It is given that<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">*</sub><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">*</sub><span class="Apple-converted-space"> </span>is defined by<img align="ABSMIDDLE" height="41" name="Object26" src="http://img1.meritnation.com/img/curr/1/12/15/230/6022/Chapter%201_html_4a5cef12.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="69" /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
One-one:</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<img align="ABSMIDDLE" height="95" name="Object27" src="http://img1.meritnation.com/img/curr/1/12/15/230/6022/Chapter%201_html_4accecb1.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="89" /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is one-one.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Onto:</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
It is clear that for<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>∈<b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>R</b><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">*</sub>, there exists<img align="ABSMIDDLE" height="44" name="Object28" src="http://img1.meritnation.com/img/curr/1/12/15/230/6022/Chapter%201_html_13279474.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="185" /> such that</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<img align="ABSMIDDLE" height="68" name="Object29" src="http://img1.meritnation.com/img/curr/1/12/15/230/6022/Chapter%201_html_fe5622f.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="112" /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is onto.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Thus, the given function (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f)</i><span class="Apple-converted-space"> </span>is one-one and onto.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, consider function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>:<b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>N<span class="Apple-converted-space"> </span></b>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">*</sub>defined by</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<img align="ABSMIDDLE" height="41" name="Object30" src="http://img1.meritnation.com/img/curr/1/12/15/230/6022/Chapter%201_html_m15c053d.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="68" /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
We have,</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<img align="ABSMIDDLE" height="45" name="Object31" src="http://img1.meritnation.com/img/curr/1/12/15/230/6022/Chapter%201_html_1baa05a9.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="229" /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>is one-one.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Further, it is clear that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>is not onto as for 1.2 ∈<b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">*</sub><span class="Apple-converted-space"> </span>there does not exit any<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<img align="ABSMIDDLE" height="41" name="Object32" src="http://img1.meritnation.com/img/curr/1/12/15/230/6022/Chapter%201_html_m1841b6ee.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="25" />.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">g</i><span class="Apple-converted-space"> </span>is one-one but not onto.</div>
<div style="background-color: white; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
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<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 2:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Check the injectivity and surjectivity of the following functions:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(i)<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>given by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(ii)<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>given by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iii)<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>given by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iv)<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N<span class="Apple-converted-space"> </span></b>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>given by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(v)<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>given by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<sup style="color: #333333; font-family: 'Times New Roman',serif; font-size: 12px;">SOLUTIONS = </sup></div>
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<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">(i)<span class="Apple-converted-space"> </span></span></span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>is given by,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is seen that for<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>∈<b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>) ⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is injective.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, 2 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>. But, there does not exist any<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>= 2.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not surjective.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is injective but not surjective.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(ii)<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>is given by,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is seen that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(−1) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(1) = 1, but −1 ≠ 1.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>is not injective.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now,−2 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b>. But, there does not exist any element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x<span class="Apple-converted-space"> </span></i>∈<b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>= −2.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not surjective.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is neither injective nor surjective.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">(iii)</span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is given by,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is seen that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(−1) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(1) = 1, but −1 ≠ 1.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>is not injective.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now,−2 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>. But, there does not exist any element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>= −2.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not surjective.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is neither injective nor surjective.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">(iv)</span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>given by,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is seen that for<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>∈<b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>) ⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is injective.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, 2 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>. But, there does not exist any element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>in domain<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>= 2.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not surjective</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is injective but not surjective.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">(v)<span class="Apple-converted-space"> </span></span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z<span class="Apple-converted-space"> </span></b>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>is given by,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is seen that for<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>) ⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is injective.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, 2 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b>. But, there does not exist any element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>in domain<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>= 2.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not surjective.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is injective but not surjective.</div>
</div>
</div>
</div>
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Question 3:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Prove that the Greatest Integer Function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>given by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = [<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>], is neither one-once nor onto, where [<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>] denotes the greatest integer less than or equal to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
SOLUTIONS =</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is given by,</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = [<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>]</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
It is seen that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(1.2) = [1.2] = 1,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(1.9) = [1.9] = 1.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(1.2) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(1.9), but 1.2 ≠ 1.9.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not one-one.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, consider 0.7 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
It is known that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = [<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>] is always an integer. Thus, there does not exist any element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = 0.7.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not onto.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, the greatest integer function is neither one-one nor onto.</div>
<div style="background-color: white; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<br /></div>
<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 4:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Show that the Modulus Function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>given by<img align="ABSMIDDLE" height="27" name="Object33" src="http://img1.meritnation.com/img/curr/1/12/15/230/6030/Chapter%201_html_m517b6130.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="68" />, is neither one-one nor onto, where<img align="ABSMIDDLE" height="27" name="Object34" src="http://img1.meritnation.com/img/curr/1/12/15/230/6030/Chapter%201_html_3549d168.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="19" /><span class="Apple-converted-space"> </span>is<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>, if<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>is positive or 0 and<img align="ABSMIDDLE" height="27" name="Object35" src="http://img1.meritnation.com/img/curr/1/12/15/230/6030/Chapter%201_html_3549d168.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="19" /> is −<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>, if<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>is negative.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
SOLUTIONS =</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<br /></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is given by,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="48" name="Object36" src="http://img1.meritnation.com/img/curr/1/12/15/230/6030/Chapter%201_html_45680a01.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="165" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is seen that<img align="ABSMIDDLE" height="27" name="Object37" src="http://img1.meritnation.com/img/curr/1/12/15/230/6030/Chapter%201_html_m2561dee5.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="188" />.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(−1) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(1), but −1 ≠ 1.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not one-one.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, consider −1 ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is known that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="27" name="Object38" src="http://img1.meritnation.com/img/curr/1/12/15/230/6030/Chapter%201_html_3549d168.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="19" /><span class="Apple-converted-space"> </span>is always non-negative. Thus, there does not exist any element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>in domain<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="27" name="Object39" src="http://img1.meritnation.com/img/curr/1/12/15/230/6030/Chapter%201_html_3549d168.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="19" /><span class="Apple-converted-space"> </span>= −1.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not onto.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, the modulus function is neither one-one nor onto.</div>
</div>
</div>
</div>
<img border="0" height="1" src="http://www.googleadservices.com/pagead/conversion/1036711378/?random=1358528915075&cv=7&fst=1358528915075&num=1&fmt=3&value=0&label=ZDPOCNTSkwMQ0uur7gM&bg=ffffff&hl=en&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=4&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6028/relations-and-functions/prove-that-the-greatest-integer-function-f&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6030/relations-and-functions/show-that-the-modulus-function-f-r-rarr-r&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><img border="0" height="1" src="http://googleads.g.doubleclick.net/pagead/viewthroughconversion/1036711378/?random=1358528915078&cv=7&fst=1358528915075&num=2&fmt=1&label=ZLO9COT-6wMQ0uur7gM&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=4&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6028/relations-and-functions/prove-that-the-greatest-integer-function-f&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6030/relations-and-functions/show-that-the-modulus-function-f-r-rarr-r&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 5:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Show that the Signum Function<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>, given by</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="75" name="Object40" src="http://img1.meritnation.com/img/curr/1/12/15/230/6032/Chapter%201_html_m312b8b2c.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="133" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
is neither one-one nor onto.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
SOLUTIONS =</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is given by,</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<img align="ABSMIDDLE" height="75" name="Object41" src="http://img1.meritnation.com/img/curr/1/12/15/230/6032/Chapter%201_html_m312b8b2c.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="133" /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
It is seen that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(1) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(2) = 1, but 1 ≠ 2.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not one-one.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) takes only 3 values (1, 0, or −1) for the element −2 in co-domain<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>, there does not exist any<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>x</i><span class="Apple-converted-space"> </span>in domain<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = −2.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not onto.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, the signum function is neither one-one nor onto.</div>
<div style="background-color: white; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<br /></div>
<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 6:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= {1, 2, 3},<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B<span class="Apple-converted-space"> </span></i>= {4, 5, 6, 7} and let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>= {(1, 4), (2, 5), (3, 6)} be a function from<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i>. Show that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is one-one.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
SOLUTIONS =</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
It is given that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= {1, 2, 3},<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B<span class="Apple-converted-space"> </span></i>= {4, 5, 6, 7}.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>is defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>= {(1, 4), (2, 5), (3, 6)}.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>(1) = 4,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>(2) = 5,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>(3) = 6</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
It is seen that the images of distinct elements of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>under<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>are distinct.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is one-one.</div>
<div style="background-color: white; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<br /></div>
<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 7:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(i) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>defined by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = 3 − 4<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(ii) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>defined by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = 1 +<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<sup style="color: #333333; font-family: 'Times New Roman',serif; font-size: 12px;">SOLUTIONS =</sup></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<sup style="color: #333333; font-family: 'Times New Roman',serif; font-size: 12px;"> </sup></div>
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<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(i) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = 3 − 4<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="27" name="Object42" src="http://img1.meritnation.com/img/curr/1/12/15/230/6034/Chapter%201_html_m267e3da6.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="247" />.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="72" name="Object43" src="http://img1.meritnation.com/img/curr/1/12/15/230/6034/Chapter%201_html_6697f990.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="124" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is one-one.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For any real number (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y)</i><span class="Apple-converted-space"> </span>in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>, there exists<img align="ABSMIDDLE" height="41" name="Object44" src="http://img1.meritnation.com/img/curr/1/12/15/230/6034/Chapter%201_html_m6c84c2ef.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="37" /> i n<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>such that</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="45" name="Object45" src="http://img1.meritnation.com/img/curr/1/12/15/230/6034/Chapter%201_html_m16b4e4d6.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="189" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>is onto.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>is bijective.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(ii) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>R</b><span class="Apple-converted-space"> </span>is defined as</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="27" name="Object46" src="http://img1.meritnation.com/img/curr/1/12/15/230/6034/Chapter%201_html_m57ec1cac.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="88" />.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="27" name="Object47" src="http://img1.meritnation.com/img/curr/1/12/15/230/6034/Chapter%201_html_m267e3da6.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="247" />.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="76" name="Object48" src="http://img1.meritnation.com/img/curr/1/12/15/230/6034/Chapter%201_html_1bbb0484.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="107" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<img align="ABSMIDDLE" height="27" name="Object49" src="http://img1.meritnation.com/img/curr/1/12/15/230/6034/Chapter%201_html_93d66d8.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="99" />does not imply that<img align="ABSMIDDLE" height="24" name="Object50" src="http://img1.meritnation.com/img/curr/1/12/15/230/6034/Chapter%201_html_m27a19131.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="51" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For instance,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="27" name="Object51" src="http://img1.meritnation.com/img/curr/1/12/15/230/6034/Chapter%201_html_1040cc0f.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="116" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not one-one.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Consider an element −2 in co-domain<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is seen that<img align="ABSMIDDLE" height="27" name="Object52" src="http://img1.meritnation.com/img/curr/1/12/15/230/6034/Chapter%201_html_m57ec1cac.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="88" /> is positive for all<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, there does not exist any<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>x</i><span class="Apple-converted-space"> </span>in domain<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = −2.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not onto.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is neither one-one nor onto.</div>
</div>
</div>
</div>
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color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; 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text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 8:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>be sets. Show that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>×<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B<span class="Apple-converted-space"> </span></i>×<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>such that (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) = (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) is bijective function.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
SOLUTIONS =</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
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<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
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<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>×<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>×<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>is defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) = (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>).</div>
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<img align="ABSMIDDLE" height="27" name="Object53" src="http://img1.meritnation.com/img/curr/1/12/15/230/6035/Chapter%201_html_3bf55b34.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="396" />.</div>
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<img align="ABSMIDDLE" height="77" name="Object54" src="http://img1.meritnation.com/img/curr/1/12/15/230/6035/Chapter%201_html_m58ad0bd6.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="139" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is one-one.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) ∈<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>×<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>be any element.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Then, there exists (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) ∈<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>×<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">B</i><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) = (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>). [By definition of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>]</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is onto.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence,<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i><span class="Apple-converted-space"> </span>is bijective.</div>
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font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; 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Question 9:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>be defined by<img align="ABSMIDDLE" height="88" name="Object55" src="http://img1.meritnation.com/img/curr/1/12/15/230/6037/Chapter%201_html_m18543bea.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="260" /></div>
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State whether the function f is bijective. Justify your answer.</div>
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SOLUTIONS=</div>
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<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>is defined as<img align="ABSMIDDLE" height="88" name="Object56" src="http://img1.meritnation.com/img/curr/1/12/15/230/6037/Chapter%201_html_m18543bea.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="260" /></div>
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It can be observed that:</div>
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<img align="ABSMIDDLE" height="41" name="Object57" src="http://img1.meritnation.com/img/curr/1/12/15/230/6037/Chapter%201_html_m5a4452fc.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="369" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="27" name="Object58" src="http://img1.meritnation.com/img/curr/1/12/15/230/6037/Chapter%201_html_2fb4345b.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="187" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not one-one.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Consider a natural number (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n)</i><span class="Apple-converted-space"> </span>in co-domain<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Case<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">I:<span class="Apple-converted-space"> </span></b><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>is odd</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>= 2<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">r</i><span class="Apple-converted-space"> </span>+ 1 for some<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">r</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N.<span class="Apple-converted-space"> </span></b>Then, there exists 4<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">r<span class="Apple-converted-space"> </span></i>+ 1∈<b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>such that</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="41" name="Object59" src="http://img1.meritnation.com/img/curr/1/12/15/230/6037/Chapter%201_html_1504181f.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="188" />.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Case<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">II:<span class="Apple-converted-space"> </span></b><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>is even</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">n</i><span class="Apple-converted-space"> </span>= 2<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">r</i><span class="Apple-converted-space"> </span>for some<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">r</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N.<span class="Apple-converted-space"> </span></b>Then,there exists 4<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">r</i><span class="Apple-converted-space"> </span>∈<b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>such that<img align="ABSMIDDLE" height="41" name="Object60" src="http://img1.meritnation.com/img/curr/1/12/15/230/6037/Chapter%201_html_m45dc46e6.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="112" />.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is onto.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not a bijective function.</div>
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</div>
</div>
<img border="0" height="1" src="http://www.googleadservices.com/pagead/conversion/1036711378/?random=1358529124990&cv=7&fst=1358529124990&num=1&fmt=3&value=0&label=ZDPOCNTSkwMQ0uur7gM&bg=ffffff&hl=en&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=9&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6035/relations-and-functions/let-a-and-b-be-sets-show-that-f-a-times&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6037/relations-and-functions/let-f-n-rarr-n-be-defined-by-state-whet&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><img border="0" height="1" src="http://googleads.g.doubleclick.net/pagead/viewthroughconversion/1036711378/?random=1358529124993&cv=7&fst=1358529124990&num=2&fmt=1&label=ZLO9COT-6wMQ0uur7gM&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=9&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6035/relations-and-functions/let-a-and-b-be-sets-show-that-f-a-times&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6037/relations-and-functions/let-f-n-rarr-n-be-defined-by-state-whet&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 10:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let A =<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>− {3} and B =<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>− {1}. Consider the function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>: A → B defined by</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="45" name="Object61" src="http://img1.meritnation.com/img/curr/1/12/15/230/6039/Chapter%201_html_2dbf1edd.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="103" />. Is<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>one-one and onto? Justify your answer.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
SOLUTIONS=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
A =<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>− {3}, B =<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>− {1}</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>: A → B is defined as<img align="ABSMIDDLE" height="45" name="Object62" src="http://img1.meritnation.com/img/curr/1/12/15/230/6039/Chapter%201_html_2dbf1edd.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="103" />.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="27" name="Object63" src="http://img1.meritnation.com/img/curr/1/12/15/230/6039/Chapter%201_html_m38b158f8.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="228" />.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="168" name="Object64" src="http://img1.meritnation.com/img/curr/1/12/15/230/6039/Chapter%201_html_23f1897b.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="237" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is one-one.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>∈B =<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>− {1}. Then,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>≠ 1.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
The function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>is onto if there exists<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>∈A such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y.</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="165" name="Object65" src="http://img1.meritnation.com/img/curr/1/12/15/230/6039/Chapter%201_html_m63b9e71a.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="239" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Thus, <span class="Apple-converted-space"></span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"></span></i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="117" name="Object67" src="http://img1.meritnation.com/img/curr/1/12/15/230/6039/Chapter%201_html_59a7ba34.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="348" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, function<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i><span class="Apple-converted-space"> </span>is one-one and onto.</div>
</div>
</div>
</div>
<img border="0" height="1" src="http://www.googleadservices.com/pagead/conversion/1036711378/?random=1358529166266&cv=7&fst=1358529166266&num=1&fmt=3&value=0&label=ZDPOCNTSkwMQ0uur7gM&bg=ffffff&hl=en&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=10&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6037/relations-and-functions/let-f-n-rarr-n-be-defined-by-state-whet&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6039/relations-and-functions/let-a-r-minus-3-and-b-r-minus-1&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><img border="0" height="1" src="http://googleads.g.doubleclick.net/pagead/viewthroughconversion/1036711378/?random=1358529166279&cv=7&fst=1358529166266&num=2&fmt=1&label=ZLO9COT-6wMQ0uur7gM&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=10&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6037/relations-and-functions/let-f-n-rarr-n-be-defined-by-state-whet&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6039/relations-and-functions/let-a-r-minus-3-and-b-r-minus-1&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 11:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>be defined as<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">4</sup>. Choose the correct answer.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(A) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is one-one onto (B) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is many-one onto</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(C) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is one-one but not onto (D) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is neither one-one nor onto</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
SOLUTIONS=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is defined as<img align="ABSMIDDLE" height="27" name="Object68" src="http://img1.meritnation.com/img/curr/1/12/15/230/6041/Chapter%201_html_477698f8.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="72" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>).</div>
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<img align="ABSMIDDLE" height="48" name="Object69" src="http://img1.meritnation.com/img/curr/1/12/15/230/6041/Chapter%201_html_4824c8c1.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="69" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<img align="ABSMIDDLE" height="27" name="Object70" src="http://img1.meritnation.com/img/curr/1/12/15/230/6041/Chapter%201_html_93d66d8.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="99" />does not imply that<img align="ABSMIDDLE" height="24" name="Object71" src="http://img1.meritnation.com/img/curr/1/12/15/230/6041/Chapter%201_html_m6849815e.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="47" />.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For instance,</div>
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<img align="ABSMIDDLE" height="27" name="Object72" src="http://img1.meritnation.com/img/curr/1/12/15/230/6041/Chapter%201_html_6189c965.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="113" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not one-one.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Consider an element 2 in co-domain<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>. It is clear that there does not exist any<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>in domain<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>such that<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = 2.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is not onto.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is neither one-one nor onto.</div>
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The correct answer is D.</div>
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<img border="0" height="1" src="http://www.googleadservices.com/pagead/conversion/1036711378/?random=1358529203177&cv=7&fst=1358529203177&num=1&fmt=3&value=0&label=ZDPOCNTSkwMQ0uur7gM&bg=ffffff&hl=en&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=11&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6039/relations-and-functions/let-a-r-minus-3-and-b-r-minus-1&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6041/relations-and-functions/let-f-r-rarr-r-be-defined-as-f-x-x4&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><img border="0" height="1" src="http://googleads.g.doubleclick.net/pagead/viewthroughconversion/1036711378/?random=1358529203179&cv=7&fst=1358529203177&num=2&fmt=1&label=ZLO9COT-6wMQ0uur7gM&guid=ON&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=32&u_his=11&u_tz=330&u_java=true&u_nplug=22&u_nmime=119&ref=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6039/relations-and-functions/let-a-r-minus-3-and-b-r-minus-1&url=http%3A//cbse.meritnation.com/study-online/ncert-solutions/math/15/6041/relations-and-functions/let-f-r-rarr-r-be-defined-as-f-x-x4&frm=0" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border: none; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" width="1" /><div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 12:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
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Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>be defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = 3<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>. Choose the correct answer.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(A) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is one-one onto (B) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is many-one onto</div>
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(C) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is one-one but not onto (D) <i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is neither one-one nor onto</div>
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SOLUTIONS=</div>
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>:<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>→<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>is defined as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) = 3<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y<span class="Apple-converted-space"> </span></i>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>such that<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i>(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>).</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒ 3<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>= 3<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>is one-one.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Also, for any real number (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y)</i><span class="Apple-converted-space"> </span>in co-domain<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>, there exists<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="41" name="Object73" src="http://img1.meritnation.com/img/curr/1/12/15/230/6043/Chapter%201_html_5cfc4828.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="17" /><span class="Apple-converted-space"> </span>in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>such that<img align="ABSMIDDLE" height="45" name="Object74" src="http://img1.meritnation.com/img/curr/1/12/15/230/6043/Chapter%201_html_5269aead.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="123" />.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f<span class="Apple-converted-space"> </span></i>is onto.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, function<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">f</i><span class="Apple-converted-space"> </span>is one-one and onto.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
The correct answer is A.</div>
<div style="background-color: white; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
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Anonymoushttp://www.blogger.com/profile/17104854705255350948noreply@blogger.com0tag:blogger.com,1999:blog-2679372747285705478.post-86819067465374516172013-01-18T08:46:00.000-08:002013-01-18T08:56:04.382-08:00excersie 1.1 continued <div dir="ltr" style="text-align: left;" trbidi="on">
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Question 6:</div>
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Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.</div>
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solution = </div>
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Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= {1, 2, 3}.</div>
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A relation R on<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>is defined as R = {(1, 2), (2, 1)}.</div>
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It is seen that (1, 1), (2, 2), (3, 3) ∉R.</div>
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∴ R is not reflexive.</div>
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Now, as (1, 2) ∈ R and (2, 1) ∈ R, then R is symmetric.</div>
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Now, (1, 2) and (2, 1) ∈ R</div>
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However,</div>
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(1, 1) ∉ R</div>
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∴ R is not transitive.</div>
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Hence, R is symmetric but neither reflexive nor transitive.</div>
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Question 7:</div>
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Show that the relation R in the set<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>of all the books in a library of a college, given by R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>have same number of pages} is an equivalence relation.</div>
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solution = </div>
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Set<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>is the set of all books in the library of a college.</div>
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R = {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>have the same number of pages}</div>
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Now, R is reflexive since (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) ∈ R as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>has the same number of pages.</div>
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Let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>) ∈ R ⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>have the same number of pages.</div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>have the same number of pages.</div>
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⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>) ∈ R</div>
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∴R is symmetric.</div>
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Now, let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>) ∈R and (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</i>) ∈ R.</div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>and have the same number of pages and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</i><span class="Apple-converted-space"> </span>have the same number of pages.</div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</i><span class="Apple-converted-space"> </span>have the same number of pages.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</i>) ∈ R</div>
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∴R is transitive.</div>
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Hence, R is an equivalence relation.</div>
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Question 8:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Show that the relation R in the set<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= {1, 2, 3, 4, 5} given by</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" alt="" height="29" name="Object7" src="http://img1.meritnation.com/img/curr/1/12/15/230/6005/Chapter%201_html_m169bdf1e.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="180" />, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solution = </div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= {1, 2, 3, 4, 5}</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<img align="ABSMIDDLE" alt="" height="29" name="Object8" src="http://img1.meritnation.com/img/curr/1/12/15/230/6005/Chapter%201_html_m169bdf1e.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="180" /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
It is clear that for any element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>∈<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>, we have<img align="ABSMIDDLE" alt="" height="27" name="Object9" src="http://img1.meritnation.com/img/curr/1/12/15/230/6005/Chapter%201_html_459ac6b9.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="65" /><span class="Apple-converted-space"> </span>(which is even).</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is reflexive.</div>
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Let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) ∈ R.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<img align="ABSMIDDLE" alt="" height="83" name="Object10" src="http://img1.meritnation.com/img/curr/1/12/15/230/6005/Chapter%201_html_m33537aa1.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="213" /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is symmetric.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) ∈ R and (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) ∈ R.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<img align="ABSMIDDLE" alt="" height="107" name="Object11" src="http://img1.meritnation.com/img/curr/1/12/15/230/6005/Chapter%201_html_1d3cb8e0.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="515" /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) ∈ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, R is an equivalence relation.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, all elements of the set {1, 3, 5} are related to each other as all the elements of this subset are odd. Thus, the modulus of the difference between any two elements will be even.</div>
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Similarly, all elements of the set {2, 4} are related to each other as all the elements of this subset are even.</div>
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Also, no element of the subset {1, 3, 5} can be related to any element of {2, 4} as all elements of {1, 3, 5} are odd and all elements of {2, 4} are even. Thus, the modulus of the difference between the two elements (from each of these two subsets) will not be even.</div>
<div style="background-color: white; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<br /></div>
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Question 9:</div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Show that each of the relation R in the set<img align="ABSMIDDLE" height="27" name="Object12" src="http://img1.meritnation.com/img/curr/1/12/15/230/6007/Chapter%201_html_m2cf95aba.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="151" />, given by</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(i) <img align="ABSMIDDLE" height="29" name="Object13" src="http://img1.meritnation.com/img/curr/1/12/15/230/6007/Chapter%201_html_1e9a99ef.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="239" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(ii) <img align="ABSMIDDLE" height="27" name="Object14" src="http://img1.meritnation.com/img/curr/1/12/15/230/6007/Chapter%201_html_m7a54ba86.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="125" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
is an equivalence relation. Find the set of all elements related to 1 in each case.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solution=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<img align="ABSMIDDLE" height="27" name="Object15" src="http://img1.meritnation.com/img/curr/1/12/15/230/6007/Chapter%201_html_43efc75.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="404" /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(i) <img align="ABSMIDDLE" height="29" name="Object16" src="http://img1.meritnation.com/img/curr/1/12/15/230/6007/Chapter%201_html_1e9a99ef.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="239" /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
For any element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>∈A, we have (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) ∈ R as<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="27" name="Object17" src="http://img1.meritnation.com/img/curr/1/12/15/230/6007/Chapter%201_html_459ac6b9.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="65" />is a multiple of 4.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is reflexive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) ∈ R ⇒<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="27" name="Object18" src="http://img1.meritnation.com/img/curr/1/12/15/230/6007/Chapter%201_html_m5a4ace0b.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="40" />is a multiple of 4.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<img align="ABSMIDDLE" height="29" name="Object19" src="http://img1.meritnation.com/img/curr/1/12/15/230/6007/Chapter%201_html_6c903344.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="252" /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) ∈ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is symmetric.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) ∈ R.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<img align="ABSMIDDLE" height="107" name="Object20" src="http://img1.meritnation.com/img/curr/1/12/15/230/6007/Chapter%201_html_2da5cd47.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="375" /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) ∈R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴ R is transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, R is an equivalence relation.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
The set of elements related to 1 is {1, 5, 9} since</div>
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(ii) R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>}</div>
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For any element<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>∈A, we have (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) ∈ R, since<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>.</div>
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∴R is reflexive.</div>
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Now, let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) ∈ R.</div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i></div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i></div>
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⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) ∈ R</div>
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∴R is symmetric.</div>
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Now, let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) ∈ R and (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) ∈ R.</div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i></div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i></div>
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⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) ∈ R</div>
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∴ R is transitive.</div>
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Hence, R is an equivalence relation.</div>
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The elements in R that are related to 1 will be those elements from set A which are equal to 1.</div>
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Hence, the set of elements related to 1 is {1}.</div>
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Question 10:</div>
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Given an example of a relation. Which is</div>
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(i) Symmetric but neither reflexive nor transitive.</div>
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(ii) Transitive but neither reflexive nor symmetric.</div>
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(iii) Reflexive and symmetric but not transitive.</div>
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(iv) Reflexive and transitive but not symmetric.</div>
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(v) Symmetric and transitive but not reflexive.</div>
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solutions= </div>
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<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(i) Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= {5, 6, 7}.</div>
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Define a relation R on<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>A</i><span class="Apple-converted-space"> </span>as R = {(5, 6), (6, 5)}.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Relation R is not reflexive as (5, 5), (6, 6), (7, 7) ∉ R.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, as (5, 6) ∈ R and also (6, 5) ∈ R, R is symmetric.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="16" name="Object22" src="http://img1.meritnation.com/img/curr/1/12/15/230/6008/Chapter%201_html_74849e29.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="20" />(5, 6), (6, 5) ∈ R, but (5, 5) ∉ R</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴R is not transitive.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, relation R is symmetric but not reflexive or transitive.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(ii)Consider a relation R in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>defined as:</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span><<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>}</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
For any<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>∈ R, we have (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) ∉ R since<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>cannot be strictly less than<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>itself. In fact,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴ R is not reflexive.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now,</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(1, 2) ∈ R (as 1 < 2)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
But, 2 is not less than 1.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴ (2, 1) ∉ R</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴ R is not symmetric.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) ∈ R.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span><<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span><<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span><<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) ∈ R</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴R is transitive.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, relation R is transitive but not reflexive and symmetric.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iii)Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= {4, 6, 8}.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Define a relation R on A as:</div>
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= {(4, 4), (6, 6), (8, 8), (4, 6), (6, 4), (6, 8), (8, 6)}</div>
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Relation R is reflexive since for every<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i>, (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) ∈R i.e., (4, 4), (6, 6), (8, 8)} ∈ R.</div>
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Relation R is symmetric since (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>) ∈ R ⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) ∈ R for all<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>∈ R.</div>
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Relation R is not transitive since (4, 6), (6, 8) ∈ R, but (4, 8) ∉ R.</div>
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Hence, relation R is reflexive and symmetric but not transitive.</div>
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(iv) Define a relation R in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>as:</div>
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R = {<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>≥<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup>}</div>
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Clearly (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) ∈ R as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup>.</div>
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∴R is reflexive.</div>
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Now,</div>
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(2, 1) ∈ R (as 2<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>≥ 1<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup>)</div>
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But,</div>
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(1, 2) ∉ R (as 1<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>< 2<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup>)</div>
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∴ R is not symmetric.</div>
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Now,</div>
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Let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) ∈ R.</div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>≥<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>≥<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup></div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>≥<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup></div>
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⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) ∈ R</div>
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∴R is transitive.</div>
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Hence, relation R is reflexive and transitive but not symmetric.</div>
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(v) Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= {−5, −6}.</div>
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Define a relation R on<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>as:</div>
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R = {(−5, −6), (−6, −5), (−5, −5)}</div>
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Relation R is not reflexive as (−6, −6) ∉ R.</div>
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Relation R is symmetric as (−5, −6) ∈ R and (−6, −5}∈R.</div>
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It is seen that (−5, −6), (−6, −5) ∈ R. Also, (−5, −5) ∈ R.</div>
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∴The relation R is transitive.</div>
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Hence, relation R is symmetric and transitive but not reflexive.</div>
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Question 11:</div>
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Show that the relation R in the set<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>of points in a plane given by R = {(P, Q): distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all point related to a point P ≠ (0, 0) is the circle passing through P with origin as center.</div>
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solutions =</div>
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<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
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R = {(P, Q): distance of point P from the origin is the same as the distance of point Q from the origin}</div>
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Clearly, (P, P) ∈ R since the distance of point P from the origin is always the same as the distance of the same point P from the origin.</div>
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∴R is reflexive.</div>
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Now,</div>
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Let (P, Q) ∈ R.</div>
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⇒ The distance of point P from the origin is the same as the distance of point Q from the origin.</div>
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⇒ The distance of point Q from the origin is the same as the distance of point P from the origin.</div>
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⇒ (Q, P) ∈ R</div>
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∴R is symmetric.</div>
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Now,</div>
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Let (P, Q), (Q, S) ∈ R.</div>
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⇒ The distance of points P and Q from the origin is the same and also, the distance of points Q and S from the origin is the same.</div>
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⇒ The distance of points P and S from the origin is the same.</div>
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⇒ (P, S) ∈ R</div>
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∴R is transitive.</div>
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Therefore, R is an equivalence relation.</div>
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The set of all points related to P ≠ (0, 0) will be those points whose distance from the origin is the same as the distance of point P from the origin.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
In other words, if O (0, 0) is the origin and OP =<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">k</i>, then the set of all points related to P is at a distance of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">k</i><span class="Apple-converted-space"> </span>from the origin.</div>
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Hence, this set of points forms a circle with the centre as the origin and this circle passes through point P.</div>
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Question 12:</div>
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Show that the relation R defined in the set<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>of all triangles as R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>is similar to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>}, is equivalence relation. Consider three right angle triangles<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>with sides 3, 4, 5,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub><span class="Apple-converted-space"> </span>with sides 5, 12, 13 and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sub><span class="Apple-converted-space"> </span>with sides 6, 8, 10. Which triangles among<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sub><span class="Apple-converted-space"> </span>are related?</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions = </div>
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<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
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R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>is similar to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>}</div>
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R is reflexive since every triangle is similar to itself.</div>
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Further, if (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>) ∈ R, then<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>is similar to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2.</sub></div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub><span class="Apple-converted-space"> </span>is similar to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1.</sub></div>
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⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>) ∈R</div>
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∴R is symmetric.</div>
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Now,</div>
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Let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sub>) ∈ R.</div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>is similar to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub><span class="Apple-converted-space"> </span>is similar to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3.</sub></div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>is similar to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3.</sub></div>
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⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sub>) ∈ R</div>
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∴ R is transitive.</div>
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Thus, R is an equivalence relation.</div>
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Now, we can observe that:</div>
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∴The corresponding sides of triangles<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sub><span class="Apple-converted-space"> </span>are in the same ratio.</div>
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Then, triangle<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>is similar to triangle<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3.</sub></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub> is related to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sub>.</div>
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</div>
</div>
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color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; 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Question 13:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Show that the relation R defined in the set<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>of all polygons as R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub><span class="Apple-converted-space"> </span>have same number of sides}, is an equivalence relation. What is the set of all elements in<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>related to the right angle triangle<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><span class="Apple-converted-space"> </span>with sides 3, 4 and 5?</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions = </div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub><span class="Apple-converted-space"> </span>have same the number of sides}</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
R is reflexive since (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>) ∈ R as the same polygon has the same number of sides with itself.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>) ∈ R.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub><span class="Apple-converted-space"> </span>have the same number of sides.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>have the same number of sides.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>) ∈ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is symmetric.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now,</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sub>) ∈ R.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub><span class="Apple-converted-space"> </span>have the same number of sides. Also,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sub><span class="Apple-converted-space"> </span>have the same number of sides.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sub><span class="Apple-converted-space"> </span>have the same number of sides.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">P</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sub>) ∈ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, R is an equivalence relation.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
The elements in<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>related to the right-angled triangle (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T)</i><span class="Apple-converted-space"> </span>with sides 3, 4, and 5 are those polygons which have 3 sides (since<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><span class="Apple-converted-space"> </span>is a polygon with 3 sides).</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, the set of all elements in<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>related to triangle<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">T</i><span class="Apple-converted-space"> </span>is the set of all triangles.</div>
<div style="background-color: white; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<br /></div>
<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 14:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><span class="Apple-converted-space"> </span>be the set of all lines in XY plane and R be the relation in<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><span class="Apple-converted-space"> </span>defined as R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</i></sub>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;"><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</i></sub><span class="Apple-converted-space"> </span>is parallel to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>}. Show that R is an equivalence relation. Find the set of all lines related to the line<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>= 2<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>+ 4.</div>
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solutions =</div>
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<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
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R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>): L<sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>is parallel to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>}</div>
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R is reflexive as any line<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>is parallel to itself i.e., (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>) ∈ R.</div>
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Now,</div>
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Let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>) ∈ R.</div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>is parallel to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2.</sub></div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub><span class="Apple-converted-space"> </span>is parallel to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1.</sub></div>
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⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>) ∈ R</div>
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∴ R is symmetric.</div>
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Now,</div>
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Let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sub>) ∈R.</div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>is parallel to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub>. Also,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sub><span class="Apple-converted-space"> </span>is parallel to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3.</sub></div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">1</sub><span class="Apple-converted-space"> </span>is parallel to<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">L</i><sub style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3.</sub></div>
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∴R is transitive.</div>
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Hence, R is an equivalence relation.</div>
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The set of all lines related to the line<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>= 2<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>+ 4 is the set of all lines that are parallel to the line<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>= 2<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>+ 4.</div>
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Slope of line<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>= 2<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>+ 4 is<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">m</i><span class="Apple-converted-space"> </span>= 2</div>
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It is known that parallel lines have the same slopes.</div>
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The line parallel to the given line is of the form<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>= 2<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>+<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>, where<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i><span class="Apple-converted-space"> </span>∈<b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>.</div>
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Hence, the set of all lines related to the given line is given by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>= 2<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>+<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>, where<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b>.</div>
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Question 15:</div>
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Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.</div>
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(A) R is reflexive and symmetric but not transitive.</div>
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(B) R is reflexive and transitive but not symmetric.</div>
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(C) R is symmetric and transitive but not reflexive.</div>
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(D) R is an equivalence relation.</div>
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solutions =</div>
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R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}</div>
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It is seen that (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R,</b><span class="Apple-converted-space"> </span>for every<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>∈{1, 2, 3, 4}.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴ R is reflexive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
It is seen that (1, 2) ∈ R, but (2, 1) ∉ R.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is not symmetric.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Also, it is observed that (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) ∈ R ⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) ∈ R for all<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i><span class="Apple-converted-space"> </span>∈ {1, 2, 3, 4}.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴ R is transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, R is reflexive and transitive but not symmetric.</div>
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The correct answer is B.</div>
<div style="background-color: white; color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<br /></div>
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Question 16:</div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Let R be the relation in the set<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N<span class="Apple-converted-space"> </span></b>given by R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>− 2,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b<span class="Apple-converted-space"> </span></i>> 6}. Choose the correct answer.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(A) (2, 4) ∈ R (B) (3, 8) ∈R (C) (6, 8) ∈R (D) (8, 7) ∈ R</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions=</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div class="qaContent" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; border-bottom-color: rgb(142, 198, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(142, 198, 0); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(142, 198, 0); border-right-style: solid; border-right-width: 1px; color: black; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding: 12px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>− 2,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>> 6}</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, since<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>> 6, (2, 4) ∉ R</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Also, as 3 ≠ 8 − 2, (3, 8) ∉ R</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
And, as 8 ≠ 7 − 2</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
<img align="ABSMIDDLE" height="13" name="Object24" src="http://img1.meritnation.com/img/curr/1/12/15/230/6018/Chapter%201_html_4dd19828.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="15" /><span class="Apple-converted-space"> </span>(8, 7) ∉ R</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now, consider (6, 8).</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
We have 8 > 6 and also, 6 = 8 − 2.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴(6, 8) ∈ R</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
The correct answer is C.</div>
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-webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; 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Anonymoushttp://www.blogger.com/profile/17104854705255350948noreply@blogger.com0Jodhpur, Rajasthan, India26.2389469 73.02430939999999326.0111109 72.7015859 26.4667829 73.347032899999988tag:blogger.com,1999:blog-2679372747285705478.post-64695704139585752192013-01-18T08:42:00.003-08:002013-01-18T08:42:52.055-08:00excersie 1.1 <div dir="ltr" style="text-align: left;" trbidi="on">
<div class="question" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; border-bottom-color: rgb(102, 102, 102); border-bottom-style: dotted; border-bottom-width: 1px; color: #999999; font-family: 'Trebuchet MS'; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Question 2:</div>
<div class="html" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; padding-bottom: 25px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Show that the relation R in the set<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R<span class="Apple-converted-space"> </span></b>of real numbers, defined as</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>≤<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup>} is neither reflexive nor symmetric nor transitive.</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions =</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
<br /></div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>≤<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup>}</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
It can be observed that<span class="Apple-converted-space"> </span><img align="ABSMIDDLE" height="49" name="Object3" src="http://img1.meritnation.com/img/curr/1/12/15/230/5979/Chapter%201_html_m28016277.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="224" /></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is not reflexive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, (1, 4) ∈ R as 1 < 4<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup></div>
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But, 4 is not less than 1<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup>.</div>
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∴(4, 1) ∉ R</div>
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∴R is not symmetric.</div>
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Now,</div>
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(3, 2), (2, 1.5) ∈ R</div>
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(as 3 < 2<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>= 4 and 2 < (1.5)<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>= 2.25)</div>
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But, 3 > (1.5)<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">2</sup><span class="Apple-converted-space"> </span>= 2.25</div>
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∴(3, 1.5) ∉ R</div>
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∴ R is not transitive.</div>
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Hence, R is neither reflexive, nor symmetric, nor transitive.</div>
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<br /></div>
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Question 3:</div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as</div>
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R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>+ 1} is reflexive, symmetric or transitive.</div>
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solution =</div>
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Let<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= {1, 2, 3, 4, 5, 6}.</div>
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A relation R is defined on set<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>as:</div>
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R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>+ 1}</div>
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∴R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}</div>
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We can find (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) ∉ R, where<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>∈ A.</div>
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For instance,</div>
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(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) ∉ R</div>
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∴R is not reflexive.</div>
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It can be observed that (1, 2) ∈ R, but (2, 1) ∉ R.</div>
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∴R is not symmetric.</div>
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Now, (1, 2), (2, 3) ∈<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b></div>
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But,</div>
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(1, 3) ∉ R</div>
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∴R is not transitive</div>
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Hence</div>
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Question 4:</div>
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Show that the relation R in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>defined as R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>≤<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>}, is reflexive and transitive but not symmetric.</div>
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solution=</div>
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</div>
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R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>);<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>≤<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>}</div>
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Clearly (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>) ∈ R as<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></i>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>.</div>
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∴R is reflexive.</div>
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Now,</div>
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(2, 4) ∈ R (as 2 < 4)</div>
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But, (4, 2) ∉ R as 4 is greater than 2.</div>
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∴ R is not symmetric.</div>
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Now, let (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>), (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) ∈ R.</div>
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Then,</div>
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<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>≤<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><span class="Apple-converted-space"> </span>≤<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i></div>
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⇒<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>≤<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i></div>
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⇒ (<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">c</i>) ∈ R</div>
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∴R is transitive.</div>
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Hence,R is reflexive and transitive but not symmetric.</div>
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, R is neither reflexive, nor symmetric, nor transitive.</div>
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Question 5:</div>
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Check whether the relation R in<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">R</b><span class="Apple-converted-space"> </span>defined as R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</i><span class="Apple-converted-space"> </span>≤<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</i><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup>} is reflexive, symmetric or transitive.</div>
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solution = </div>
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<div class="tabsClass ui-tabs ui-widget ui-widget-content ui-corner-all" id="tabs" style="background-color: #eeeeee; background-image: url(http://css.meritnation.com/css/images/ui-bg_highlight-soft_100_eeeeee_1x100.png); background-position: 50% 0%; background-repeat: repeat no-repeat; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; border: 1px solid rgb(221, 221, 221); color: #333333; font-family: 'Trebuchet MS', Tahoma, Verdana, Arial, sans-serif; font-size: 1.1em; padding: 0.2em; position: relative; zoom: 1;">
<div class="answerContent html ui-tabs-panel ui-widget-content ui-corner-bottom" id="optionContent1" name="answer" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-color: rgb(221, 221, 221) rgb(232, 232, 232); border-style: solid; border-width: 0px 1px; color: #333333; display: block; font-family: 'Times New Roman', serif; font-size: 17px; padding: 12px 12px 2px; text-align: left;">
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
R = {(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</em>):<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">a<span class="Apple-converted-space"> </span></em>≤<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">b</em><sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup>}</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
It is observed that<img align="absmiddle" alt="" height="49" name="Object4" src="http://img1.meritnation.com/img/curr/1/12/15/230/5993/Chapter%201_html_m17570f67.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="195" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴ R is not reflexive.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Now,</div>
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(1, 2) ∈ R (as 1 < 2<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>= 8)</div>
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But,</div>
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(2, 1) ∉ R (as 2 > 1<sup style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12px;">3</sup><span class="Apple-converted-space"> </span>= 1)</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴ R is not symmetric.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
We have<img align="absmiddle" alt="" height="49" name="Object5" src="http://img1.meritnation.com/img/curr/1/12/15/230/5993/Chapter%201_html_29fcd0f.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="319" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
But<span class="Apple-converted-space"> </span><img align="absmiddle" alt="" height="49" name="Object6" src="http://img1.meritnation.com/img/curr/1/12/15/230/5993/Chapter%201_html_3ca0076b.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="159" /></div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
∴ R is not transitive.</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Hence, R is<span class="Apple-converted-space"> </span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">neither reflexive, nor symmetric, nor transitive.</span></div>
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-webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span class="Apple-converted-space"> </span></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></span><span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; 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Anonymoushttp://www.blogger.com/profile/17104854705255350948noreply@blogger.com0tag:blogger.com,1999:blog-2679372747285705478.post-72260462335526966372013-01-18T08:33:00.000-08:002013-01-18T08:35:40.625-08:00relations and functions <div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: #f1f1f1; color: black; display: inline !important; float: none; font-family: Verdana, Geneva, sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: nowrap; widows: 2; word-spacing: 0px;">Exercise 1.1</span></div>
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Question 1:</div>
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<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
Determine whether each of the following relations are reflexive, symmetric and transitive:</div>
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(i)Relation R in the set<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= {1, 2, 3…13, 14} defined as</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>): 3<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>= 0}</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(ii) Relation R in the set<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">N</b><span class="Apple-converted-space"> </span>of natural numbers defined as</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>+ 5 and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>< 4}</div>
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(iii) Relation R in the set<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>= {1, 2, 3, 4, 5, 6} as</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>is divisible by<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>}</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(iv) Relation R in the set<span class="Apple-converted-space"> </span><b style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</b><span class="Apple-converted-space"> </span>of all integers defined as</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>is as integer}</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(v) Relation R in the set<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</i><span class="Apple-converted-space"> </span>of human beings in a town at a particular time given by</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(a) R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x<span class="Apple-converted-space"> </span></i>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>work at the same place}</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(b) R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i><span class="Apple-converted-space"> </span>live in the same locality}</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(c) R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x<span class="Apple-converted-space"> </span></i>is exactly 7 cm taller than<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>}</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(d) R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>is wife of<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>}</div>
<div style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; line-height: 22px;">
(e) R = {(<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i>,<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</i>):<span class="Apple-converted-space"> </span><i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</i><span class="Apple-converted-space"> </span>is father of<i style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;"><span class="Apple-converted-space"> </span>y</i>}</div>
<div style="color: #333333; font-family: 'Times New Roman',serif; font-size: 17px; line-height: 22px;">
solutions = </div>
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(i)<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</em><span class="Apple-converted-space"> </span>= {1, 2, 3 … 13, 14}</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
R = {(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>): 3<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>= 0}</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R = {(1, 3), (2, 6), (3, 9), (4, 12)}</div>
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R is not reflexive since (1, 1), (2, 2) … (14, 14) ∉ R.</div>
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Also, R is not symmetric as (1, 3) ∈R, but (3, 1) ∉ R. [3(3) − 1 ≠ 0]</div>
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Also, R is not transitive as (1, 3), (3, 9) ∈R, but (1, 9) ∉ R.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
[3(1) − 9 ≠ 0]</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, R is neither reflexive, nor symmetric, nor transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(ii) R = {(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>):<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>=<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>+ 5 and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>< 4} = {(1, 6), (2, 7), (3, 8)}</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
It is seen that (1, 1) ∉ R.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is not reflexive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(1, 6) ∈R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
But,</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(6, 1) ∉ R.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is not symmetric.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, since there is no pair in R such that (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>) and (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) ∈R, then (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) cannot belong to R.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴ R is not transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, R is neither reflexive, nor symmetric, nor transitive.</div>
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(iii)<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">A</em><span class="Apple-converted-space"> </span>= {1, 2, 3, 4, 5, 6}</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
R = {(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>):<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>is divisible by<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>}</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
We know that any number (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x)</em><span class="Apple-converted-space"> </span>is divisible by itself.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<img align="absmiddle" alt="" height="16" name="Object1" src="http://img1.meritnation.com/img/curr/1/12/15/230/5971/Exercise2_html_74849e29.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="20" /><span class="Apple-converted-space"> </span>(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>) ∈R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is reflexive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now,</div>
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(2, 4) ∈R [as 4 is divisible by 2]</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
But,</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(4, 2) ∉ R. [as 2 is not divisible by 4]</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is not symmetric.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Let (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>), (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) ∈ R. Then,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>is divisible by<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em><span class="Apple-converted-space"> </span>is divisible by<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em><span class="Apple-converted-space"> </span>is divisible by<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒ (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) ∈R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, R is reflexive and transitive but not symmetric.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(iv) R = {(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>):<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>is an integer}</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, for every<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><strong style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</strong>, (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>) ∈R as<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>= 0 is an integer.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is reflexive.</div>
<div align="justify" style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, for every<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><strong style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</strong><span class="Apple-converted-space"> </span>if (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>) ∈ R, then<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>is an integer.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒ −(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>) is also an integer.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒ (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>) is an integer.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴ (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>) ∈ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is symmetric.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now,</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Let (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>) and (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) ∈R, where<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em><span class="Apple-converted-space"> </span>∈<span class="Apple-converted-space"> </span><strong style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">Z</strong>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒ (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>) and (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) are integers.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x<span class="Apple-converted-space"> </span></em>−<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em><span class="Apple-converted-space"> </span>= (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>) + (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>−<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) is an integer.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴ (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) ∈R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, R is reflexive, symmetric, and transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(v) (a) R = {(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>):<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>work at the same place}</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<img align="absmiddle" alt="" height="16" name="Object2" src="http://img1.meritnation.com/img/curr/1/12/15/230/5971/Exercise2_html_74849e29.gif" style="border: none; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;" width="20" /><span class="Apple-converted-space"> </span>(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>) ∈ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴ R is reflexive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
If (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>) ∈ R, then<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>work at the same place.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>work at the same place.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒ (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>) ∈ R.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is symmetric.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, let (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>), (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) ∈ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>work at the same place and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em><span class="Apple-converted-space"> </span>work at the same place.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em><span class="Apple-converted-space"> </span>work at the same place.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒ (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) ∈R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴ R is transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, R is reflexive, symmetric, and transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(b) R = {(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>):<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>live in the same locality}</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Clearly (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>) ∈ R as<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is the same human being.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴ R is reflexive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
If (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>) ∈R, then<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>live in the same locality.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>live in the same locality.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒ (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>) ∈ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is symmetric.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, let (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>) ∈ R and (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) ∈ R.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>live in the same locality and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em><span class="Apple-converted-space"> </span>live in the same locality.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em><span class="Apple-converted-space"> </span>live in the same locality.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒ (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x,</em><span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) ∈ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴ R is transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, R is reflexive, symmetric, and transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(c) R = {(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>):<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is exactly 7 cm taller than<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>}</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now,</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>) ∉ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Since human being<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x<span class="Apple-converted-space"> </span></em>cannot be taller than himself.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is not reflexive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, let (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>) ∈R.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is exactly 7 cm taller than<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Then,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>is not taller than<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴ (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>) ∉R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Indeed if<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is exactly 7 cm taller than<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>, then<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>is exactly 7 cm shorter than<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is not symmetric.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now,</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Let (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>), (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) ∈ R.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is exactly 7 cm taller than<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y<span class="Apple-converted-space"> </span></em>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>is exactly 7 cm taller than z.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is exactly 14 cm taller than<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z<span class="Apple-converted-space"> </span></em>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) ∉R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴ R is not transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, R is neither reflexive, nor symmetric, nor transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(d) R = {(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>):<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is the wife of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>}</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now,</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>) ∉ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Since<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>cannot be the wife of herself.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is not reflexive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, let (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>) ∈ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is the wife of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y.</em></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Clearly<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>is not the wife of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>) ∉ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Indeed if<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is the wife of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>, then<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>is the husband of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴ R is not transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Let (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>), (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) ∈ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is the wife of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>is the wife of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
This case is not possible. Also, this does not imply that<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is the wife of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) ∉ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is not transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, R is neither reflexive, nor symmetric, nor transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(e) R = {(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>):<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is the father of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>}</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>) ∉ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
As<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>cannot be the father of himself.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is not reflexive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, let (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>) ∈R.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is the father of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y.</em></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>cannot be the father of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y.</em></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Indeed,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>is the son or the daughter of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y.</em></div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴(<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>) ∉ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴ R is not symmetric.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Now, let (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>) ∈ R and (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) ∈ R.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is the father of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>and<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">y</em><span class="Apple-converted-space"> </span>is the father of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
⇒<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is not the father of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Indeed<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em><span class="Apple-converted-space"> </span>is the grandfather of<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴ (<em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">x</em>,<span class="Apple-converted-space"> </span><em style="color: #333333; font-family: 'Times New Roman', serif; font-size: 17px;">z</em>) ∉ R</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
∴R is not transitive.</div>
<div style="-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; background-color: white; color: #333333; font-family: 'Times New Roman', serif; font-size: 17px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 22px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
Hence, R is neither reflexive, nor symmetric, nor transitive.</div>
</div>
</div>
Anonymoushttp://www.blogger.com/profile/17104854705255350948noreply@blogger.com1Jodhpur, Rajasthan, India26.2389469 73.02430939999999326.0111109 72.7015859 26.4667829 73.347032899999988