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<html lang="en"><head><title>Lecture 3</title><meta charset="utf-8"/><link rel="preconnect" href="https://fonts.googleapis.com"/><link rel="preconnect" href="https://fonts.gstatic.com"/><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=IBM Plex Mono&family=Schibsted Grotesk:wght@400;700&family=Source Sans Pro:ital,wght@0,400;0,600;1,400;1,600&display=swap"/><link rel="preconnect" href="https://cdnjs.cloudflare.com" crossorigin="anonymous"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><meta name="og:site_name" content="ACIT4330 Lecture Notes"/><meta property="og:title" content="Lecture 3"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:title" content="Lecture 3"/><meta name="twitter:description" content="Proposition 1.1.4 Each real number is a limit of a sequence of rational numbers. \mathbb{Q} \subset \mathbb{R} As an ordered Number Field \mathbb{R} is complete, meaning that every Cauchy Sequence in \mathbb{R} converges to a real number."/><meta property="og:description" content="Proposition 1.1.4 Each real number is a limit of a sequence of rational numbers. \mathbb{Q} \subset \mathbb{R} As an ordered Number Field \mathbb{R} is complete, meaning that every Cauchy Sequence in \mathbb{R} converges to a real number."/><meta property="og:image:type" content="image/webp"/><meta property="og:image:alt" content="Proposition 1.1.4 Each real number is a limit of a sequence of rational numbers. \mathbb{Q} \subset \mathbb{R} As an ordered Number Field \mathbb{R} is complete, meaning that every Cauchy Sequence in \mathbb{R} converges to a real number."/><meta property="og:image:width" content="1200"/><meta property="og:image:height" content="630"/><meta property="og:image:url" content="https://https://acit4330.pages.anthonyberg.io//static/og-image.png"/><meta name="twitter:image" 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xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Lectures"><button class="folder-button"><span class="folder-title">Lectures</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Lectures"><li><a href="../Lectures/Lecture-1---1.1-Sets-and-Numbers" data-for="Lectures/Lecture-1---1.1-Sets-and-Numbers">Lecture 1 - 1.1 Sets and Numbers</a></li><li><a href="../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../Lectures/">Lectures</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Lecture 3</a></div></nav><h1 class="article-title">Lecture 3</h1><p show-comma="true" class="content-meta"><time datetime="2025-01-13T00:00:00.000Z">13 Jan 2025</time><span>2 min read</span></p></div></div><article class="popover-hint"><h1 id="proposition-114">Proposition 1.1.4<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#proposition-114" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
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<p>Each real number is a limit of a sequence of rational numbers.<br/>
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<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8556em;vertical-align:-0.1667em;"></span><span class="mord mathbb">Q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span><br/>
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As an ordered <a href="../Definitions/Number-Field" class="internal alias" data-slug="Definitions/Number-Field">Number Field</a> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> is <strong>complete</strong>, meaning that every <a href="../Definitions/Cauchy-Sequence" class="internal alias" data-slug="Definitions/Cauchy-Sequence">Cauchy Sequence</a> in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> converges to a real number.<br/>
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Equivalently, the real numbers have the <strong><a href="../Definitions/Least-Upper-Bound-Property" class="internal alias" data-slug="Definitions/Least-Upper-Bound-Property">Least Upper Bound Property</a></strong>; <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord">∀</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> bounded above has a least upper bound denoted by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">u</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span>. Eq. an <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">in</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">R</span></span></span></span> if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> bounded below.</p>
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<h1 id="functions-and-cardinality">Functions and Cardinality<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#functions-and-cardinality" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
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<p>A function <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> is <strong><a href="../Definitions/Terminology/Injective" class="internal" data-slug="Definitions/Terminology/Injective">Injective</a></strong> if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>.</p>
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<p><a href="../Definitions/Terminology/Surjective" class="internal" data-slug="Definitions/Terminology/Surjective">Surjective</a> if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>.</p>
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<p><a href="../Definitions/Terminology/Bijective" class="internal" data-slug="Definitions/Terminology/Bijective">Bijective</a> if it is both injective and surjective.</p>
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<p>Then we write <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≃</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span>.<br/>
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<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord text"><span class="mord">(cardinality)</span></span></span></span></span><br/>
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Say <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> is <a href="../Definitions/Terminology/Countable" class="internal" data-slug="Definitions/Terminology/Countable">Countable</a> if <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathbb">N</span><span class="mord">∣</span></span></span></span>; this means that the members of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> can be listed as a sequence with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>, where <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">IN</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> is some bijection.</p>
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<h2 id="cantors-diagonal-argument">Cantor’s Diagonal Argument<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#cantors-diagonal-argument" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h2>
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<p>The real numbers cannot be listed, or they are <strong>uncountable</strong>.</p>
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<p>Indeed, present a list of the real numbers in <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">⟩</span></span></span></span> written as binary expansions. Then the number that has as its <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>-th digit, the opposite value to the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>-th digit of the <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>-th number of the list, will never be in the list.</p>
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<div class="table-container"><table><thead><tr><th><strong>\</strong></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>0</td><td><strong>1</strong></td><td>0</td><td>1</td><td>1</td><td>0</td><td>0</td><td>1</td><td>1</td><td>1</td><td>…</td></tr><tr><td>0</td><td>1</td><td><strong>1</strong></td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>…</td></tr><tr><td>0</td><td>0</td><td>0</td><td><strong>0</strong></td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>…</td></tr><tr><td>0</td><td>1</td><td>0</td><td>1</td><td><strong>0</strong></td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>…</td></tr><tr><td>…</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>So here the bold numbers going diagonally from the <strong>\</strong> shows that they cannot be countable as they are not the same number.</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Cantor: 0.0011…</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table></div>
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<h1 id="axiom-of-choice">Axiom of Choice<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#axiom-of-choice" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
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<p>Any <strong><a href="../Definitions/Functions/Direct-Product" class="internal alias" data-slug="Definitions/Functions/Direct-Product">Direct Product</a></strong><br/>
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<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8607em;vertical-align:-0.1774em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1774em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mbin">∪</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1774em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span><br/>
|
||
is non-empty when all <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">∅</span></span></span></span>.</p>
|
||
<p>Any <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8607em;vertical-align:-0.1774em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1774em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is called a choice function.</p>
|
||
<p>The <a href="../Definitions/Functions/Power-Set" class="internal alias" data-slug="Definitions/Functions/Power-Set">Power Set</a> <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">℘</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span> of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> consists of all the subsets of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span>.</p>
|
||
<p><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">∃</span></span></span></span> bijection <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">℘</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">Π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">{</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">}</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">}}</span></span></span></span> that sends <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> to its <a href="../Definitions/Functions/Characteristic-Function" class="internal alias" data-slug="Definitions/Functions/Characteristic-Function">Characteristic Function</a>.</p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{"drag":true,"zoom":true,"depth":1,"scale":1.1,"repelForce":0.5,"centerForce":0.3,"linkDistance":30,"fontSize":0.6,"opacityScale":1,"showTags":true,"removeTags":[],"focusOnHover":false,"enableRadial":false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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s-2-0.897-2-2s0.897-2,2-2S47,39.897,47,41z M49,10c-2.206,0-4-1.794-4-4s1.794-4,4-4s4,1.794,4,4S51.206,10,49,10z"></path></svg></button></div><div id="global-graph-outer"><div id="global-graph-container" data-cfg="{"drag":true,"zoom":true,"depth":-1,"scale":0.9,"repelForce":0.5,"centerForce":0.3,"linkDistance":30,"fontSize":0.6,"opacityScale":1,"showTags":true,"removeTags":[],"focusOnHover":true,"enableRadial":true}"></div></div></div><div class="toc desktop-only"><button type="button" id="toc" class aria-controls="toc-content" aria-expanded="true"><h3>Table of Contents</h3><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="fold"><polyline points="6 9 12 15 18 9"></polyline></svg></button><div id="toc-content" class><ul class="overflow"><li class="depth-0"><a href="#proposition-114" data-for="proposition-114">Proposition 1.1.4</a></li><li class="depth-0"><a href="#functions-and-cardinality" data-for="functions-and-cardinality">Functions and Cardinality</a></li><li class="depth-1"><a href="#cantors-diagonal-argument" data-for="cantors-diagonal-argument">Cantor’s Diagonal Argument</a></li><li class="depth-0"><a href="#axiom-of-choice" data-for="axiom-of-choice">Axiom of Choice</a></li></ul></div></div><div class="backlinks"><h3>Backlinks</h3><ul class="overflow"><li><a href="../" class="internal">ACIT4330 Table of Contents</a></li></ul></div></div><footer class><p>Created with <a href="https://quartz.jzhao.xyz/">Quartz v4.4.0</a> © 2025</p><ul><li><a href="https://git.anthonyberg.io/smyalygames/ACIT4330-Page">Gitea</a></li></ul></footer></div></div></body><script type="application/javascript">function c(){let t=this.parentElement;t.classList.toggle("is-collapsed");let l=t.classList.contains("is-collapsed")?this.scrollHeight:t.scrollHeight;t.style.maxHeight=l+"px";let o=t,e=t.parentElement;for(;e;){if(!e.classList.contains("callout"))return;let n=e.classList.contains("is-collapsed")?e.scrollHeight:e.scrollHeight+o.scrollHeight;e.style.maxHeight=n+"px",o=e,e=e.parentElement}}function i(){let t=document.getElementsByClassName("callout is-collapsible");for(let s of t){let l=s.firstElementChild;if(l){l.addEventListener("click",c),window.addCleanup(()=>l.removeEventListener("click",c));let e=s.classList.contains("is-collapsed")?l.scrollHeight:s.scrollHeight;s.style.maxHeight=e+"px"}}}document.addEventListener("nav",i);window.addEventListener("resize",i);
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