arguments.xml

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<!-- Logic and Proof for Teachers                                                                -->
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<!-- Copyright (C) 2019  Lesa L. Beverly, Kimberly M. Childs, Deborah A. Pace, Thomas W. Judson  -->
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<chapter xml:id="Arguments" xmlns:xi="http://www.w3.org/2001/XInclude">

<title>Arguments and Proofs</title>

    <introduction>
        <p>An argument may be described as a group of statements, one of which is claimed to follow from the others. Arguments have structure. In mathematics, the statement which is supposedly validated by the others is called the conclusion; those statements which are claimed to provide justification for the conclusion are called the hypotheses.</p>

        <p>The type of reasoning used in arguments is traditionally divided into two basic types, deductive and inductive. It is often said that deductive reasoning involves moving from the general to the specific, whereas inductive reasoning involves moving from specific observations to claims of general principles. However, this description is a generalization that is not always the case. The major distinction might better be described in terms of whether or not the conclusion must always follow from the hypotheses. In a <em><term>deductive argument</term></em> it is claimed that the conclusion must be true if the hypotheses are true; that is, it is impossible for the conclusion to fail if the hypotheses hold true.</p>

        <p>In contrast, an <em><term>inductive argument</term></em> involves the claim that the conclusion probably follows from the hypotheses. Deductive arguments do not become <q>more valid</q> by adding hypotheses, whereas inductive arguments may become stronger or weaker by adding hypotheses.</p>

    </introduction>

    <section xml:id="arguments-section-deductive-reasoning">
        <title>Deductive Reasoning</title>

        <p>Deductive or direct reasoning is a process of reaching a conclusion from one (or more) statements, called the hypothesis (or hypotheses). This somewhat informal definition can be rephrased using the language and symbolism of the preceding sections. An <em><term>argument</term></em> is a set of statements in which one of the statements is called the conclusion and the rest make up the hypothesis. A <em><term>valid argument</term></em> is an argument in which the conclusion must be true whenever the hypotheses are true. In the case of a valid argument we say the conclusion follows from the hypothesis. For example, consider the following argument: <q>If it is snowing, then it is cold. It is snowing. Therefore, it is cold.</q> In this argument, when the two statements in the hypothesis, namely, <q>if it is snowing, then it is cold</q> and <q>It is snowing</q> are both true, then one can conclude that <q>It is cold.</q> That is, this argument is valid since the conclusion follows necessarily from the hypotheses.</p><idx><h>Argument</h></idx><idx><h>Valid argument</h></idx>

        <p>It is important to distinguish between the notions of truth and validity. While individual statements may be either true or false, arguments cannot. Similarly, arguments may be described as valid or invalid, but statements cannot. An argument is said to be an invalid argument if its conclusion can be false when its hypothesis is true. An example of an invalid argument is the following: <q>If it is raining, then the streets are wet. The streets are wet. Therefore, it is raining.</q> For convenience, we will represent this argument symbolically as <m>[(p \rightarrow q) \wedge p] \rightarrow p</m>. This is an invalid argument since the streets could be wet from a variety of causes (e.g., fire hydrant open, sprinkler system malfunction, etc.) without having had any rain. It is possible for valid arguments to contain either true or false hypotheses, as indicated in the two valid arguments in <xref ref="arguments-example-deductive" />.</p>

        <example xml:id="arguments-example-deductive">
            <p><ul>
                <li><p>Arguement 1:
                    <ul>

                        <li>All counting numbers are positive.</li>

                        <li>All positive numbers are larger than negative <m>2</m>.</li>

                        <li>Therefore, all counting numbers are larger than negative <m>2</m>.</li>

                    </ul></p></li>

                <li><p>Arguement 2:
                    <ul>
                        <li>All numbers are positive.</li>

                        <li>All positive numbers are larger than <m>5</m>.</li>

                        <li>Therefore, all numbers are larger than <m>5</m>.</li>

                    </ul></p></li>

                </ul>
            Note that in both Arguments 1 and 2, the conclusions follow necessarily from the hypotheses. Thus, Argument 2 is considered valid even though both hypotheses are false. It should also be noted that an argument may be invalid even though the hypotheses and the conclusion are true. In Argument 3 below, even though both hypotheses may be true, it is possible for the conclusion to be either true or false; thus, the argument is invalid.</p>

            <p><ul>
                <li><p>Arguement 3:
                    <ul>

                        <li>If it is raining outside, then the lawn gets wet.</li>

                        <li>It is not raining outside.</li>

                        <li>Therefore, the lawn is not wet.</li>

                    </ul></p></li>

                </ul></p>

            <p>The truth table below (<xref ref="arguments-table-deductive" />) shows that Arguement 3 is invalid, since it is possible to have the hypotheses, <m>(p \rightarrow q) \wedge \negate p</m>, true with the conclusion, <m>\negate q</m>, false.  This situation, of course, makes the statement <m>[(p \rightarrow q) \wedge \negate p] \rightarrow \negate q</m> false, and the argument is invalid.</p>
        </example>

        <table xml:id="arguments-table-deductive">
            <title>Truth table for <m>[(p \rightarrow q) \wedge \negate p] \rightarrow \negate q</m></title>
            <tabular halign="center" top="medium">
                <row bottom="medium">
                    <cell><m>p</m></cell><cell><m>q</m></cell><cell><m>\negate p</m></cell><cell><m>\negate q</m></cell><cell><m>p \rightarrow q</m></cell><cell><m>(p \rightarrow q) \wedge \negate p</m></cell><cell><m>[(p \rightarrow q) \wedge \negate p] \rightarrow \negate q</m></cell>
                </row>
                <row>
                    <cell>T</cell><cell>T</cell><cell>F</cell><cell>F</cell><cell>T</cell><cell>F</cell><cell>T</cell>
                </row>                            <row>
                    <cell>T</cell><cell>F</cell><cell>F</cell><cell>T</cell><cell>F</cell><cell>F</cell><cell>T</cell>
                </row>                                        
                <row>
                    <cell>F</cell><cell>T</cell><cell>T</cell><cell>F</cell><cell>T</cell><cell>T</cell><cell>F</cell>
                </row>                
                <row bottom="medium">
                    <cell>F</cell><cell>F</cell><cell>T</cell><cell>T</cell><cell>T</cell><cell>T</cell><cell>T</cell>
                </row>
           </tabular>      
        </table>

        <assemblage>
            <title>TAKS CONNECTION</title>
            <p>How might a student apply deductive reasoning to answer the following question taken from the 2006 Texas Assessment of Knowledge and Skills (TAKS) Grade 5 Mathematics test?</p>

            <p>Sue is taller than Bianca and shorter than Colette. If Colette is shorter than Dora, who is the shortest person?
                <ul>

                    <li>F. Sue</li>

                    <li>G. Bianca</li>

                    <li>H. Colette</li>

                    <li>J. Dora</li>

                </ul></p>
        </assemblage>

        <example>
            <p>Charles Dodgson (1832<ndash />1898) was an English mathematician who taught logic at Oxford University. As a teacher of logic and a lover of nonsense, he designed entertaining puzzles to train people in systematic reasoning. In these puzzles he would string together a list of implications, purposefully nonsensical so that his students would not influenced by any preconceived opinions. The task presented to the student was   to use all the listed implications to arrive at an inescapable conclusion. You may know Charles Dodgson better by his pen name Lewis Carroll, author of <em>Alice's Adventures in Wonderland</em> and <em>Through the Looking-Glass</em>.</p> <idx><h>Charles Dodgson</h></idx><idx><h>Lewis Carroll</h></idx>

            <p>For example, consider the following statements.
                <ol>

                    <li>All babies are illogical.</li>

                    <li>Nobody is despised who can manage a crocodile.</li>

                    <li>Illogical persons are despised.</li>

                </ol>
            Let 
                <md>
                    <mrow>p &amp; \text{  it is a baby}</mrow>
                    <mrow>q &amp; \text{  it is logical}</mrow>
                    <mrow>r &amp; \text{  it can manage a crocodile}</mrow>
                    <mrow>s &amp; \text{  it is despised}.</mrow>
                </md>
            The statements now translate to 
                <ol>

                    <li><m>p \rightarrow \negate q</m> (All babies are illogical.)</li>

                    <li><m> r \rightarrow \negate s</m>  or <m>s \rightarrow \negate r</m> (Nobody is despised who can manage a crocodile.)</li>

                    <li><m>\negate q \rightarrow s</m> (Illogical persons are despised.)</li>

                </ol>
                Linking these statements together, we see that <m>p \rightarrow \negate q \rightarrow s \rightarrow \negate r</m>.  In other words, <m>p \rightarrow \negate r</m> or <q>babies cannot manage crocodiles.</q></p>

            </example>

            <p>Translating a conditional statement into <q>if-then</q> form can be quite confusing.  The statement <q>All babies are illogical</q> is not in a very useful form; however, we can write an equivalent <q>if-then</q> statement: <q>If it is a baby, then it is not logical.</q>  Consider the following examples.
            	<ul>

            		<li><q>It rains only if I carry an umbrella</q> can be rewritten as <q>If it rains, then I carry an umbrella.</q></li>

            		<li><q>All citizens of Egypt speak Arabic.</q> can be rewritten as <q>If someone is a citizen of Egypt, then they speak Arabic.</q></li>

            		<li><q>Unless it is sunny, I carry an umbrella.</q> can be rewritten as <q>If it is not sunny, I carry an umbrella.</q></li>

					<li><q>No one in MTH 300 speaks Chinese.</q> can be rewritten as <q>If you are in MTH 300, then you do not speak Chinese.</q></li>

					<li><q>For cows to fly it is sufficient that 3 + 4 = 8.</q> can be rewritten as <q>If 3 + 4 = 8, then cows fly.</q></li>

					<li><q>For cows to fly it is necessary that 3 + 4 = 8.</q> can be rewritten as <q>If cows fly, then 3 + 4 = 8.</q></li>

					<li><q>When it rains, I carry an umbrella.</q> can be rewritten as <q>If it rains, I carry an umbrella.</q></li>

            	</ul></p>

        <exercises>

            <exercise>
                <statement>
                    <p>Rewrite the following conditional statements as <q>if-then</q> statements.
                        <ol>

                            <li>All citizens of Egypt speak Arabic.</li>

                            <li>Dallas is the capital of Texas only if <m>2 + 3 \neq 7</m>.</li>

                            <li>Nacogdoches is the oldest city in Texas unless mermaids exist.</li>

                            <li>No resident of Boston likes hot peppers.</li>

                            <li>For <m>3 + 7 = 10</m> it is necessary that cows fly.</li>

                            <li>For <m>3 + 7 = 10</m> it is sufficient that cows fly.</li>

                            <li>I carry and umbrella when it rains.</li>

                            <li>I carry an umbrella only if it rains.</li>

                        </ol></p>
                </statement>
            </exercise>



            <p>See how many of the following Lewis Carroll puzzles you can solve.</p>

            <exercise>
                <statement>
                    <p><ul>

                        <li>All babies are illogical.</li>

                        <li>Nobody is despised who can manage a crocodile.</li>

                        <li>Illogical persons are dispised.</li>

                    </ul></p> 
                </statement>
            </exercise>


            <exercise>
                <statement>
                    <p><ul>

                        <li>None of the unnoticed things, met with at sea, are mermaids.</li>

                        <li>Things entered in the log, as met with at sea, are sure to be worth remembering.</li>

                        <li>I have never met with anything worth remembering, when on a voyage.</li>

                        <li>Things met with at sea, that are noticed, are sure to be recorded in the log.</li>

                    </ul></p> 
                </statement>
            </exercise>


            <exercise>
                <statement>
                    <p><ul>

                        <li>No ducks waltz.</li>

                        <li>No officers ever decline to waltz.</li>

                        <li>All my poultry are ducks.</li>

                    </ul></p> 
                </statement>
            </exercise>


                        <exercise>
                <statement>
                    <p><ul>

                        <li>No birds, except ostriches, are 9 feet high.</li>

                        <li>There are no birds in this aviary that belong to anyone but me.</li>

                        <li>No ostrich lives on mince pies.</li>

                        <li>I have no birds less than 9 feet high.</li>

                    </ul></p> 
                </statement>
            </exercise>


            <exercise>
                <statement>
                    <p><ul>

                        <li>All writers, who understand human nature, are clever.</li>

                        <li>No one is a true poet unless he can stir the hearts of men.</li>

                        <li>Shakespeare wrote <q>Hamlet.</q></li>

                        <li>No writer, who does not understand human nature, can stir the hearts of men.</li>

                        <li>None but a true poet could have written <q>Hamlet.</q></li>

                    </ul></p> 
                </statement>
            </exercise>


                        <exercise>
                <statement>
                    <p><ul>

                        <li>No kitten, that loves fish, is unteachable.</li>

                        <li>No kitten without a tail will play with a gorilla.</li>

                        <li>Kittens with whiskers always love fish.</li>

                        <li>No teachable kitten has green eyes.</li>

                        <li>No kittens have tails unless they have whiskers</li>

                    </ul></p> 
                </statement>
            </exercise>


                        <exercise>
                <statement>
                    <p><ul>

                        <li>No shark ever doubts that he is well fitted out.</li>

                        <li>A fish, that cannot dance a minuet, is contemptible.</li>

                        <li>No fish is quite certain that it is well fitted out, unless it has three rows of teeth.</li>

                        <li>All fishes, except sharks, are kind to children.</li>

                        <li>No heavy fish can dance a minuet.</li>

                        <li>A fish with three rows of teeth is not to be despised.</li>

                    </ul></p> 
                </statement>
            </exercise>
        </exercises>

    </section>



    <section xml:id="arguments-section-forms-of-arguments">
        <title>Three Forms of Valid Arguments</title>

        <introduction>
            <p>Three especially important forms of valid arguments, used repeatedly in logic, are discussed next.</p>
        </introduction>

        <subsection>
            <title>Law of Detachment (Direct Reasoning): <m>[((p \rightarrow q) \wedge p)] \rightarrow q</m></title>

            <p>The Law of Detachment is the most commonly used principle of deductive reasoning. In words, this law says that whenever a conditional statement and its hypothesis are true, the conclusion is also true. That is, the conclusion can be <q>detached</q> from the conditional (see <xref ref="arguments-example-detachment" />).</p> <idx><h>Law of Detachment</h></idx><idx><h>Direct reasoning</h></idx>

            <example xml:id="arguments-example-detachment">
                <p><ul>

                    <li>If the units digit of a number is zero, then the number is a multiple of <m>10</m>.</li>

                    <li>The units digit in the number <m>40</m> is zero.</li>

                    <li>Therefore, <m>40</m> is a multiple of <m>10</m>.</li>

                </ul></p>
            </example>

            <p>Special types of diagrams, called Euler (pronounced <q>oiler</q>) diagrams, can also be used to help determine the validity of arguments. The argument in <xref ref="arguments-example-detachment" /> can be visualized using an Euler diagram as indicated in <xref ref="figure-arguments-euler" />.</p>
            <idx><h>Euler diagrams</h></idx>

            <figure xml:id="figure-arguments-euler">
                <caption>Euler diagram for direct reasoning</caption>
                <image width="40%" source="images/arguments-euler.png">
                    <description>Euler diagram for direct reasoning</description>
                </image>
            </figure>

        </subsection>

        <subsection>
            <title>Law of Syllogism (Transitive Reasoning): <m>[(p \rightarrow q) \wedge (q \rightarrow r)] \rightarrow (p \rightarrow r)</m></title>

            <p>The Law of Syllogism is also called transitive reasoning or the chain rule. Examples of the Law of Syllogism occur repeatedly in mathematics. The following argument is an application of this law.</p>
            <idx><h>Law of Syllogism</h></idx><idx><h>Transtive reasoning</h></idx>

            <example xml:id="arguments-example-syllogism">
                <p><ul>

                    <li>If a number is a multiple of eight, then it is a multiple of four.</li>

                    <li>If a number is a multiple of four, then it is a multiple of two.</li>

                    <li>Therefore, if a number is a multiple of eight, then it is a multiple of two.</li>

                </ul>
                An Euler diagram for this argument is given in <xref ref="figure-arguments-euler-syllogism" />. Notice that if <m>x</m> is any number that is a multiple of eight, then <m>x</m> is also a multiple of four. Then, since <m>x</m> is a multiple of four, <m>x</m> must also be a multiple of two.</p>
            </example>

            <figure xml:id="figure-arguments-euler-syllogism">
                <caption>Euler diagram for transitive reasoning</caption>
                <image width="40%" source="images/arguments-euler-syllogism.png">
                    <description>Euler diagram for transitive reasoning</description>
                </image>
            </figure>


        </subsection>

        <subsection>
            <title>Law of Contraposition (Indirect Reasoning): <m>[(p \rightarrow q) \wedge \negate q] \rightarrow \negate p</m></title>

            <p>Since the contrapositive of a conditional is logically equivalent to the original conditional, <m>\negate q \rightarrow \negate p</m> is logically equivalent to <m>p \rightarrow q</m>. Then, by applying the Law of Detachment to the contrapositive of <m>p \rightarrow q</m>, we may deduce <m>\negate p</m>.</p>
            <idx><h>Law of Contraposition</h></idx><idx><h>Indirect reasoning</h></idx>

            <example xml:id="arguments-example-contraposition">
                <p><ul>

                    <li>If a number is a power of <m>3</m>, then its units digit is <m>1</m>, <m>3</m>, <m>7</m>, or <m>9</m>.</li>

                    <li>The units digit ins <m>3{,}124</m> is not <m>1</m>, <m>3</m>, <m>7</m>, or <m>9</m>.</li>

                    <li>Therefore, <m>3{,}124</m> is not a power of <m>3</m>.</li>

                </ul>
                In words, the Law of Contraposition says that whenever a conditional is true and its conclusion is false, then the hypothesis is also false. (In other words, if a conditional is true and the negation of its conclusion is also true, then the negation of its hypothesis is true.) Again, an Euler diagram may be used to help determine the validity of the argument (<xref ref="figure-arguments-euler-contraposition" />).</p>
            </example>

            <figure xml:id="figure-arguments-euler-contraposition">
                <caption>Euler diagram indirect reasoning</caption>
                <image width="50%" source="images/arguments-euler-contraposition.png">
                    <description>Euler diagram for indirect reasoning</description>
                </image>
            </figure>

        </subsection>

        <exercises>

            <exercise>
                <statement>
                    <p>Determine the validity of the following arguments. Justify your thinking.
                        <ol>
                            
                            <li><ul>
                                <li>All equilateral triangles are equiangular.</li>

                                <li>All equiangular triangles are isosceles.</li> 

                                <li>Therefore, all equilateral triangles are isosceles.</li>
                            </ul></li>
                            
                            <li><ul>
                                <li>All equilateral triangles are equiangular.</li>

                                <li>All equiangular triangles are isosceles.</li> 

                                <li>Therefore all isosceles triangles are equilateral.</li>
                            </ul></li>
                            
                            <li><ul>
                                <li>If you study every day, then you will be successful.</li>

                                <li>You do not study every day.</li>

                                <li>Therefore, you will not be successful.</li>
                            </ul></li>
                            
                            <li><ul>
                                <li>If you study every day, then you will be successful.</li>

                                <li>You are not successful.</li>

                                <li>Therefore, you did not study every day.</li>
                            </ul></li>
                            
                            <li><ul>
                                <li>Some females are doctors.</li>

                                <li>All doctors are college graduates.</li>

                                <li>Therefore, all females are college graduates.</li>
                            </ul></li>
                            
                            <li><ul>
                                <li>If the alarm goes off, then I will call the police.</li> 

                                <li>I called the police.</li>

                                <li>So the alarm went off.</li>
                            </ul></li>
                            
                            <li><ul>
                                <li>If the alarm goes off, then I will call the police.</li> 

                                <li>If I call the police, then I will file a report.</li>

                                <li>The alarm went off, so I will file a report.</li>
                            </ul></li>
                            
                            <li><ul>
                                <li>If today is Friday, then tomorrow is Saturday.</li> 

                                <li>Tomorrow is Monday, so today is not Friday.</li>
                            </ul></li>
                            
                            <li><ul>
                                <li>All teachers are smart.</li> 

                                <li>Some teachers are funny.</li>

                                <li>Therefore, some smart people are funny.</li>
                            </ul></li>
                            
                            <li><ul>
                                <li>If a student is a freshman, then the student takes English.</li> 

                                <li>Mark is a junior.</li>

                                <li>Therefore, Mark does not take English.</li>
                            </ul></li>



                        </ol></p>
                </statement>
            </exercise>
            
            <exercise>
                <statement>
                     <p>Determine a valid conclusion that follows from each of the following statements and explain your reasoning.
                        <ol>
                            
                            <li>If you come to class every day, then you will be successful. You come to class every day.</li>

                            <li>If Jana does not fall, then she will win the race. Jana does not win the race.</li>

                            <li>Every square is a rectangle. Some parallelograms are rhombuses. Every rectangle is a parallelogram.</li>

                        </ol></p>
                </statement>
            </exercise>  

            <exercise>
                <statement>
                    <p>Suppose that <m>r \rightarrow s</m> is false. Determine the truth values (true, false, or cannot be determined) of each of the following statements.
                        <ol cols="2">

                            <li><m>s \rightarrow r</m></li>

 
                            <li><m>s \vee r</m></li>

                            <li><m>s \wedge r</m></li>

                            <li><m>\negate r</m></li>

                            <li><m>\negate s \rightarrow r</m></li>

                            <li><m>\negate s \wedge r</m></li>


                        </ol></p>
                </statement>
            </exercise>                      
          
            <exercise>
                <statement>
                    <p>What conclusions can be deduced from these sets of hypotheses? (Let <m>f</m> stand for a statement that is false.)
                        <ol>

                            <li><tabular halign="left">
                                    <row>
                                        <cell>Hypotheses:</cell><cell><m>p</m> or <m>q</m></cell>
                                    </row>
                                                
                                    <row bottom="medium">
                                        <cell></cell><cell><m>\negate p</m></cell>
                                    </row>                
                                    <row top="medium">
                                        <cell>Conclusion</cell><cell>?</cell>
                                    </row>
                               </tabular></li>

                           <li><tabular halign="left">

                                                
                                    <row bottom="medium">
                                        <cell>Hypotheses:</cell><cell><m>\negate p \rightarrow f</m></cell>
                                    </row>                
                                    <row top="medium">
                                        <cell>Conclusion</cell><cell>?</cell>
                                    </row>
                               </tabular></li>

                           <li><tabular halign="left">
                                    <row>
                                        <cell>Hypotheses:</cell><cell><m>(p \wedge q) \rightarrow r</m></cell>
                                    </row>
                                                
                                    <row bottom="medium">
                                        <cell></cell><cell><m>p</m></cell>
                                    </row>                
                                    <row top="medium">
                                        <cell>Conclusion</cell><cell>? (Give a conditional statement.)</cell>
                                    </row>
                               </tabular></li>

                        </ol></p>
                </statement>
            </exercise>

            <exercise>
                <statement>
                    <p>There are three forms of invalid reasoning which commonly occur.
                        <ul>   
                            <li><line><em>Fallacy of the converse.</em></line>
                                <tabular halign="left">
                                    <row>
                                        <cell>If <m>p</m>, then <m>q</m>.</cell><cell></cell>
                                    </row>
                                                
                                    <row bottom="medium">
                                        <cell><m>q</m></cell><cell></cell>
                                    </row>                
                                    <row top="medium">
                                        <cell><m>p</m></cell><cell>(invalid)</cell>
                                    </row>
                               </tabular></li>

                            <li><line><em>Fallacy of the inverse.</em></line>
                                <tabular halign="left">
                                    <row>
                                        <cell>If <m>p</m>, then <m>q</m>.</cell><cell></cell>
                                    </row>
                                                
                                    <row bottom="medium">
                                        <cell><m>\negate p</m></cell><cell></cell>
                                    </row>                
                                    <row top="medium">
                                        <cell><m>\negate q</m></cell><cell>(invalid)</cell>
                                    </row>
                               </tabular></li>

                            <li><line><em>False transitivity.</em></line>
                                <tabular halign="left">
                                    <row>
                                        <cell>If <m>p</m>, then <m>q</m>.</cell><cell></cell>
                                    </row>
                                                
                                    <row bottom="medium">
                                        <cell>If <m>p</m>, then <m>r</m>.</cell><cell></cell>
                                    </row>                
                                    <row top="medium">
                                        <cell>If <m>q</m>, then <m>r</m>.</cell><cell>(invalid)</cell>
                                    </row>
                               </tabular></li>


                        </ul></p>

                        <p>Which fallacies occur in the following arguments?
                            <ol>

                                <li>If I am a good person, nothing bad will happen to me.  Nothing happened to me.  Therefore, I am a good person.</li>


                                <li>If you work hard, you will be wealthy and wise. Therefore, if you are wealthy, then you will be wise.</li>

                            </ol></p>
                </statement>
            </exercise>        
          
        </exercises>

    </section>

    <section xml:id="proofs-section">

        <title>Proofs</title>

         <introduction>
                <p>As we stated both in the preface as well as earlier in this chapter, our working definition of mathematics is that it is the application of inductive and deductive logic to a system of axioms. It is not our purpose in this text to formalize the logical procedure required to provide formalistic proofs. Rather, we wish to arm the student with the basic logic and methods of attack used to form convincing arguments of the validity of the statements encountered in a reasonably careful study of the foundations of mathematics.</p>

                <p>Since we will need a working definition of the word <q>proof,</q> we agree that a proof is a logical sequence of steps which validate the truth of the proposition in question. In this vein the reader should review those statements which we have <a>proven</a> and note that usually we merely showed that certain definitions were satisfied. For example, when we proposed certain statements were equivalent, we established that they had the same truth value. Surely, as we proceed further, we will be forced to provide proofs which require longer and at times more subtle sequences of logical statements. Our endeavor, as well as yours, will be to convince the reader of the truth of the propositions in question.</p>

                <p>There are, however, some general approaches to proofs which are based on the various tautologies and contradictions presented in <xref ref="logic-section-tautologies" />. Most theorems are merely conditional statements of the form, <q>If <m>p</m>, then <m>q</m>.</q> Certainly, <m>p</m> and <m>q</m> might themselves be complicated compound statements, but that should not be allowed to cloud the issue at this time, so let us consider a typical theorem and a few general types of proof.</p>
            </introduction>

            <theorem>
                <statement>
                    <p>If <m>p</m>, then <m>q</m>.</p>
                </statement>
            </theorem>

            <subsection>
                <title>Method 1: Direct Proof</title>

                <p>Recall from the truth table of a conditional sentence that when <m>p</m> is false, <m>q</m> can have any truth value and the conditional will still be true. Thus, we need only consider the case when <m>p</m> is true and argue that <m>q</m> must also be true. Hence, we assume <m>p</m> is true and by applying various known tautologies and apparent implications, we argue <m>q</m> is also true.</p>
                <idx><h>Direct proof</h></idx>

                <example xml:id="arguments-example-direct-proof">
                    <p>We will prove the statement: <q>Let <m>m</m> and <m>n</m> be integers. If <m>m</m> is even and <m>n</m> is even, then <m>m + n</m> is even.</q></p>

                    <p><em>Proof.</em> Assume the hypothesis, <q><m>m</m> is even and <m>n</m> is even</q> is true. By definition of conjunction, it follows that the component statements <q><m>m</m> is even</q> and <q><m>n</m> is even</q> are true. But since even numbers are by definition multiples of <m>2</m>, there must exist integers <m>r</m> and <m>s</m> such that <m>m = 2r</m> and <m>n = 2s</m>. Then substitution yields
                        <me>m + n = 2r + 2s = 2(r + s).</me>
                    Since the set of integers is closed under addition, the number <m>r + s</m> is also an integer. Thus, we have written <m>m + n </m> as <m>2</m> times an integer, so we have shown <m>m + n</m> is even. That is, the conclusion <q><m>m + n</m> is even</q> is true whenever the hypothesis <q><m>m</m> is even and <m>n</m> is even</q> is true.</p>
                </example>

            </subsection>

             <subsection>
                <title>Method 2: Proof by Contradiction</title>

                <p>We know that <m>[\negate(p \rightarrow q)]  \leftrightarrow [p \, \wedge (\negate q)] </m> by part (5) of <xref ref="logic-theorem-tautology" />. If we begin by assuming <m>p \, \wedge (\negate q)</m> is true and reach a contradiction, it must be the case that <m>p \, \wedge (\negate q)</m> is false.  But <m>p \, \wedge (\negate q)</m> being false and yet equivalent to <m>\negate(p \rightarrow q)</m> implies that <m>\negate(p \rightarrow q)</m> is also false or <m>p \rightarrow q</m> is true. Therefore, in proving <m><m>p \rightarrow q</m></m> by contradiction, we assume <m>p</m> and <m>\negate q</m> are both true and reach a contradiction. This logically shows that the statement <m>p \rightarrow q</m> as argued above.</p>

                <idx><h>Proof by contradiction</h></idx>

                <example xml:id="arguments-example-contradiction-proof">
                    <p>We will prove the statement: <q>Let <m>x</m> and <m>y</m> be positive real numbers. If <m>x \neq y</m>, then <m>x^2 \neq y^2</m>.</q></p>

                    <p><em>Proof.</em> For positive real numbers <m>x</m> and <m>y</m>, assume <m>x \neq y</m> and <m>x^2 = y^2</m>. Then
                        <me>x^2 - y^2 = (x - y)(x + y) = 0.</me>
                    Hence, either <m>x - y = 0</m> or <m>x + y = 0</m>.  We assumed that <m>x \neq y</m>, so <m>x - y \neq 0</m>.  Consequently, <m>x + y = 0</m> or <m>x = -y</m>, which contradicts the assumption that both <m>x</m> and <m>y</m> are positive.  Therefore, <m>x^2 \neq y^2</m> if <m>x \neq y</m>.</p>
                </example>

            </subsection>

             <subsection>
                <title>Method 3: Proof by Contrapositive (Indirect Proof)</title>

                <p>Since we know that <m>(p \rightarrow q)  \leftrightarrow ( \negate q \rightarrow \negate p )</m> by part (8) of <xref ref="logic-theorem-tautology" />, we can prove <m>\negate q \rightarrow \negate p</m> instead of <m>p \rightarrow q</m>. That is, we assume that <m>\negate q</m> is true and argue that <m>\negate p</m> is true. So essentially, a proof by contraposition is a direct proof applied to a statement that is equivalent to the one we wish to prove.</p>

                <idx><h>Proof by contrapositive</h></idx>
                <idx><h>Indirect proof</h></idx>

                <example xml:id="arguments-example-contrapositive-proof">
                    <p>As in <xref ref="arguments-example-contradiction-proof" />, we will prove the statement: <q>Let <m>x</m> and <m>y</m> be positive real numbers. If <m>x \neq y</m>, then <m>x^2 \neq y^2</m>.</q> However, we will directly prove the equivalent statement, <q>If <m>x^2 = y^2</m>, the <m>x = y</m>.</q></p>

                    <p><em>Proof.</em> Assume that <m>x^2 = y^2</m>.  Then
                        <me>x^2 - y^2 = (x - y)(x + y) = 0.</me>
                    Consequently,  <m>x - y = 0</m> or <m>x + y = 0</m>. Since both <m>x</m> and <m>y</m> are positive, <m>x + y</m> must also be positive.  Hence, <m>x + y \neq 0</m>.  Therefore, <m>x - y = 0</m> or <m>x = y</m>.</p>
                </example>

            </subsection>

            <subsection>
                <title>Counterexamples</title>

                <p>We can use a counterexample to prove that a statement is false. In considering the truth value of a conditional statement <m>p \rightarrow q</m>, we would know the statement is false if we could find a single example for which <m>p</m> is true but <m>q</m> is false.</p>

                <idx><h>Counterexample</h></idx>

                <example xml:id="arguments-example-counterexample">
                    <p>Consider the statement: <q>Let <m>m</m> and <m>n</m> be integers. If <m>m</m> is even, then <m>m + n</m> is even.</q>  By considering a single example such as <m>m = 2</m> and <m>n = 3</m>, and observing that
                        <me>m + n = 2 + 3 = 5</me>
                    is not even, we have established the conditional statement is false.</p>
                </example>

            </subsection>

        <exercises>

             <exercise xml:id="exercise-arguments-proof">
                <statement>
                    <p>Let <m>m</m> and <m>n</m> represent integers. Prove by the direct method.
                        <ol>

                            <li>If <m>n</m> is even, then <m>-n</m> is even.</li>

                            <li>If <m>n</m> is odd, then <m>-n</m> is odd.</li>

                            <li>If <m>m</m> is even and <m>n</m> is odd, then <m>m + n</m> is odd.</li>

                            <li>If <m>m</m> is odd, then <m>m^2 + 1</m> is even.</li>

                    </ol></p>
                </statement>
            </exercise>

            <exercise>
                <statement>
                    <p>Using facts from <xref ref="exercise-arguments-proof" />, prove the given statement by the method indicated.
                    <blockquote>
                        <p>If <m>m + n</m> is even and <m>m</m> is odd, then <m>n</m> is odd.</p>
                    </blockquote>
                        <ol>

                            <li>Direct method.</li>

                            <li>Contraposition.</li>

                            <li>Contradiction.</li>

                    </ol></p>
                </statement>
            </exercise>

        </exercises>


    </section>

</chapter>