File "HL-paper3.html"
Path: /IB QUESTIONBANKS/4 Fourth Edition - PAPER/HTML/Mathematics HL/Topic 8/HL-paper3html
File size: 263.58 KB
MIME-type: application/octet-stream
Charset: utf-8
<!DOCTYPE html>
<html>
<meta http-equiv="content-type" content="text/html;charset=utf-8">
<head>
<meta charset="utf-8">
<title>IB Questionbank</title>
<link rel="stylesheet" media="all" href="css/application-212ef6a30de2a281f3295db168f85ac1c6eb97815f52f785535f1adfaee1ef4f.css">
<link rel="stylesheet" media="print" href="css/print-6da094505524acaa25ea39a4dd5d6130a436fc43336c0bb89199951b860e98e9.css">
<script src="js/application-13d27c3a5846e837c0ce48b604dc257852658574909702fa21f9891f7bb643ed.js"></script>
<script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML-full"></script>
<!--[if lt IE 9]>
<script src='https://cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv.min.js'></script>
<![endif]-->
<meta name="csrf-param" content="authenticity_token">
<meta name="csrf-token" content="iHF+M3VlRFlNEehLVICYgYgqiF8jIFlzjGNjIwqOK9cFH3ZNdavBJrv/YQpz8vcspoICfQcFHW8kSsHnJsBwfg==">
<link href="favicon.ico" rel="shortcut icon">
</head>
<body class="teacher questions-show">
<div class="navbar navbar-fixed-top">
<div class="navbar-inner">
<div class="container">
<div class="brand">
<div class="inner"><a href="http://ibo.org/">ibo.org</a></div>
</div>
<ul class="nav">
<li>
<a href="../../index.html">Home</a>
</li>
<li class="active dropdown">
<a class="dropdown-toggle" data-toggle="dropdown" href="#">
Questionbanks
<b class="caret"></b>
</a><ul class="dropdown-menu">
<li>
<a href="../../geography.html" target="_blank">DP Geography</a>
</li>
<li>
<a href="../../physics.html" target="_blank">DP Physics</a>
</li>
<li>
<a href="../../chemistry.html" target="_blank">DP Chemistry</a>
</li>
<li>
<a href="../../biology.html" target="_blank">DP Biology</a>
</li>
<li>
<a href="../../furtherMath.html" target="_blank">DP Further Mathematics HL</a>
</li>
<li>
<a href="../../mathHL.html" target="_blank">DP Mathematics HL</a>
</li>
<li>
<a href="../../mathSL.html" target="_blank">DP Mathematics SL</a>
</li>
<li>
<a href="../../mathStudies.html" target="_blank">DP Mathematical Studies</a>
</li>
</ul></li>
<!-- - if current_user.is_language_services? && !current_user.is_publishing? -->
<!-- %li= link_to "Language services", tolk_path -->
</ul>
<ul class="nav pull-right">
<li>
<a href="https://06082010.xyz">IB Documents (2) Team</a>
</li></ul>
</div>
</div>
</div>
<div class="page-content container">
<div class="row">
<div class="span24">
</div>
</div>
<div class="page-header">
<div class="row">
<div class="span16">
<p class="back-to-list">
</p>
</div>
<div class="span8" style="margin: 0 0 -19px 0;">
<img style="width: 100%;" class="qb_logo" src="images/logo.jpg" alt="Ib qb 46 logo">
</div>
</div>
</div><h2>HL Paper 3</h2><div class="specification">
<p class="p1">A relation \(S\) is defined on \(\mathbb{R}\) by \(aSb\) if and only if \(ab > 0\).</p>
</div>
<div class="specification">
<p class="p1">A relation \(R\) is defined on a non-empty set \(A\). \(R\) is symmetric and transitive but not reflexive.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(S\) is</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>not reflexive;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>symmetric;</p>
<p class="p2">(iii) <span class="Apple-converted-space"> </span>transitive.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain why there exists an element \(a \in A\) <span class="s1">that is not related to itself.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence prove that there is at least one element of \(A\) <span class="s1">that is not related to any other element of \(A\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f:G \to H\) be a homomorphism between groups \(\{ G,{\text{ }} * \} \) and \(\{ H,{\text{ }} \circ \} \) with identities \({e_G}\) and \({e_H}\) respectively.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that \(f({e_G}) = {e_H}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that \({\text{Ker}}(f)\) is a subgroup of \(\{ G,{\text{ }} * \} \).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>\(A\), \(B\) and \(C\) are three subsets of a universal set.</p>
</div>
<div class="specification">
<p>Consider the sets \(P = \{ 1,{\text{ }}2,{\text{ }}3\} ,{\text{ }}Q = \{ 2,{\text{ }}3,{\text{ }}4\} \) and \(R = \{ 1,{\text{ }}3,{\text{ }}5\} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Represent the following set on a Venn diagram,</p>
<p>\(A\Delta B\), the symmetric difference of the sets \(A\) and \(B\);</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Represent the following set on a Venn diagram,</p>
<p>\(A \cap (B \cup C)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For sets \(P\), \(Q\) and \(R\), verify that \(P \cup (Q\Delta R) \ne (P \cup Q)\Delta (P \cup R)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>In the context of the distributive law, describe what the result in part (b)(i) illustrates.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The function \(f\,{\text{: }}\mathbb{Z} \to \mathbb{Z}\) is defined by \(f\left( n \right) = n + {\left( { - 1} \right)^n}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that \(f \circ f\) is the identity function.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(f\) is injective.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(f\) is surjective.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(\{ G,{\text{ }} \circ \} \) be the group of all permutations of \(1,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }}5,{\text{ }}6\) under the operation of composition of permutations.</p>
</div>
<div class="specification">
<p class="p1">Consider the following Venn diagram, where \(A = \{ 1,{\text{ }}2,{\text{ }}3,{\text{ }}4\} ,{\text{ }}B = \{ 3,{\text{ }}4,{\text{ }}5,{\text{ }}6\} \).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-02_om_09.47.40.png" alt="N16/5/MATHL/HP3/ENG/TZ0/SG/01.f"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">(i) <span class="Apple-converted-space"> </span>Write the permutation \(\alpha = \left( {\begin{array}{*{20}{c}} 1&2&3&4&5&6 \\ 3&4&6&2&1&5 \end{array}} \right)\) </span>as a composition of disjoint cycles.</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>State the order of \(\alpha \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">(i) <span class="Apple-converted-space"> </span>Write the permutation \(\beta = \left( {\begin{array}{*{20}{c}} 1&2&3&4&5&6 \\ 6&4&3&5&1&2 \end{array}} \right)\) </span>as a composition of disjoint cycles.</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>State the order of \(\beta \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write the permutation \(\alpha \circ \beta \) as a composition of disjoint cycles.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write the permutation \(\beta \circ \alpha \) as a composition of disjoint cycles.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State the order of \(\{ G,{\text{ }} \circ \} \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Find the number of permutations in \(\{ G,{\text{ }} \circ \} \) </span>which will result in \(A\), \(B\) and \(A \cap B\) remaining unchanged.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The binary operations \( \odot \) and \( * \) are defined on \({\mathbb{R}^ + }\) by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[a \odot b = \sqrt {ab} {\text{ and }}a * b = {a^2}{b^2}.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine whether or not</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \odot \) is commutative;</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( * \) is associative;</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( * \) is distributive over \( \odot \) ;</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \odot \) has an identity element.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(\{ G,{\text{ }} * \} \) be a finite group that contains an element <em>a</em> (that is not the identity element) and \(H = \{ {a^n}|n \in {\mathbb{Z}^ + }\} \), where \({a^2} = a * a,{\text{ }}{a^3} = a * a * a\) etc.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\{ H,{\text{ }} * \} \) is a subgroup of \(\{ G,{\text{ }} * \} \).</span></p>
</div>
<br><hr><br><div class="specification">
<p>The set \(A\) contains all positive integers less than 20 that are congruent to 3 modulo 4.</p>
<p>The set \(B\) contains all the prime numbers less than 20.</p>
</div>
<div class="specification">
<p>The set \(C\) is defined as \(C = \{ 7,{\text{ }}9,{\text{ }}13,{\text{ }}19\} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down all the elements of \(A\) and all the elements of \(B\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the symmetric difference, \(A\Delta B\), of the sets \(A\) and \(B\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down all the elements of \(A \cap B,{\text{ }}A \cap C\) and \(B \cup C\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence by considering \(A \cap (B \cup C)\), verify that in this case the operation \( \cap \) is distributive over the operation \( \cup \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The relation <em>R</em> is defined on \(\mathbb{R} \times \mathbb{R}\) such that \(({x_1},{\text{ }}{y_1})R({x_2},{\text{ }}{y_2})\) if and only if \({x_1}{y_1} = {x_2}{y_2}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <em>R</em> is an equivalence relation.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the equivalence class of <em>R</em> containing the element \((1,{\text{ }}2)\) and illustrate this graphically.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The group \(\{ G,{\text{ }}{ \times _7}\} \) is defined on the set {1, 2, 3, 4, 5, 6} where \({ \times _7}\) denotes multiplication modulo 7.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down the Cayley table for \(\{ G,{\text{ }}{ \times _7}\} \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine whether or not \(\{ G,{\text{ }}{ \times _7}\} \) is cyclic.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Find the subgroup of <em>G</em> of order 3, denoting it by <em>H</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) Identify the element of order 2 in <em>G</em> and find its coset with respect to <em>H</em> .</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The group \(\{ K,{\text{ }} \circ \} \) is defined on the six permutations of the integers 1, 2, 3 and \( \circ \) denotes composition of permutations.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that \(\{ K,{\text{ }} \circ \} \) is non-Abelian.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Giving a reason, state whether or not \(\{ G,{\text{ }}{ \times _7}\} \) and \(\{ K,{\text{ }} \circ \} \) are isomorphic.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">The set of all permutations of the elements \(1,{\text{ }}2,{\text{ }} \ldots 10\) </span>is denoted by \(H\) and the binary operation \( \circ \) represents the composition of permutations.</p>
<p class="p1">The permutation \(p = (1{\text{ }}2{\text{ }}3{\text{ }}4{\text{ }}5{\text{ }}6)(7{\text{ }}8{\text{ }}9{\text{ }}10)\) <span class="s1">generates the subgroup \(\{ G,{\text{ }} \circ \} \) of the group \(\{ H,{\text{ }} \circ \} \)</span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the order of \(\{ G,{\text{ }} \circ \} \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State the identity element in \(\{ G,{\text{ }} \circ \} \).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(p \circ p\);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>the inverse of \(p \circ p\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the maximum possible order of an element in \(\{ H,{\text{ }} \circ \} \).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Give an example of an element with this order.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The relation <em>R </em>is defined on the set \(\mathbb{N}\) such that for \(a{\text{ }},{\text{ }}b \in \mathbb{N}{\text{ }},{\text{ }}aRb\) if and only if \({a^3} \equiv {b^3}(\bmod 7)\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that <em>R </em>is an equivalence relation.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equivalence class containing 0.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="line-height: normal; font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">Denote the equivalence class containing <em>n</em> by C<em><sub>n</sub></em> .</span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">List the first six elements of \({C_1}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="line-height: normal; font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">Denote the equivalence class containing <em>n</em> by C<em><sub>n</sub></em> .</span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that \({C_n} = {C_{n + 7}}\) for all \(n \in \mathbb{N}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The function \(f\) is defined by \(f:{\mathbb{R}^ + } \times {\mathbb{R}^ + } \to {\mathbb{R}^ + } \times {\mathbb{R}^ + }\) where \(f(x,{\text{ }}y) = \left( {\sqrt {xy} ,{\text{ }}\frac{x}{y}} \right)\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that \(f\) is an injection.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Prove that \(f\) is a surjection.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence, or otherwise, write down the inverse function \({f^{ - 1}}\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The relation \(R=\) is defined on \({\mathbb{Z}^ + }\) such that \(aRb\) if and only if \({b^n} - {a^n} \equiv 0(\bmod p)\) where \(n,{\text{ }}p\) <span class="s1">are fixed positive integers greater than 1.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(R\) is an equivalence relation.</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(n = 2\) and \(p = 7\), determine the first four members of each of the four equivalence classes of \(R\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>c </em>be a positive, real constant. Let <em>G </em>be the set \(\{ \left. {x \in \mathbb{R}} \right| - c < x < c\} \) . The binary </span><span style="font-family: 'times new roman', times; font-size: medium;">operation \( * \) is defined on the set <em>G </em>by \(x * y = \frac{{x + y}}{{1 + \frac{{xy}}{{{c^2}}}}}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 44.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Simplify \(\frac{c}{2} * \frac{{3c}}{4}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">State the identity element for <em>G </em>under \( * \).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">For \(x \in G\) find an expression for \({x^{ - 1}}\) (the inverse of <em>x </em>under \( * \)).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the binary operation \( * \) is commutative on <em>G </em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the binary operation \( * \) is associative on <em>G </em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) If \(x,{\text{ }}y \in G\) explain why \((c - x)(c - y) > 0\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence show that \(x + y < c + \frac{{xy}}{c}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that <em>G </em>is closed under \( * \).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Explain why \(\{ G, * \} \) is an Abelian group.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Below are the graphs of the two functions \(F:P \to Q{\text{ and }}g:A \to B\) .</span></p>
<p><img src="data:image/png;base64,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" alt></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">Determine, with reference to features of the graphs, whether the functions are injective and/or surjective.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given two functions \(h:X \to Y{\text{ and }}k:Y \to Z\) . </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) if both <em>h</em> and <em>k</em> are injective then so is the composite function \(k \circ h\) ;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) if both <em>h</em> and <em>k</em> are surjective then so is the composite function \(k \circ h\) .</span></p>
<div> </div>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the group \(\{ G,{\text{ }}{ \times _{18}}\} \) defined on the set \(\{ 1,{\text{ }}5,{\text{ }}7,{\text{ }}11,{\text{ }}13,{\text{ }}17\} \) where \({ \times _{18}}\) denotes multiplication modulo 18. The group \(\{ G,{\text{ }}{ \times _{18}}\} \) is shown in the following Cayley table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-10_om_09.24.56.png" alt="N17/5/MATHL/HP3/ENG/TZ0/SG/01"></p>
</div>
<div class="specification">
<p>The subgroup of \(\{ G,{\text{ }}{ \times _{18}}\} \) of order two is denoted by \(\{ K,{\text{ }}{ \times _{18}}\} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the order of elements 5, 7 and 17 in \(\{ G,{\text{ }}{ \times _{18}}\} \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State whether or not \(\{ G,{\text{ }}{ \times _{18}}\} \) is cyclic, justifying your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the elements in set \(K\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the left cosets of \(K\) in \(\{ G,{\text{ }}{ \times _{18}}\} \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A group \(\{ D,{\text{ }}{ \times _3}\} \) <span class="s1">is defined so that \(D = \{ 1,{\text{ }}2\} \) </span>and \({ \times _3}\) is multiplication modulo \(3\)<span class="s1">.</span></p>
<p class="p2">A function \(f:\mathbb{Z} \to D\) is defined as \(f:x \mapsto \left\{ {\begin{array}{*{20}{c}} {1,{\text{ }}x{\text{ is even}}} \\ {2,{\text{ }}x{\text{ is odd}}} \end{array}} \right.\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that the function \(f\) is a homomorphism from the group \(\{ \mathbb{Z},{\text{ }} + \} {\text{ to }}\{ D,{\text{ }}{ \times _3}\} \).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the kernel of \(f\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that \(\{ {\text{Ker}}(f),{\text{ }} + \} \) is a subgroup of \(\{ \mathbb{Z},{\text{ }} + \} \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Associativity and commutativity are two of the five conditions for a set <em>S </em>with the binary operation \( * \) to be an Abelian group; state the other three conditions.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The Cayley table for the binary operation \( \odot \) defined on the set <em>T </em>= {<em>p</em>, <em>q</em>, <em>r</em>, <em>s</em>, <em>t</em>} is given below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that exactly three of the conditions for {<em>T </em>, \( \odot \)} to be an Abelian group are satisfied, but that neither associativity nor commutativity are satisfied.</span><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the proper subsets of <em>T </em>that are groups of order 2, and comment on your result in the context of Lagrange’s theorem.</span><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Find the solutions of the equation \((p \odot x) \odot x = x \odot p\)<em> </em>.</span></p>
<div class="marks">[15]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The binary operation \( * \) is defined by</p>
<p style="text-align: center;">\(a * b = a + b - 3\) for \(a,{\text{ }}b \in \mathbb{Z}\).</p>
</div>
<div class="specification">
<p>The binary operation \( \circ \) is defined by</p>
<p style="text-align: center;">\(a \circ b = a + b + 3\) for \(a,{\text{ }}b \in \mathbb{Z}\).</p>
<p>Consider the group \(\{ \mathbb{Z},{\text{ }} \circ {\text{\} }}\) and the bijection \(f:\mathbb{Z} \to \mathbb{Z}\) given by \(f(a) = a - 6\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\{ \mathbb{Z},{\text{ }} * \} \) is an Abelian group.</p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that there is no element of order 2.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a proper subgroup of \(\{ \mathbb{Z},{\text{ }} * \} \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the groups \(\{ \mathbb{Z},{\text{ }} * \} \) and \(\{ \mathbb{Z},{\text{ }} \circ \} \) are isomorphic.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The set \(S\) is defined as the set of real numbers greater than 1.</p>
<p>The binary operation \( * \) is defined on \(S\) by \(x * y = (x - 1)(y - 1) + 1\) for all \(x,{\text{ }}y \in S\).</p>
</div>
<div class="specification">
<p>Let \(a \in S\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(x * y \in S\) for all \(x,{\text{ }}y \in S\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the operation \( * \) on the set \(S\) is commutative.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the operation \( * \) on the set \(S\) is associative.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that 2 is the identity element.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that each element \(a \in S\) has an inverse.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The elements of sets <em>P </em>and <em>Q </em>are taken from the universal set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. <em>P </em>= {1, 2, 3} and <em>Q </em>= {2, 4, 6, 8, 10}.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(R = (P \cap Q')'\) , list the elements of <em>R </em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">For a set <em>S </em>, let \({S^ * }\)<span style="font: 7.0px Times;"> </span>denote the set of all subsets of <em>S </em>,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) find \({P^ * }\)<span style="font: 7.0px Times;"> </span>;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) find \(n({R^ * })\)<span style="font: 12.5px Times;"> </span>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The relation \(R\) is defined such that \(aRb\) if and only if \({4^a} - {4^b}\) is divisible by 7, where \(a,{\text{ }}b \in {\mathbb{Z}^ + }\).</p>
</div>
<div class="specification">
<p>The equivalence relation \(S\) is defined such that \(cSd\) if and only if \({4^c} - {4^d}\) is divisible by 6, where \(c,{\text{ }}d \in {\mathbb{Z}^ + }\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(R\) is an equivalence relation.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the equivalence classes of \(R\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the number of equivalence classes of \(S\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">An Abelian group, \(\{ G,{\text{ }} * \} \), has <span class="s1">12 </span>different elements which are of the form \({a^i} * {b^j}\) where \(i \in \{ 1,{\text{ }}2,{\text{ }}3,{\text{ }}4\} \) and \(j \in \{ 1,{\text{ }}2,{\text{ }}3\} \). The elements \(a\) and \(b\) satisfy \({a^4} = e\) and \({b^3} = e\) where \(e\) is the identity.</p>
</div>
<div class="specification">
<p class="p1">Let \(\{ H,{\text{ }} * \} \) be the proper subgroup of \(\{ G,{\text{ }} * \} \) having the maximum possible order.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">State the possible orders of an element of \(\{ G,{\text{ }} * \} \) </span>and for each order give an example of an element of that order.</p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>State a generator for \(\{ H,{\text{ }} * \} \).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Write down the elements of \(\{ H,{\text{ }} * \} \).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Write down the elements of the coset of \(H\) containing \(a\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The relation \(R\) is defined such that \(xRy\) if and only if \(\left| x \right| + \left| y \right| = \left| {x + y} \right|\) for \(x\), \(y\), \(y \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(R\) is reflexive.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(R\) is symmetric.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show, by means of an example, that \(R\) is not transitive.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The group <em>G </em>has a unique element, <em>h </em>, of order 2.</span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that \(gh{g^{ - 1}}\) has order 2 for all \(g \in G\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Deduce that <em>gh </em>= <em>hg </em>for all \(g \in G\).</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Two functions, <em>F</em> and <em>G</em> , are defined on \(A = \mathbb{R}\backslash \{ 0,{\text{ }}1\} \) by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[F(x) = \frac{1}{x},{\text{ }}G(x) = 1 - x,{\text{ for all }}x \in A.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that under the operation of composition of functions each function is its own inverse.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <em>F</em> and <em>G</em> together with four other functions form a closed set under the operation of composition of functions.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find these four functions.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The binary operation \( * \) is defined for \(x,{\text{ }}y \in S = \{ 0,{\text{ }}1,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }}5,{\text{ }}6\} \) by</p>
<p class="p1">\[x * y = ({x^3}y - xy)\bmod 7.\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the element \(e\) such that \(e * y = y\), for all \(y \in S\)<span class="s1">.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the least solution of \(x * x = e\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Deduce that \((S,{\text{ }} * )\) is not a group.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine whether or not \(e\) is an identity element.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">All of the relations in this question are defined on \(\mathbb{Z}\backslash \{ 0\} \).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Decide, giving a proof or a counter-example, whether \(xRy \Leftrightarrow x + y > 7\) is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) reflexive;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) symmetric;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) transitive.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Decide, giving a proof or a counter-example, whether \(xRy \Leftrightarrow - 2 < x - y < 2\) is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) reflexive;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) symmetric;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) transitive.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Decide, giving a proof or a counter-example, whether \(xRy \Leftrightarrow xy > 0\) is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) reflexive;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) symmetric;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) transitive.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Decide, giving a proof or a counter-example, whether \(xRy \Leftrightarrow \frac{x}{y} \in \mathbb{Z}\) is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) reflexive;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) symmetric;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) transitive.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">One of the relations from parts (a), (b), (c) and (d) is an equivalence relation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">For this relation, state what the equivalence classes are.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 39.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(A = \left\{ {a,{\text{ }}b} \right\}\).</span></p>
</div>
<div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Let the set of all these subsets be denoted by \(P(A)\) . The binary operation symmetric difference, \(\Delta\) , is defined on \(P(A)\) by \(X\Delta Y = (X\backslash Y) \cup (Y\backslash X)\) where \(X\) , \(Y \in P(A)\).</span></p>
</div>
<div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Let \({\mathbb{Z}_4} = \left\{ {0,{\text{ }}1,{\text{ }}2,{\text{ }}3} \right\}\) and \({ + _4}\) denote addition modulo \(4\).</span></p>
</div>
<div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Let \(S\) be any non-empty set. Let \(P(S)\) be the set of all subsets of \(S\) . For the following parts, you are allowed to assume that \(\Delta\), \( \cup \) and \( \cap \) are associative.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down all four subsets of <em>A </em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Construct the Cayley table for \(P(A)\) under \(\Delta \) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that </span><span style="font-family: 'times new roman', times; font-size: medium;">\(\left\{ {P(A),{\text{ }}\Delta } \right\}\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> is a group. You are allowed to assume that \(\Delta \) is associative.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Is \(\{ P(A){\text{, }}\Delta \} \) isomorphic to \(\{ {\mathbb{Z}_4},{\text{ }}{ + _4}\} \) ? Justify your answer.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State the identity element for \(\{ P(S){\text{, }}\Delta \} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Write down \({X^{ - 1}}\) for \(X \in P(S)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Hence prove that \(\{ P(S){\text{, }}\Delta \} \) is a group.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Explain why \(\{ P(S){\text{, }} \cup \} \) is not a group.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Explain why \(\{ P(S){\text{, }} \cap \} \) is not a group.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The binary operation \( * \) is defined on the set \(T = \{ 0,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }}5,{\text{ }}6\} \) by \(a * b = (a + b - ab)(\bmod 7),{\text{ }}a,{\text{ }}b \in T\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Copy and complete the following Cayley table for \(\{ T,{\text{ }} * \} \).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-21_om_14.49.34.png" alt></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that \(\{ T,{\text{ }} * \} \) forms an Abelian group.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the order of each element in \(T\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(\{ H,{\text{ }} * \} \) is the subgroup of \(\{ T,{\text{ }} * \} \) of order \(2\)<span class="s1">, partition \(T\) into the left cosets with respect to \(H\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The function \(f:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}\) is defined by \(f(x,{\text{ }}y) = (2{x^3} + {y^3},{\text{ }}{x^3} + 2{y^3})\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(f\) is a bijection.</p>
<div class="marks">[12]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence write down the inverse function \({f^{ - 1}}(x,{\text{ }}y)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">Let \(A\) </span>be the set \(\{ x|x \in \mathbb{R},{\text{ }}x \ne 0\} \). Let \(B\) be the set \(\{ x|x \in ] - 1,{\text{ }} + 1[,{\text{ }}x \ne 0\} \).</p>
<p class="p1">A function \(f:A \to B\) is defined by \(f(x) = \frac{2}{\pi }\arctan (x)\).</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">Let \(D\) </span>be the set \(\{ x|x \in \mathbb{R},{\text{ }}x > 0\} \).</p>
<p class="p1">A function \(g:\mathbb{R} \to D\) is defined by \(g(x) = {{\text{e}}^x}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Sketch the graph of \(y = f(x)\) and hence justify whether or not \(f\) <span class="s1">is a bijection.</span></p>
<p class="p2"><span class="s2">(ii) <span class="Apple-converted-space"> </span>Show that \(A\) </span>is a group under the binary operation of multiplication.</p>
<p class="p2"><span class="s2">(iii) <span class="Apple-converted-space"> </span>Give a reason why \(B\) </span>is not a group under the binary operation of multiplication.</p>
<p class="p1">(iv) <span class="Apple-converted-space"> </span>Find an example to show that \(f(a \times b) = f(a) \times f(b)\) is not satisfied for all \(a,{\text{ }}b \in A\).</p>
<div class="marks">[13]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Sketch the graph of \(y = g(x)\) and hence justify whether or not \(g\) <span class="s1">is a bijection.</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Show that \(g(a + b) = g(a) \times g(b)\) for all \(a,{\text{ }}b \in \mathbb{R}\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Given that \(\{ \mathbb{R},{\text{ }} + \} \) and \(\{ D,{\text{ }} \times \} \) are both groups, explain whether or not they are isomorphic.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \(f:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}\) defined by \(f(x,{\text{ }}y) = (2x + y,{\text{ }}x - y)\) is a bijection.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the inverse of <em>f</em> .</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The binary operation \( * \) is defined on \(\mathbb{R}\) as follows. For any elements <em>a</em> , \(b \in \mathbb{R}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[a * b = a + b + 1.\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that \( * \) is commutative.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the identity element.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Find the inverse of the element <em>a</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The binary operation \( \cdot \) is defined on \(\mathbb{R}\) as follows. For any elements <em>a</em> , \(b \in \mathbb{R}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a \cdot b = 3ab\) . The set <em>S</em> is the set of all ordered pairs \((x,{\text{ }}y)\) of real numbers and the binary operation \( \odot \) is defined on the set <em>S</em> as</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(({x_1},{\text{ }}{y_1}) \odot ({x_2},{\text{ }}{y_2}) = ({x_1} * {x_2},{\text{ }}{y_1} \cdot {y_2}){\text{ }}.\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine whether or not \( \odot \) is associative.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Draw the Cayley table for the set of integers <em>G</em> = {0, 1, 2, 3, 4, 5} under addition modulo 6, \({ + _6}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that \(\{ G,{\text{ }}{ + _6}\} \) is a group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find the order of each element.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Show that \(\{ G,{\text{ }}{ + _6}\} \) is cyclic and state its generators.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Find a subgroup with three elements. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(f) Find the other proper subgroups of \(\{ G,{\text{ }}{ + _6}\} \).</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f:[0,{\text{ }}\infty [ \to [0,{\text{ }}\infty [\) is defined by \(f(x) = 2{{\text{e}}^x} + {{\text{e}}^{ - x}} - 3\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find \(f'(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that <em>f</em> is a bijection.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find an expression for \({f^{ - 1}}(x)\) .</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The universal set contains all the positive integers less than 30. The set <em>A</em> contains all prime numbers less than 30 and the set <em>B</em> contains all positive integers of the form \(3 + 5n{\text{ }}(n \in \mathbb{N})\) that are less than 30. Determine the elements of</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>A</em> \ <em>B</em> ;</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 37.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(A\Delta B\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The binary operation \( * \) is defined on \(\mathbb{N}\) by \(a * b = 1 + ab\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine whether or not \( * \)</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">is closed;</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">is commutative;</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">is associative;</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">has an identity element.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A group with the binary operation of multiplication modulo 15 is shown in the following Cayley table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-13_om_06.41.36.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the values represented by each of the letters in the table.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the order of each of the elements of the group.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the three sets that form subgroups of order 2.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the three sets that form subgroups of order 4.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that <em>p</em> , <em>q</em> and <em>r</em> are elements of a group, prove the left-cancellation rule, <em>i.e.</em> \(pq = pr \Rightarrow q = r\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Your solution should indicate which group axiom is used at each stage of the proof.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the group <em>G</em> , of order 4, which has distinct elements <em>a</em> , <em>b</em> and <em>c</em> and the identity element <em>e</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Giving a reason in each case, explain why <em>ab</em> cannot equal <em>a</em> or <em>b</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Given that <em>c</em> is self inverse, determine the two possible Cayley tables for <em>G</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Determine which one of the groups defined by your two Cayley tables is isomorphic to the group defined by the set {1, −1, i, −i} under multiplication of complex numbers. Your solution should include a correspondence between <em>a</em>, <em>b</em>, <em>c</em>, <em>e</em> and 1, −1, i, −i .</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A binary operation is defined on {−1, 0, 1} by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[A \odot B = \left\{ {\begin{array}{*{20}{c}}<br> { - 1,}&{{\text{if }}\left| A \right| < \left| B \right|} \\ <br> {0,}&{{\text{if }}\left| A \right| = \left| B \right|} \\ <br> {1,}&{{\text{if }}\left| A \right| > \left| B \right|{\text{.}}} <br>\end{array}} \right.\]<br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Construct the Cayley table for this operation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Giving reasons, determine whether the operation is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) closed;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) commutative;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) associative.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Sets <em>X </em>and <em>Y </em>are defined by \({\text{ }}X = \left] {0,{\text{ }}1} \right[;{\text{ }}Y = \{ 0,{\text{ }}1,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }}5\} \).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Sketch the set \(X \times Y\) in the Cartesian plane.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Sketch the set \(Y \times X\) in the Cartesian plane.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) State \((X \times Y) \cap (Y \times X)\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the function \(f:X \times Y \to \mathbb{R}\) defined by \(f(x,{\text{ }}y) = x + y\) and the function \(g:X \times Y \to \mathbb{R}\) defined by \(g(x,{\text{ }}y) = xy\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the range of the function <em>f</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the range of the function <em>g</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Show that \(f\) is an injection.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) Find \({f^{ - 1}}(\pi )\), expressing your answer in exact form.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(v) Find all solutions to \(g(x,{\text{ }}y) = \frac{1}{2}\).</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f:G \to H\) be a homomorphism of finite groups.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that \(f({e_G}) = {e_H}\), where \({e_G}\) is the identity element in \(G\) and \({e_H}\) is the identity</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">element in \(H\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Prove that the kernel of \(f,{\text{ }}K = {\text{Ker}}(f)\), is closed under the group operation.</p>
<p>(ii) Deduce that \(K\) is a subgroup of \(G\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Prove that \(gk{g^{ - 1}} \in K\) for all \(g \in G,{\text{ }}k \in K\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Deduce that each left coset of <em>K </em>in <em>G </em>is also a right coset.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(X\) and \(Y\) be sets. The functions \(f:X \to Y\) and \(g:Y \to X\) are such that \(g \circ f\) is the identity function on \(X\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that: </p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(f\) is an injection,</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(g\) is a surjection.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(X = {\mathbb{R}^ + } \cup \{ 0\} \) and \(Y = \mathbb{R}\), choose a suitable pair of functions \(f\) and \(g\) to show that \(g\) is not necessarily a bijection.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \((H,{\text{ }} * {\text{)}}\) be a subgroup of the group \((G,{\text{ }} * {\text{)}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the relation \(R\) defined in \(G\) by \(xRy\) if and only if \({y^{ - 1}} * x \in H\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \(R\) is an equivalence relation on \(G\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Determine the equivalence class containing the identity element.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the set \(A\) consisting of all the permutations of the integers \(1,2,3,4,5\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Two members of \(A\) are given by \(p = (1{\text{ }}2{\text{ }}5)\) and \(q = (1{\text{ }}3)(2{\text{ }}5)\).</p>
<p class="p1">Find the single permutation which is equivalent to \(q \circ p\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State a permutation belonging to<em> </em>\(A\) of order</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(4\);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(6\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(P = \) {all permutations in \(A\) where exactly two integers change position},</p>
<p>and \(Q = \) {all permutations in \(A\) where the integer \(1\) changes position}.</p>
<p>(i) List all the elements in \(P \cap Q\).</p>
<p>(ii) Find \(n(P \cap Q')\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Given the sets \(A\) and \(B\), use the properties of sets to prove that \(A \cup (B' \cup A)' = A \cup B\), justifying each step of the proof.</p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Write down why the table below is a Latin square.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\begin{gathered}<br> \begin{array}{*{20}{c}}<br> {}&d&e&b&a&c <br>\end{array} \\<br> \begin{array}{*{20}{c}}<br> d \\ <br> e \\ <br> b \\ <br> a \\ <br> c <br>\end{array}\left[ {\begin{array}{*{20}{c}}<br> c&d&e&b&a \\ <br> d&e&b&a&c \\ <br> a&b&d&c&e \\ <br> b&a&c&e&d \\ <br> e&c&a&d&b <br>\end{array}} \right] \\ <br>\end{gathered} \]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Use Lagrange’s theorem to show that the table is not a group table.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(p = {2^k} + 1,{\text{ }}k \in {\mathbb{Z}^ + }\) be a prime number and let <em>G </em>be the group of integers 1, 2, ..., <em>p </em>− 1 under multiplication defined modulo <em>p</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">By first considering the elements \({2^1},{\text{ }}{2^2},{\text{ ..., }}{2^k}\) and then the elements \({2^{k + 1}},{\text{ }}{2^{k + 2}},{\text{ …,}}\) show that the order of the element 2 is 2<em>k</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Deduce that \(k = {2^n}{\text{ for }}n \in \mathbb{N}\) .</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that \((A \cap B)\backslash (A \cap C) = A \cap (B\backslash C)\) where <em>A</em>, <em>B</em> and <em>C</em> are three subsets of the universal set <em>U</em>.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(\{ G,{\text{ }} * \} \) be a finite group and let <em>H</em> be a non-empty subset of <em>G</em> . Prove that \(\{ H,{\text{ }} * \} \) is a group if <em>H</em> is closed under \( * \).</span></p>
</div>
<br><hr><br><div class="specification">
<p>The group \(\{ G,{\rm{ }} * {\rm{\} }}\) has identity \({e_G}\) and the group \(\{ H,{\text{ }} \circ \} \) has identity \({e_H}\). A homomorphism \(f\) is such that \(f:G \to H\). It is given that \(f({e_G}) = {e_H}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that for all \(a \in G,{\text{ }}f({a^{ - 1}}) = {\left( {f(a)} \right)^{ - 1}}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(\{ H,{\text{ }} \circ \} \) be the cyclic group of order seven, and let \(p\) be a generator.</p>
<p class="p1">Let \(x \in G\) such that \(f(x) = {p^{\text{2}}}\).</p>
<p class="p1">Find \(f({x^{ - 1}})\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(f(x * y) = p\), find \(f(y)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>H</em> and <em>K</em> are subgroups of a group <em>G</em>. By considering the four group axioms, prove that \(H \cap K\) is also a subgroup of <em>G</em>.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that set difference is not associative.</span></p>
</div>
<br><hr><br><div class="specification">
<p>Define \(f:\mathbb{R}\backslash \{ 0.5\} \to \mathbb{R}\) by \(f(x) = \frac{{4x + 1}}{{2x - 1}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that <em>\(f\) </em>is an injection.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that <em>\(f\) </em>is not a surjection.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the sets</p>
<p class="p1">\[G = \left\{ {\frac{n}{{{6^i}}}|n \in \mathbb{Z},{\text{ }}i \in \mathbb{N}} \right\},{\text{ }}H = \left\{ {\frac{m}{{{3^j}}}|m \in \mathbb{Z},{\text{ }}j \in \mathbb{N}} \right\}.\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \((G,{\text{ }} + )\) forms a group where \( + \) denotes addition on \(\mathbb{Q}\). Associativity may be assumed.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Assuming that \((H,{\text{ }} + )\) forms a group, show that it is a proper subgroup of \((G,{\text{ }} + )\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The mapping \(\phi :G \to G\) is given by \(\phi (g) = g + g\), for \(g \in G\).</p>
<p class="p1">Prove that \(\phi \) is an isomorphism.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the following functions</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(f:\left] {1,{\text{ }} + \infty } \right[ \to {\mathbb{R}^ + }\) where \(f(x) = (x - 1)(x + 2)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(g:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}\) where \(g(x,{\text{ }}y) = \left( {\sin (x + y),{\text{ }}x + y} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(h:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}\) where \(h(x,{\text{ }}y) = (x + 3y,{\text{ }}2x + y)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \(f\) is bijective.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Determine, with reasons, whether</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(g\) is injective;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(g\)<em> </em>is surjective.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find an expression for \({h^{ - 1}}(x,{\text{ }}y)\) and hence justify that \(h\)<em> </em>has an inverse function.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f:\mathbb{Z} \times \mathbb{R} \to \mathbb{R},{\text{ }}f(m,{\text{ }}x) = {( - 1)^m}x\). Determine whether <em>f</em> is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) surjective;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) injective.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>P</em> is the set of all polynomials such that \(P = \left\{ {\sum\limits_{i = 0}^n {{a_i}{x^i}|n \in \mathbb{N}} } \right\}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(g:P \to P,{\text{ }}g(p) = xp\). Determine whether <em>g</em> is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) surjective;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) injective.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(h:\mathbb{Z} \to {\mathbb{Z}^ + }\), \(h(x) = \left\{ {\begin{array}{*{20}{c}}<br> {2x,}&{x > 0} \\ <br> {1 - 2x,}&{x \leqslant 0} <br>\end{array}} \right\}\). Determine whether <em>h</em> is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) surjective;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) injective.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The function \(f:\mathbb{R} \to \mathbb{R}\) is defined as \(f:x \to \left\{ {\begin{array}{*{20}{c}} {1,{\text{ }}x \ge 0} \\ { - 1,{\text{ }}x < 0} \end{array}} \right.\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that \(f\) is</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>not injective;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>not surjective.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The relation \(R\) is defined for \(a,{\text{ }}b \in \mathbb{R}\) so that \(aRb\) if and only if \(f(a) \times f(b) = 1\)<span class="s1">.</span></p>
<p class="p1">Show that \(R\) is an equivalence relation.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The relation \(R\) is defined for \(a,{\text{ }}b \in \mathbb{R}\) so that \(aRb\) if and only if \(f(a) \times f(b) = 1\)<span class="s1">.</span></p>
<p class="p1">State the equivalence classes of \(R\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f:{\mathbb{R}^ + } \times {\mathbb{R}^ + } \to {\mathbb{R}^ + } \times {\mathbb{R}^ + }\) is defined by \(f(x,{\text{ }}y) = \left( {x{y^2},\frac{x}{y}} \right)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that <em>f</em> is a bijection.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>G</em> be a finite cyclic group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Prove that <em>G</em> is Abelian.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Given that <em>a</em> is a generator of <em>G</em>, show that \({a^{ - 1}}\) is also a generator.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Show that if the order of <em>G</em> is five, then all elements of <em>G</em>, apart from the identity, are generators of <em>G</em>.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f:\mathbb{R} \to \mathbb{R}\) is defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = 2{{\text{e}}^x} - {{\text{e}}^{ - x}}.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that <em>f</em> is a bijection.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find an expression for \({f^{ - 1}}(x)\).</span></p>
</div>
<br><hr><br><div class="specification">
<p>The set of all permutations of the list of the integers 1, 2, 3 4 is a group, <em>S</em><sub>4</sub>, under the operation of function composition.</p>
</div>
<div class="specification">
<p>In the group <em>S</em><sub>4</sub> let \({p_1} = \left( \begin{gathered}<br> \begin{array}{*{20}{c}}<br> 1&2&3&4 <br>\end{array} \hfill \\<br> \begin{array}{*{20}{c}}<br> 2&3&1&4 <br>\end{array} \hfill \\ <br>\end{gathered} \right)\) and \({p_2} = \left( \begin{gathered}<br> \begin{array}{*{20}{c}}<br> 1&2&3&4 <br>\end{array} \hfill \\<br> \begin{array}{*{20}{c}}<br> 2&1&3&4 <br>\end{array} \hfill \\ <br>\end{gathered} \right)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the order of <em>S</em><sub>4</sub>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the proper subgroup <em>H</em> of order 6 containing \({p_1}\), \({p_2}\) and their compositions. Express each element of <em>H</em> in cycle form.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(f{\text{:}}\,{S_4} \to {S_4}\) be defined by \(f\left( p \right) = p \circ p\) for \(p \in {S_4}\).</p>
<p>Using \({p_1}\) and \({p_2}\), explain why \(f\) is not a homomorphism.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The relation <em>aRb</em> is defined on {1, 2, 3, 4, 5, 6, 7, 8, 9} if and only if <em>ab</em> is the square of a positive integer. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that <em>R</em> is an equivalence relation. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the equivalence classes of <em>R</em> that contain more than one element.</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given the group \((G,{\text{ }} * )\), a subgroup \((H,{\text{ }} * )\) and \(a,{\text{ }}b \in G\), we define \(a \sim b\) if and only if \(a{b^{ - 1}} \in H\). Show that \( \sim \) is an equivalence relation.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Set \(S = \{ {x_0},{\text{ }}{x_1},{\text{ }}{x_2},{\text{ }}{x_3},{\text{ }}{x_4},{\text{ }}{x_5}\} \) and a binary operation \( \circ \) on <em>S</em> is defined as \({x_i} \circ {x_j} = {x_k}\), where \(i + j \equiv k(\bmod 6)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) Construct the Cayley table for \(\{ S,{\text{ }} \circ \} \) and hence show that it is a group.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 28px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Show that \(\{ S,{\text{ }} \circ \} \) is cyclic.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Let \(\{ G,{\text{ }} * \} \) be an Abelian group of order 6. The element \(a \in {\text{G}}\) has order 2 and the element \(b \in {\text{G}}\) has order 3.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 28px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) Write down the six elements of \(\{ G,{\text{ }} * \} \).</span></p>
<p style="margin: 0px 0px 0px 30px; font: 28px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Find the order of \({\text{a}} * b\) and hence show that \(\{ G,{\text{ }} * \} \) is isomorphic to \(\{ S,{\text{ }} \circ \} \).</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f </em>is defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \frac{{1 - {{\text{e}}^{ - x}}}}{{1 + {{\text{e}}^{ - x}}}},{\text{ }}x \in \mathbb{R}{\text{ .}}\]</span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the range of <em>f </em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Prove that <em>f </em>is an injection.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Taking the codomain of <em>f </em>to be equal to the range of <em>f </em>, find an expression for \({f^{ - 1}}(x)\) .</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The relation <em>R</em> is defined on \(\mathbb{Z} \times \mathbb{Z}\) such that \((a,{\text{ }}b)R(c,{\text{ }}d)\) if and only if <em>a</em> − <em>c</em> is divisible by 3 and <em>b</em> − <em>d</em> is divisible by 2.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Prove that <em>R</em> is an equivalence relation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the equivalence class for (2, 1) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Write down the five remaining equivalence classes.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The binary operation \( * \) is defined on the set <em>S</em> = {0, 1, 2, 3} by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[a * b = a + 2b + ab(\bmod 4){\text{ .}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) Construct the Cayley table.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Write down, with a reason, whether or not your table is a Latin square.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Write down, with a reason, whether or not \( * \) is commutative.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Determine whether or not \( * \) is associative, justifying your answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find all solutions to the equation \(x * 1 = 2 * x\) , for \(x \in S\) .</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the six roots of the equation \({z^6} - 1 = 0\) , giving your answers in the form \(r\,{\text{cis}}\,\theta {\text{, }}r \in {\mathbb{R}^ + }{\text{, }}0 \leqslant \theta < 2\pi \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Show that these six roots form a group <em>G </em>under multiplication of complex numbers.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Show that <em>G </em>is cyclic and find all the generators.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (iii) Give an example of another group that is isomorphic to <em>G</em>, stating clearly the corresponding elements in the two groups.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The relation <em>R</em> is defined on \({\mathbb{Z}^ + }\) by <em>aRb</em> if and only if <em>ab</em> is even. Show that only one of the conditions for <em>R</em> to be an equivalence relation is satisfied.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The relation <em>S</em> is defined on \({\mathbb{Z}^ + }\) by <em>aSb</em> if and only if \({a^2} \equiv {b^2}(\bmod 6)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that <em>S</em> is an equivalence relation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) For each equivalence class, give the four smallest members.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The groups \(\{ K,{\text{ }} * \} \) and \(\{ H,{\text{ }} \odot \} \) are defined by the following Cayley tables.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">G </span><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">H </span><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">By considering a suitable function from <em>G</em> to <em>H</em> , show that a surjective homomorphism exists between these two groups. State the kernel of this homomorphism.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Three functions mapping \(\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}\) are defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{f_1}(m,{\text{ }}n) = m - n + 4;\,\,\,{f_2}(m,{\text{ }}n) = \left| m \right|;\,\,\,{f_3}(m,{\text{ }}n) = {m^2} - {n^2}.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Two functions mapping \(\mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}\) are defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{g_1}(k) = (2k,{\text{ }}k);\,\,\,{g_2}(k) = \left( {k,{\text{ }}\left| k \right|} \right).\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the range of</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({f_1} \circ {g_1}\) ;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({f_3} \circ {g_2}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find all the solutions of \({f_1} \circ {g_2}(k) = {f_2} \circ {g_1}(k)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find all the solutions of \({f_3}(m,{\text{ }}n) = p\) in each of the cases <em>p</em> =1 and <em>p</em> = 2 .</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">\(\{ G,{\text{ }} * \} \) is a group with identity element \(e\). Let \(a,{\text{ }}b \in G\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State Lagrange’s theorem.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Verify that the inverse of \(a * {b^{ - 1}}\) is equal to \(b * {a^{ - 1}}\).</p>
<p class="p1"> </p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(\{ H,{\rm{ }} * {\rm{\} }}\) be a subgroup of \(\{ G,{\rm{ }} * {\rm{\} }}\). Let \(R\) be a relation defined on \(G\) by</p>
<p class="p1">\[aRb \Leftrightarrow a * {b^{ - 1}} \in H.\]</p>
<p class="p1">Prove that \(R\) is an equivalence relation, indicating clearly whenever you are using one of the four properties required of a group.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(\{ H,{\rm{ }} * {\rm{\} }}\) be a subgroup of \(\{ G,{\rm{ }} * {\rm{\} }}\) .Let \(R\) be a relation defined on \(G\) by</p>
<p class="p1">\[aRb \Leftrightarrow a * {b^{ - 1}} \in H.\]</p>
<p class="p1">Show that \(aRb \Leftrightarrow a \in Hb\), where \(Hb\) is the right coset of \(H\) containing \(b\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(\{ H,{\rm{ }} * {\rm{\} }}\) be a subgroup of \(\{ G,{\rm{ }} * {\rm{\} }}\) .Let \(R\) be a relation defined on \(G\) by</p>
<p class="p1">\[aRb \Leftrightarrow a * {b^{ - 1}} \in H.\]</p>
<p class="p1">It is given that the number of elements in any right coset of \(H\) is equal to the order of \(H\).</p>
<p class="p1">Explain how this fact together with parts (c) and (d) prove Lagrange’s theorem.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Given a set \(U\), and two of its subsets \(A\) and \(B\), prove that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[(A\backslash B) \cup (B\backslash A) = (A \cup B)\backslash (A \cap B),{\text{ where }}A\backslash B = A \cap B'.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Let \(S = \{ A,{\text{ }}B,{\text{ }}C,{\text{ }}D\} \) where \(A = \emptyset ,{\text{ }}B = \{ 0\} ,{\text{ }}C = \{ 0,{\text{ }}1\} \) and \(D = \{ {\text{0, 1, 2}}\} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">State, with reasons, whether or not each of the following statements is true.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) The operation \ is closed in \(S\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) The operation \( \cap \) has an identity element in \(S\) but not all elements have an inverse.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Given \(Y \in S\), the equation \(X \cup Y = Y\) always has a unique solution for \(X\) in \(S\).</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The relation \(R\) is defined on \(\mathbb{Z}\) by \(xRy\) if and only if \({x^2}y \equiv y\bmod 6\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the product of three consecutive integers is divisible by \(6\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence prove that \(R\) is reflexive.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the set of all \(y\) for which \(5Ry\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the set of all \(y\) for which \(3Ry\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Using your answers for (c) and (d) show that \(R\) is not symmetric.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine, giving reasons, which of the following sets form groups under the operations given below. Where appropriate you may assume that multiplication is associative.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\mathbb{Z}\) under subtraction.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) The set of complex numbers of modulus 1 under multiplication.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) The set {1, 2, 4, 6, 8} under multiplication modulo 10.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) The set of rational numbers of the form</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\frac{{3m + 1}}{{3n + 1}},{\text{ where }}m,{\text{ }}n \in \mathbb{Z}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">under multiplication.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the set <em>S </em>defined by \(S = \{ s \in \mathbb{Q}:2s \in \mathbb{Z}\} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">You may assume that \( + \) (addition) and \( \times \) (multiplication) are associative binary operations</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">on \(\mathbb{Q}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down the six smallest non-negative elements of \(S\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Show that \(\{ S,{\text{ }} + \} \) is a group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Give a reason why \(\{ S,{\text{ }} \times \} \) is not a group. Justify your answer.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The relation \(R\) is defined on \(S\) by \({s_1}R{s_2}\) if \(3{s_1} + 5{s_2} \in \mathbb{Z}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that \(R\) is an equivalence relation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine the equivalence classes.</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The binary operation \(\Delta\) is defined on the set \(S =\) {1, 2, 3, 4, 5} by the following Cayley table.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-10_om_13.21.35.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) State whether <em>S </em>is closed under the operation Δ and justify your answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) State whether Δ is commutative and justify your answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(c) State whether there is an identity element and justify your answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Determine whether Δ is associative and justify your answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Find the solutions of the equation \(a\Delta b = 4\Delta b\), for \(a \ne 4\).</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The binary operation \( * \) is defined for \(a{\text{, }}b \in {\mathbb{Z}^ + }\) by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\[a * b = a + b - 2.\]</span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Determine whether or not \( * \) is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) closed,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) commutative,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) associative.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Find the identity element.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the set of positive integers having an inverse under \( * \).</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(A\), \(B\), \(C\) and \(D\) are subsets of \(\mathbb{Z}\) .</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(A = \{ \left. m \right|m{\text{ is a prime number less than 15}}\}\)</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(B = \{ \left. m \right|{m^4} = 8m\} \)</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(C = \{ \left. m \right|(m + 1)(m - 2) < 0\} \)</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(D = \{ \left. m \right|{m^2} < 2m + 4\} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) List the elements of each of these sets.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Determine, giving reasons, which of the following statements are true and which are false.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) \(n(D) = n(B) + n(B \cup C)\)</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) \(D\backslash B \subset A\)</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (iii) \(B \cap A' = \emptyset \)</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (iv) \(n(B\Delta C) = 2\)</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Consider the set <em>A</em> = {1, 3, 5, 7} under the binary operation \( * \), where \( * \) denotes multiplication modulo 8.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) Write down the Cayley table for \(\{ A,{\text{ }} * \} \).</span></p>
<p style="margin: 0px 0px 0px 30px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Show that \(\{ A,{\text{ }} * \} \) is a group.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (iii) Find all solutions to the equation \(3 * x * 7 = y\). Give your answers in the form \((x,{\text{ }}y)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Now consider the set <em>B</em> = {1, 3, 5, 7, 9} under the binary operation \( \otimes \), where \( \otimes \) denotes multiplication modulo 10. Show that \(\{ B,{\text{ }} \otimes \} \) is not a group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Another set <em>C</em> can be formed by removing an element from <em>B</em> so that \(\{ C,{\text{ }} \otimes \} \) is a group.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 25px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) State which element has to be removed.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 25px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Determine whether or not \(\{ A,{\text{ }} * \} \) and \(\{ C,{\text{ }} \otimes \} \) are isomorphic.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let {<em>G</em> , \( * \)} be a finite group of order <em>n</em> and let <em>H</em> be a non-empty subset of <em>G</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that any element \(h \in H\) has order smaller than or equal to <em>n</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) If <em>H</em> is closed under \( * \), show that {<em>H</em> , \( * \)} is a subgroup of {<em>G</em> , \( * \)}.</span></p>
</div>
<br><hr><br><div class="question">
<p class="p1"><span class="s1">The group \(\{ G,{\text{ }} * \} \) </span>is Abelian and the bijection \(f:{\text{ }}G \to G\) is defined by \(f(x) = {x^{ - 1}},{\text{ }}x \in G\).</p>
<p class="p1">Show that \(f\) is an isomorphism.</p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The group <em>G</em> has a subgroup <em>H</em>. The relation <em>R</em> is defined on <em>G</em> by <em>xRy</em> if and only if \(x{y^{ - 1}} \in H\), for \(x,{\text{ }}y \in G\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that <em>R</em> is an equivalence relation.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The Cayley table for <em>G</em> is shown below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The subgroup <em>H</em> is given as \(H = \{ e,{\text{ }}{a^2}b\} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the equivalence class with respect to <em>R</em> which contains <em>ab</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Another equivalence relation \(\rho \) is defined on <em>G</em> by \(x\rho y\) if and only if \({x^{ - 1}}y \in H\), for \(x,{\text{ }}y \in G\). Find the equivalence class with respect to \(\rho \) which contains <em>ab</em>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The relation <em>R</em> is defined on {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} by <em>aRb</em> if and only if \(a(a + 1) \equiv b(b + 1)(\bmod 5)\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that <em>R</em> is an equivalence relation.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the equivalence defining <em>R</em> can be written in the form</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[(a - b)(a + b + 1) \equiv 0(\bmod 5).\]</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, or otherwise, determine the equivalence classes.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the set \({S_3} = \{ {\text{ }}p,{\text{ }}q,{\text{ }}r,{\text{ }}s,{\text{ }}t,{\text{ }}u\} \) of permutations of the elements of the set \(\{ 1,{\text{ }}2,{\text{ }}3\} \), defined by</p>
<p class="p1"><span class="Apple-converted-space"> </span>\(p = \left( {\begin{array}{*{20}{c}} 1&2&3 \\ 1&2&3 \end{array}} \right),{\text{ }}q = \left( {\begin{array}{*{20}{c}} 1&2&3 \\ 1&3&2 \end{array}} \right),{\text{ }}r = \left( {\begin{array}{*{20}{c}} 1&2&3 \\ 3&2&1 \end{array}} \right),{\text{ }}s = \left( {\begin{array}{*{20}{c}} 1&2&3 \\ 2&1&3 \end{array}} \right),{\text{ }}t = \left( {\begin{array}{*{20}{c}} 1&2&3 \\ 2&3&1 \end{array}} \right),{\text{ }}u = \left( {\begin{array}{*{20}{c}} 1&2&3 \\ 3&1&2 \end{array}} \right).\)</p>
<p class="p1">Let \( \circ \) denote composition of permutations, so \(a \circ b\) means \(b\) followed by \(a\). You may assume that \(({S_3},{\text{ }} \circ )\) forms a group.</p>
<p class="p1"> </p>
<p class="p1"> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Complete the following Cayley table</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-08_om_09.28.14.png" alt></p>
<p class="p1" style="text-align: left;"><em><strong>[5 marks]</strong></em></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) State the inverse of each element.</p>
<p class="p1">(ii) Determine the order of each element.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the subgroups containing</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(r\),</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(u\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The permutation \({p_1}\) of the set {1, 2, 3, 4} is defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{p_1} = \left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 2&4&1&3 <br>\end{array}} \right)\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) State the inverse of \({p_1}\).</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Find the order of \({p_1}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Another permutation \({p_2}\) is defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{p_2} = \left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 3&2&4&1 <br>\end{array}} \right)\]</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) Determine whether or not the composition of \({p_1}\) and \({p_2}\) is commutative.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Find the permutation \({p_3}\) which satisfies</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{p_1}{p_3}{p_2} = \left( {\begin{array}{*{20}{c}}<br> 1&2&3&4 \\ <br> 1&2&3&4 <br>\end{array}} \right){\text{.}}\]</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>R</em> be a relation on the set \(\mathbb{Z}\) such that \(aRb \Leftrightarrow ab \geqslant 0\), for <em>a</em>, <em>b</em> \( \in \mathbb{Z}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Determine whether <em>R</em> is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) reflexive;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) symmetric;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) transitive.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Write down with a reason whether or not <em>R</em> is an equivalence relation.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The relation <em>R</em> is defined for <em>a</em> , \(b \in {\mathbb{Z}^ + }\) such that <em>aRb</em> if and only if \({a^2} - {b^2}\) is divisible by 5.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that <em>R</em> is an equivalence relation.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Identify the three equivalence classes.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(G\) be a group of order 12 with identity element <em>e</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(a \in G\) such that \({a^6} \ne e\) and \({a^4} \ne e\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Prove that \(G\) is cyclic and state two of its generators.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Let \(H\) be the subgroup generated by \({a^4}\). Construct a Cayley table for \(H\).</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">State, with a reason, whether or not it is necessary that a group is cyclic given that all its proper subgroups are cyclic.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that {1, −1, i, −i} forms a group of complex numbers <em>G</em> under multiplication.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Consider \(S = \{ e,{\text{ }}a,{\text{ }}b,{\text{ }}a * b\} \) under an associative operation \( * \) where <em>e</em> is the identity element. If \(a * a = b * b = e\) and \(a * b = b * a\) , show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(a * b * a = b\) ,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(a * b * a * b = e\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) Write down the Cayley table for \(H = \{ S{\text{ , }} * \} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Show that <em>H</em> is a group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Show that <em>H</em> is an Abelian group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) For the above groups, <em>G</em> and <em>H</em> , show that one is cyclic and write down why the other is not. Write down all the generators of the cyclic group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Give a reason why </span><em style="font-family: 'times new roman', times; font-size: medium;">G</em><span style="font-family: 'times new roman', times; font-size: medium;"> and </span><em style="font-family: 'times new roman', times; font-size: medium;">H</em><span style="font-family: 'times new roman', times; font-size: medium;"> are not isomorphic.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The relations <em>R</em> and <em>S</em> are defined on quadratic polynomials <em>P</em> of the form</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[P(z) = {z^2} + az + b{\text{ , where }}a{\text{ , }}b \in \mathbb{R}{\text{ , }}z \in \mathbb{C}{\text{ .}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) The relation <em>R</em> is defined by \({P_1}R{P_2}\) if and only if the sum of the two zeros of \({P_1}\) is equal to the sum of the two zeros of \({P_2}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that <em>R</em> is an equivalence relation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine the equivalence class containing \({z^2} - 4z + 5\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) The relation <em>S</em> is defined by \({P_1}S{P_2}\) if and only if \({P_1}\) and \({P_2}\) have at least one zero in common. Determine whether or not <em>S</em> is transitive.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The relation <em>R </em>is defined on ordered pairs by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\[(a,{\text{ }}b)R(c,{\text{ }}d){\text{ if and only if }}ad = bc{\text{ where }}a,{\text{ }}b,{\text{ }}c,{\text{ }}d \in {\mathbb{R}^ + }.\]</span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that <em>R </em>is an equivalence relation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Describe, geometrically, the equivalence classes.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the set <em>S</em> = {1, 3, 5, 7, 9, 11, 13} under the binary operation multiplication modulo 14 denoted by \({ \times _{14}}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Copy and complete the following Cayley table for this binary operation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><img src="data:image/png;base64,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" alt></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Give one reason why \(\{ S,{\text{ }}{ \times _{14}}\} \) is not a group.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that a new set <em>G</em> can be formed by removing one of the elements of <em>S</em> such that \(\{ G,{\text{ }}{ \times _{14}}\} \) is a group.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the order of each element of \(\{ G,{\text{ }}{ \times _{14}}\} \).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the proper subgroups of \(\{ G,{\text{ }}{ \times _{14}}\} \).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The binary operator multiplication modulo 14, denoted by \( * \), is defined on the set <em>S</em> = {2, 4, 6, 8, 10, 12}.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Copy and complete the following operation table.</span><br><img src="data:image/png;base64,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" alt></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that {<em>S</em> , \( * \)} is a group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the order of each element of {<em>S</em> , \( * \)}.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Hence show that {<em>S</em> , \( * \)} is cyclic and find all the generators.</span></p>
<div class="marks">[11]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The set <em>T</em> is defined by \(\{ x * x:x \in S\} \). Show that {<em>T</em> , \( * \)} is a subgroup of {<em>S</em> , \( * \)}.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">The group \(\{ G,{\text{ }} * \} \) <span class="s1">is defined on the set \(G\) </span>with binary operation \( * \)<span class="s1">. \(H\) is a subset of \(G\) defined by \(H = \{ x:{\text{ }}x \in G,{\text{ }}a * x * {a^{ - 1}} = x{\text{ for all }}a \in G\} \)</span>. Prove that \(\{ H,{\text{ }} * \} \) is a subgroup of \(\{ G,{\text{ }} * \} \).</p>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">The following Cayley table for the binary operation multiplication modulo 9, denoted by \( * \)</span>, is defined on the set \(S = \{ 1,{\text{ }}2,{\text{ }}4,{\text{ }}5,{\text{ }}7,{\text{ }}8\} \).</p>
<p class="p1"><img src="data:image/png;base64,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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Copy and complete the table.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Show that \(\{ S,{\text{ }} * \} \) </span>is an Abelian group.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine the orders of all the elements of \(\{ S,{\text{ }} * \} \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the two proper subgroups of \(\{ S,{\text{ }} * \} \).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the coset of each of these subgroups with respect to the element 5.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Solve the equation \(2 * x * 4 * x * 4 = 2\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The binary operation multiplication modulo 10, denoted by ×<sub>10</sub>, is defined on the set <em>T</em> = {2 , 4 , 6 , 8} and represented in the following Cayley table.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that {<em>T</em>, ×<sub>10</sub>} is a group. (You may assume associativity.)</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By making reference to the Cayley table, explain why<em> T</em> is Abelian.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the order of each element of {<em>T</em>, ×<sub>10</sub>}.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that {<em>T</em>, ×<sub>10</sub>} is cyclic and write down all its generators.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The binary operation multiplication modulo 10, denoted by ×<sub>10</sub> , is defined on the set <em>V</em> = {1, 3 ,5 ,7 ,9}.</p>
<p>Show that {<em>V</em>, ×<sub>10</sub>} is not a group.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the sets <em>A</em> = {1, 3, 5, 7, 9} , <em>B</em> = {2, 3, 5, 7, 11} and <em>C</em> = {1, 3, 7, 15, 31} .</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\left( {A \cup B} \right) \cap \left( {A \cup C} \right)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <em>A</em> \ <em>C</em> ≠ <em>C </em>\ <em>A</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <em>S</em> be a set containing \(n\) elements where \(n \in \mathbb{N}\).</p>
<p>Show that S has \({2^n}\) subsets.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the following Cayley table for the set <em>G</em> = {1, 3, 5, 7, 9, 11, 13, 15} under the operation \({ \times _{16}}\), where \({ \times _{16}}\) denotes multiplication modulo 16.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the values of <em>a</em>, <em>b</em>, <em>c</em>, <em>d</em>, <em>e</em>, <em>f</em>, <em>g</em>, <em>h</em>, <em>i</em> and <em>j</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Given that \({ \times _{16}}\) is associative, show that the set <em>G</em>, together with the operation \({ \times _{16}}\), forms a group.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The Cayley table for the set \(H = \{ e,{\text{ }}{a_1},{\text{ }}{a_2},{\text{ }}{a_3},{\text{ }}{b_1},{\text{ }}{b_2},{\text{ }}{b_3},{\text{ }}{b_4}\} \) under the operation \( * \), is shown below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Given that \( * \) is associative, show that <em>H</em> together with the operation \( * \) forms a group.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find two subgroups of order 4.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\{ G,{\text{ }}{ \times _{16}}\} \) and \(\{ H,{\text{ }} * \} \) are not isomorphic.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\{ H,{\text{ }} * \} \) is not cyclic.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(g:\mathbb{Z} \to \mathbb{Z}\) is defined by \(g(n) = \left| n \right| - 1{\text{ for }}n \in \mathbb{Z}\) . Show that <em>g </em>is neither surjective nor injective.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The set <em>S </em>is finite. If the function \(f:S \to S\) is injective, show that <em>f </em>is surjective.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the set \({\mathbb{Z}^ + }\) as both domain and codomain, give an example of an injective function that is not surjective.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the functions \(f:A \to B\) and \(g:B \to C\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that if both <em>f</em> and <em>g</em> are injective, then \(g \circ f\) is also injective.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that if both <em>f</em> and <em>g</em> are surjective, then \(g \circ f\) is also surjective.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show, using a single counter example, that both of the converses to the results in part (a) and part (b) are false.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f:\mathbb{R} \to \mathbb{R}\) is defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \left\{ {\begin{array}{*{20}{c}}<br> {2x + 1}&{{\text{for }}x \leqslant 2} \\ <br> {{x^2} - 2x + 5}&{{\text{for }}x > 2.} <br>\end{array}} \right.\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Sketch the graph of <em>f</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) By referring to your graph, show that <em>f</em> is a bijection.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({f^{ - 1}}(x)\).</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine, using Venn diagrams, whether the following statements are true.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(A' \cup B' = (A \cup B)'\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \((A\backslash B) \cup (B\backslash A) = (A \cup B)\backslash (A \cap B)\)</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove, without using a Venn diagram, that \(A\backslash B\) and \(B\backslash A\) are disjoint sets.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br>