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source/backmatter.ptx

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<?xml version="1.0" encoding="UTF-8"?>
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<backmatter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="backmatter" permid="mqL">
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<backmatter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="backmatter">
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<title>Backmatter</title>
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<xi:include href="peer-instruction.ptx" />
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<solutions divisional="hint" permid="SxU">
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<solutions divisional="hint">
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<title>Selected Hints</title>
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</solutions>
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<solutions divisional="solution" permid="yFd">
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<solutions divisional="solution">
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<title>Selected Solutions</title>
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</solutions>
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<appendix permid="GjC">
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<appendix>
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<title>List of Symbols</title>
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<notation-list />
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</appendix>
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</index>
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<colophon>
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<p permid="eMm"> This book was authored in <pretext />. </p>
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<p> This book was authored in <pretext />. </p>
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</colophon>
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</backmatter>
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source/ch_additionalTopics.ptx

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<?xml version="1.0" encoding="UTF-8"?>
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<chapter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="ch_additionalTopics" permid="KTv">
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<chapter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="ch_additionalTopics">
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<title>Additional Topics</title>
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<xi:include href="sec_addtops-genfun.ptx"/>
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<xi:include href="sec_addtops-numbth.ptx"/>

source/ch_counting.ptx

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<?xml version="1.0" encoding="UTF-8"?>
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<chapter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="ch_counting" permid="raE">
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<chapter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="ch_counting">
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<title>Counting</title>
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<idx><h>counting</h></idx>
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<idx><h>enumeration</h><see>counting</see></idx>
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<introduction permid="XhN">
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<p permid="Unh">
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<introduction>
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<p>
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One of the first things you learn in mathematics is how to count.
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Now we want to count large collections of things quickly and precisely.
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For example:
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<ul permid="DoW">
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<li permid="jwf">
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<p permid="bRG">
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<ul>
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<li>
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<p>
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In a group of 10 people,
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if everyone shakes hands with everyone else exactly once,
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how many handshakes took place?
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</p>
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</li>
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<li permid="PDo">
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<p permid="HYP">
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<li>
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<p>
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How many ways can you distribute <m>10</m> girl scout cookies to <m>7</m> boy scouts?
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</p>
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</li>
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<li permid="vKx">
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<p permid="ofY">
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<li>
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<p>
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How many anagrams are there of <q>anagram</q>?
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</p>
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</li>
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</ul>
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</p>
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<p permid="Auq">
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<p>
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Before tackling questions like these,
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let's look at the basics of counting.
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</p>

source/ch_graphtheory.ptx

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<?xml version="1.0" encoding="UTF-8"?>
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<chapter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="ch_graphtheory" permid="gBz">
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<chapter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="ch_graphtheory">
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<title>Graph Theory</title>
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<introduction permid="xzK">
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<investigation permid="dGT">
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<introduction>
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<investigation>
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<idx><h>Seven Bridges of Königsberg</h></idx>
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<idx><h>Königsberg, Seven Bridges of</h></idx>
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<idx><h>puzzle</h><h>seven bridges</h></idx>
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<p permid="uFe">
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<p>
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In the time of Euler, in the town of Königsberg in Prussia,
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there was a river containing two islands.
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The islands were connected to the banks of the river by seven bridges
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Are you?
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</p>
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<sidebyside width="80%" permid="YXa">
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<sidebyside width="80%">
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<image xml:id="gt-bridges-art" permid="fwg">
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<image xml:id="gt-bridges-art">
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<latex-image>
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\begin{tikzpicture}[scale=0.9]
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</sidebyside>
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</investigation>
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<p permid="aMn">
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<p>
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Graph Theory is a relatively new area of mathematics,
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first studied by the super famous mathematician Leonhard Euler in 1735.
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Since then it has blossomed in to a powerful tool used in nearly every branch of science and is currently an active area of mathematics research.
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</p>
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<p permid="GTw">
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<p>
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The problem above, known as the
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<em>Seven Bridges of Königsberg</em>,
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is the problem that originally inspired graph theory.
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</p>
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<image width = "20%" xml:id="gt-bridges-graph" permid="LDp">
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<image width = "20%" xml:id="gt-bridges-graph">
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<latex-image>
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\begin{tikzpicture}[scale=0.9, yscale=.5]
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\draw (-1,-2) \v to [out=120, in=240] (-1,0) \v to [out=120, in=240] (-1,2) \v to [out=300, in=60] (-1,0) to [out=300, in=60] (-1,-2);
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</image>
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<p permid="naF">
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<p>
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There is an obvious connection between these two problems.
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Any path in the dot and line drawing corresponds exactly to a path over the bridges of Königsberg.
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</p>
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<p permid="ThO">
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<p>
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<idx><h>graph</h></idx>
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<idx><h>graph</h><h>vertex</h></idx>
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<idx><h>graph</h><h>edge</h></idx>
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<!-- This was the great insight that Euler had. -->
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</p>
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<p permid="zoX">
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<p>
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We will return to the question of finding paths through graphs later.
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But first, here are a few other situations you can represent with graphs:
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</p>
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<example permid="MII">
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<example>
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<statement>
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<p permid="iqM">
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<p>
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Al, Bob, Cam, Dan,
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and Euler are all members of the social networking website <em>Facebook</em>.
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The site allows members to be <q>friends</q> with each other.
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Represent this situation with a graph.
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</p>
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</statement>
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<solution permid="JOc">
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<p permid="Wcu">
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<solution>
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<p>
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Each person will be represented by a vertex and each friendship will be represented by an edge.
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That is, two vertices will be adjacent
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(there will be an edge between them)
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<image width="20%" xml:id="gt-facebook" permid="rKy">
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<image width="20%" xml:id="gt-facebook">
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<latex-image>
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\begin{tikzpicture}[scale=0.7]
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</solution>
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</example>
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<example permid="sPR">
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<example>
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<idx><h>houses and utilities puzzle</h></idx>
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<idx><h>puzzle</h><h>houses and utilities</h></idx>
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<statement>
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<p permid="OxV">
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<p>
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Each of three houses must be connected to each of three utilities.
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Is it possible to do this without any of the utility lines crossing?
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</p>
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</statement>
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<solution permid="pVl">
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<p permid="CjD">
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<solution>
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<p>
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We will answer this question later.
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For now, notice how we would ask this question in the context of graph theory.
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We are really asking whether it is possible to redraw the graph below without any edges crossing
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bottom row as the utilities.
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</p>
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<image width="20%" xml:id="gt-k33" permid="XRH">
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<image width="20%" xml:id="gt-k33">
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<latex-image>
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\begin{tikzpicture}[yscale=.8]

source/ch_intro.ptx

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<?xml version="1.0" encoding="UTF-8"?>
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<chapter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="ch_intro" permid="DYQ">
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<chapter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="ch_intro">
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<title>Introduction and Preliminaries</title>
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<introduction permid="kfZ">
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<p permid="Qni">
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<introduction>
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<p>
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Welcome to Discrete Mathematics.
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If this is your first time encountering the subject,
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you will probably find discrete mathematics quite different from other math subjects.

source/ch_logic.ptx

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<?xml version="1.0" encoding="UTF-8"?>
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<chapter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="ch_logic" permid="wur">
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<chapter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="ch_logic">
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<title>Symbolic Logic and Proofs</title>
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<introduction permid="UXb">
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<p permid="Bek">
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<introduction>
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<p>
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Logic is the study of consequence.
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Given a few mathematical statements or facts,
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we would like to be able to draw some conclusions.
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and from these we find a consequence of interest, our <em>conclusion</em>.
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</p>
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<p permid="hlt">
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<p>
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The problem is, as you no doubt know from arguing with friends,
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not all arguments are <em>good</em> arguments.
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A <q>bad</q> argument is one in which the conclusion does not follow from the premises,
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a consequence of a set of premises.
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</p>
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<p permid="NsC">
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<p>
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By the way, <q>argument</q> is actually a technical term in math (and philosophy,
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another discipline which studies logic):
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</p>
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<assemblage permid="cBA">
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<assemblage>
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<title>Arguments</title>
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<p permid="tzL">
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<p>
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An <term>argument</term>
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<idx><h>argument</h></idx>
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is a set of statements, one of which is called the <term>conclusion</term>
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</p>
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</assemblage>
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<p>
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For example, consider the following two arguments:
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</p>
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<sidebyside>
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<tabular permid="yjE">
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<tabular>
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<row>
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<cell/>
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<cell>If Edith eats her vegetables, then she can have a cookie.</cell>
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</sidebyside>
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<sidebyside>
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<tabular permid="eqN">
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<tabular>
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<row>
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<cell/>
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<cell>Florence must eat her vegetables in order to get a cookie.</cell>
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</sidebyside>
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<p permid="FOd">
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<p>
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(The symbol <q><m>\therefore</m></q>
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<notation>
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means <q>therefore</q>)
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</p>
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<p>
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Are these arguments valid?
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Hopefully you agree that the first one is but the second one is not.
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Logic tells us why by analyzing the structure of the statements in the argument.
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In mathematics, we never get that luxury.
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</p>
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<p>
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We will start in <xref ref="sec_logic-statements"/> by considering what a statement even is and what form they can take.
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Then we will see what mathematical tools we can develop to better analyze these in <xref ref="sec_logic-prop"/>,
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and then see how this helps read and write proofs in <xref ref="sec_logic-proofs"/>.

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