logic.xml

<?xml version="1.0" encoding="UTF-8" ?>

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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="logic" xmlns:xi="http://www.w3.org/2001/XInclude">

<title>Logic</title>

    <assemblage>
    	<title>TEXAS STATE BOARD FOR EDUCATOR CERTIFICATION (SBEC): MATHEMATICS STANDARDS COVERED</title>
    	<p><ul>

    		<li>STANDARD III: GEOMETRY AND MEASUREMENT: The mathematics teacher understands and uses geometry, spatial reasoning, measurement concepts and principles, and technology appropriate to teach the statewide curriculum (Texas Essential Knowledge and Skills [TEKS]) in order to prepare students to use mathematics.</li>

    		<li>STANDARD V: MATHEMATICAL PROCESSES: The mathematics teacher understands and uses mathematical processes to reason mathematically, to solve mathematical problems, to make mathematical connections within and outside of mathematics, and to communicate mathematically.</li>

    		<li>STANDARD VI: MATHEMATICAL PERSPECTIVES: The mathematics teacher understands the historical development of mathematical ideas, the interrelationship between society and mathematics, the structure of mathematics, and the evolving nature of mathematics and mathematical knowledge.</li>

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

    <introduction>
        <p>As noted in the preface, one useful definition of mathematics is the body of knowledge obtained by applying logic (deductive and inductive) to a system of axioms. Our first thought is to deal with things which conform to a bivalent system of logic; that is, things which are either true or false.</p>
    </introduction>



    <section xml:id="logic-section-definitions">
    	<title>Definitions</title>
    	
        <definition xml:id="logic-definition-statement">
            <p>A <em><term>statement</term></em> is a sentence which is either true or false. We will notationally speak of a statement <m>p</m> or a statement <m>q</m>.</p><idx><h>Statement</h></idx>
        </definition>

        <example xml:id="logic-example-statements">
            <p>Each of the following sentences are statements.
                <ol>
                    <li>George Washington was the first President of the United States.</li>

                    <li><m>2 + 3 = 5</m>.</li>

                    <li>There are <m>12</m> inches in a foot.</li>

                    <li>Harry S. Truman was the second President of the United States.</li>

                    <li><m>4 \cdot 8 = 12</m>.</li>

                    <li>There are <m>30</m> inches in a yard.</li>

                </ol></p>
        </example>

        <p>Certainly we recognize some underlying knowledge is required in interpreting the sentences in <xref ref="logic-example-statements" />. For instance, in the sentence <q><m>2 + 3 = 5</m>,</q> we assume recognition of the concepts of <m>2</m>, <m>3</m>, <m>5</m>, and addition base <m>10</m>. Nevertheless, we must make some general knowledge assumptions, and certainly the six sentences of <xref ref="logic-example-statements" /> are statements. (The first three sentences are true while the last three are false.)</p>

        <example xml:id="logic-example-not-statements">
            <p>Each of the following sentences are not statements.
                <ol>

                    <li><m>2 + 3</m></li>

                    <li>Study mathematics.</li>

                    <li xml:id="logic-example-nice-number">Three is a nice number.</li>

                    <li xml:id="logic-example-handsome">Chris Hemsworth, who plays Thor in the <em>Avengers</em> series, is a handsome man.</li>

                </ol></p>
        

            <p>Clearly, the problem in each of these sentences is that their truth or falsity cannot be uniquely determined. Actually, <xref ref="logic-example-nice-number" /> and <xref ref="logic-example-handsome" /> could be statements provided that we have good definitions of <q>nice numbers</q> and <q>handsome</q>.</p>

        </example>

        <example xml:id="logic-example-paradox">
            <p>Consider the following sentence. <q>This sentence is false.</q> This is not a statement. Why? The sentence in question is an example of what is known in mathematics as a paradox. If it is true, then it is false, while on the other hand, if it is false, then it is true. Such paradoxes are not statements.</p>
        </example>

        <p>What, then, is an axiom? Surely it must be a statement, but also something more. As we study a body of mathematical knowledge, we encounter new statements, some of which can be proven from the existing system.  These statements are called lemmas, theorems, facts, etc. Other statements cannot be proven. If we can in fact prove that the truth value of the statement is independent of the existing system, we have a potential axiom. We can either assume the statement is true, adding it to our system as an axiom, or we could assume the statement is false, adding its negative (or some form of its negative) to our system as an axiom. Surely, quite different bodies of knowledge would evolve depending on what axiom we added. The best examples of this concept are Euclidean geometry and the various non-Euclidean geometries.</p>

        <p>Logicians as well as some other mathematicians are deeply concerned with such questions of creating minimal systems of axioms and developing mathematics very systematically from them. Such important but esoteric questions are beyond the scope and intent of this course. However, the remainder of this chapter will attempt to create a firm, logical base which will be used throughout this text and many subsequent courses the student will encounter.</p>

        <p>For our purposes the student needs a basic feel for the flow of mathematics and a concrete understanding of the concept of bivalent logic applied to statements.</p>

        <exercises>
            <p>Determine whether or not the following sentences represent statements. If so, state the truth value.</p>

            <exercise>
                <statement>
                    <p><m>7 \cdot 9 = 63</m>.</p>
                </statement>
            </exercise>
            
            <exercise>
                <statement>
                    <p>There are more males than females registered in this class.</p>
                </statement>
            </exercise>  

            <exercise>
                <statement>
                    <p><em>Gone with the Wind</em> is a good book.</p>
                </statement>
            </exercise>            
          

            <exercise>
                <statement>
                    <p>Eggs are a good source of calcium.</p>
                </statement>
            </exercise>            
          
            <exercise>
                <statement>
                    <p><m>64 \div 2 = 37</m>.</p>
                </statement>
            </exercise>      

            <exercise>
                <statement>
                    <p><m>ax^2 + bx + c</m>.</p>
                </statement>
            </exercise> 

            <exercise>
                <statement>
                    <p><m>ax^2 + bx + c = 0</m>.</p>
                </statement>
            </exercise>             
          
            <exercise>
                <statement>
                    <p>The metric system of measurement is difficult to learn.</p>
                </statement>
            </exercise>            
          
            <exercise>
                <statement>
                    <p>Summer is the best season of the year.</p>
                </statement>
            </exercise>            
          
            <exercise>
                <statement>
                    <p>There are <m>30</m> people registered for this class.</p>
                </statement>
            </exercise>            
          
            <exercise>
                <statement>
                    <p><m>\sqrt{64} = 9</m>.</p>
                </statement>
            </exercise>            
          
            <exercise>
                <statement>
                    <p>Today is a beautiful day.</p>
                </statement>
            </exercise>            
          
        </exercises>

    </section>

    <section xml:id="logic-section-statements">
    	<title>Compound Statements</title>
    	
        <p>In <xref ref="logic-section-definitions" /> we defined a statement to be a sentence which is either true or false. Many statements we are interested in studying are actually combinations of several simpler ones. Then the problem of determining the truth value (truth or falsity) of such statements becomes one of discovering the truth value of the statements being combined as well as understanding the methods of combination. We will at this time consider the negation, conjunction, and disjunction of statements.</p>

        <definition xml:id="logic-definition-negation">
            <p>Let <m>p</m> be a statement. The <em><term>negation</term></em> of <m>p</m>, denoted <m>\negate p</m>, is a statement forming the denial of <m>p</m>. The statement <m>\negate p</m>, read <q>not p,</q> has the opposite truth value of <m>p</m>.</p><idx><h>Negation</h></idx>
        </definition>

        <example xml:id="logic-example-negation">
            <p><ol>
                <li>Consider the statement, <q>Austin is the capital of Texas.</q> The negation of that statement would be the statement, <q>Austin is not the capital of Texas.</q></li>

                <li>The statement <q><m>2 + 3 = 5</m></q> has as its negation the statement <q><m>2 + 3 \neq 5</m>.</q></li>

            </ol></p>
        </example>

        <p>Since one of our stated concerns in this section is the determination of the truth value of a given statement based upon the truth values of its component statements, we consider the concept of a truth table. Very simply, a truth table is exactly a table which indicates the relationships between the truth values of the statements forming the table. Thus, the truth table below (<xref ref="logic-table-negation" />) indicates the relationship between the statements <m>p</m> and <m>\negate p</m>, giving us the basic table for a negation.</p><idx><h>Truth table</h></idx>

        <table xml:id="logic-table-negation">
            <title>Truth table for negation</title>
           <tabular halign="center" top="medium">
                <row bottom="medium">
                    <cell><m>p</m></cell><cell><m>\negate p</m></cell>
                </row>
                            
                <row>
                    <cell>T</cell><cell>F</cell>
                </row>                
                <row bottom="medium">
                    <cell>F</cell><cell>T</cell>
                </row>
           </tabular>
        </table>

        <p>Notice that the table shows that if <m>p</m> is true, then <m>\negate p</m> is false and if <m>p</m> is false, then <m>\negate p</m> is true. Truth tables become very useful when we deal with more complicated statements.</p>

        <p>The first type of compound statement we consider is the conjunction. When combining statements in logic, the most important aspect of the definition is the truth value of the resulting statement in terms of the component statements.</p>

        <definition xml:id="logic-definition-conjunction">
            <p>Let <m>p</m> and <m>q</m> be statements. The <em><term>conjunction</term></em> of <m>p</m> and <m>q</m>, denoted <m>p \wedge q</m>, is the compound statement obtained by connecting and with the English connective <q>and.</q> The conjunction is true only when both <m>p</m> and <m>q</m> are true.</p>
        </definition><idx><h>Conjunction</h></idx>

        <example xml:id="logic-example-conjunction">
            <p>The compound statement <q>Austin is the capital of Texas, and five is greater than two</q> is obtained by using <q>and</q> to connect the two statements <q>Austin is the capital of Texas</q> and <q>five is greater than two.</q></p>
        </example>

        <p>The key to understanding the conjunction is the truth table below (<xref ref="logic-table-conjunction" />), which systematically exhibits the four possible combinations of the truth values for <m>p</m> and <m>q</m>. Thus, we see that the conjunction of two statements is true only in the case when both statements are true.</p>

        <table xml:id="logic-table-conjunction">
            <title>Truth table for conjunction</title>
            <tabular halign="center" top="medium">
                <row bottom="medium">
                    <cell><m>p</m></cell><cell><m>q</m></cell><cell><m>p \wedge q</m></cell>
                </row>
                <row>
                    <cell>T</cell><cell>T</cell><cell>T</cell>
                </row>                            <row>
                    <cell>T</cell><cell>F</cell><cell>F</cell>
                </row>                                        
                <row>
                    <cell>F</cell><cell>T</cell><cell>F</cell>
                </row>                
                <row bottom="medium">
                    <cell>F</cell><cell>F</cell><cell>F</cell>
                </row>
            </tabular>           
        </table>

        <definition xml:id="logic-definition-disjunction">
            <p>Let <m>p</m> and <m>q</m> be statements. The <em><term>disjunction</term></em> of <m>p</m> and <m>q</m>, denoted <m>p \vee q</m>, is the compound statement obtained by connecting and with the English connective <q>or.</q> The conjunction is true when at least one of the statements is true.</p>
        </definition><idx><h>Disjunction</h></idx>

        <p>A brief comment about <q>or</q> must be noted. As used in a mathematical/logical sense, <q>or</q> is interpreted in the inclusive sense. That is, <a>or</a> is interpreted as and/or, meaning one and/or the other is true. Consider carefully the truth table for the disjunction (<xref ref="logic-table-disjunction" />). So we see the disjunction is false only when both <m>p</m> and <m>q</m> are false. (The exclusive use of <q>or</q> would yield truth only if exactly one of the two statements were true.)</p>

        <table xml:id="logic-table-disjunction">
            <title>Truth table for disjunction</title>
            <tabular halign="center" top="medium">
                <row bottom="medium">
                    <cell><m>p</m></cell><cell><m>q</m></cell><cell><m>p \vee q</m></cell>
                </row>
                <row>
                    <cell>T</cell><cell>T</cell><cell>T</cell>
                </row>                            <row>
                    <cell>T</cell><cell>F</cell><cell>T</cell>
                </row>                                        
                <row>
                    <cell>F</cell><cell>T</cell><cell>T</cell>
                </row>                
                <row bottom="medium">
                    <cell>F</cell><cell>F</cell><cell>F</cell>
                </row>
            </tabular> 
        </table>


        <example xml:id="logic-example-disjunction">
            <p>Consider the four disjunctions.
                <ol>
                    <li>Austin is the capital of Texas or five is greater than two.</li>

                    <li>Austin is the capital of Texas or five is less than two.</li>

                    <li>Austin is not the capital of Texas or two is less than five.</li>

                    <li>Austin is not the capital of Texas or five is less than two.</li>

                </ol>
            Here we see the first three compound sentences are disjunctions which are true, while the disjunction in (4) is false.</p>
        </example>

        <p>Another way of logically combining statements is the conditional statement, which is the heart of mathematical logic.</p>

        <definition xml:id="logic-definition-conditional">
            <p>Let <m>p</m> and <m>q</m> be statements. The <em><term>conditional statement</term></em> is the compound statement obtained by considering this statement: <q>if <m>p</m>, then <m>q</m></q> or <q><m>p</m> implies <m>q</m>,</q> and is denoted <m>p \rightarrow q</m>. The conditional is true unless <m>p</m> is true and <m>q</m> is false.</p>
        </definition><idx><h>Conditional Statement</h></idx>

        <p>In mathematics/logic the truth table for a conditional statement is given in <xref ref="logic-table-conditional" />.</p>

        <table xml:id="logic-table-conditional">
            <title>Truth table for a conditional statement</title>
            <tabular halign="center" top="medium">
                <row bottom="medium">
                    <cell><m>p</m></cell><cell><m>q</m></cell><cell><m>p \rightarrow q</m></cell>
                </row>
                <row>
                    <cell>T</cell><cell>T</cell><cell>T</cell>
                </row>                            <row>
                    <cell>T</cell><cell>F</cell><cell>F</cell>
                </row>                                        
                <row>
                    <cell>F</cell><cell>T</cell><cell>T</cell>
                </row>                
                <row bottom="medium">
                    <cell>F</cell><cell>F</cell><cell>T</cell>
                </row>
            </tabular>        
        </table>

        <example xml:id="logic-example-conditional">
            <p>Consider the four conditional statements.
                <ol>
                    <li>If Austin is the capital of Texas, then five is greater than two.</li>

                    <li>If Austin is the capital of Texas, then five is less than two.</li>

                    <li>If Austin is not the capital of Texas, then two is less than five.</li>

                    <li>If Austin is not the capital of Texas, then five is less than two.</li>

                </ol>
            Here we see the first three compound sentences are disjunctions which are true, while the disjunction in (4) is false. Here we see by the truth table defining the truth value of a conditional statement that (1), (3),and (4) are true conditional statements while (2) is a false conditional statement. Notice that we can determine the truth value of these statements even though the component statements appear to be totally unrelated in terms of cause and effect!</p>
        </example>

        <p>We emphasize that the student must understand that conditional statements have truth values precisely as assigned by the definition. That is, to determine truth value, we do not need to be able to <q>prove</q> or <q>disprove</q> the consequence from the hypothesis. Certainly <q>proving</q> things will be the ultimate focus of this course, but at this time we are simply discovering the ways of combining statements logically and the resulting truth values of such combinations.</p>

        <p>The last compound statement we will introduce is the biconditional statement.</p>

        <definition xml:id="logic-definition-biconditional">
            <p>Consider two statements <m>p</m> and <m>q</m>. The <em><term>biconditional statement</term></em> is the compound statement <q><m>p</m> if and only if <m>q</m></q> or <q><m>p</m> is equivalent to <m>q</m>,</q> denoted <m>p \leftrightarrow q</m>. Frequently we write <q><m>p</m> iff <m>q</m></q> as a shorthand notation for <q><m>p</m> if and only if <m>q</m>.</q></p>
        </definition><idx><h>Biconditonal Statement</h></idx>

        <p>Two statements, no matter how complicated, are equivalent when they have precisely the same truth value. You can find the truth table for the biconditional statement in <xref ref="logic-table-biconditional" />.</p>

        <table xml:id="logic-table-biconditional">
            <title>Truth table for a biconditional statement</title>
            <tabular halign="center" top="medium">
                <row bottom="medium">
                    <cell><m>p</m></cell><cell><m>q</m></cell><cell><m>p \leftrightarrow q</m></cell>
                </row>
                <row>
                    <cell>T</cell><cell>T</cell><cell>T</cell>
                </row>                            <row>
                    <cell>T</cell><cell>F</cell><cell>F</cell>
                </row>                                        
                <row>
                    <cell>F</cell><cell>T</cell><cell>F</cell>
                </row>                
                <row bottom="medium">
                    <cell>F</cell><cell>F</cell><cell>T</cell>
                </row>
            </tabular>    
        </table>

        <example xml:id="logic-example-biconditional">
            <p>Consider the four biconditional statements.
                <ol>
                    <li>If Austin is the capital of Texas if and only if five is greater than two.</li>

                    <li>If Austin is the capital of Texas if and only if five is less than two.</li>

                    <li>If Austin is not the capital of Texas if and only if two is less than five.</li>

                    <li>If Austin is not the capital of Texas if and only if five is less than two.</li>

                </ol>
            Here we see that (1) and (4) are true biconditional statements while (2) and (3) are false.</p>
        </example>

        <p>Mathematicians  often use other expressions to describe conditional type statements. A few of the most common such expressions are given below.
            <ul>

                <li><m>p \leftrightarrow q</m>: <q><m>p</m> is a necessary and sufficient condition for <m>q</m>.</q></li>

                <li><m>p \rightarrow q</m>: <q><m>p</m> is a sufficient condition for <m>q</m>.</q></li>

                <li><m>q \rightarrow p</m>: <q><m>p</m> is a necessary condition for <m>q</m>.</q></li>

            </ul>
        Since it is easy to confuse these expressions, you must always carefully identify the hypothesis and conclusion before working with any conditional type statement.</p>

        <p>Again we stress that we are not attempting to <q>prove</q> anything yet, but rather only define a compound statement and its truth value in terms of the truth values of the statements used to obtain it.</p>
            
        <p>In order to determine the truth values of more complicated statements, it is critical that you thoroughly understand and remember these five basic truth tables. That is, sufficient time must be spent digesting these tables and examples in order that you need not constantly refer back to the basic tables when working on more difficult ones.</p>

        <p>Before going on to the last definition and fact of this section, we give an example of a more involved statement along with a step-by-step approach to constructing the associated table. We note that there are several methods available for constructing truth tables. We will exhibit one in the example below and employ an alternate approach in the proof of <xref ref="logic-fact-equivalency-of-contrapositive" /> at the end of this section. You should adopt the one most comfortable and appropriate for dealing with the statement at hand.</p>

        <example xml:id="logic-example-truth-table">
            <p>Let us construct the truth table (<xref ref="logic-table-truth-table-example" />) for the statement
                <me>(q \wedge p) \vee [q \wedge (\negate p)].</me>
                After listing the component statements and all possible combinations of truth values associated with them in the table, the remaining compound statements should be given in the order in which they will be considered. This is done much like ordering of operations in an arithmetic problem or an algebraic expression.</p>

            <table xml:id="logic-table-truth-table-example">
                <title>Truth table for <m>(q \wedge p) \vee [q \wedge (\negate p)]</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>q \wedge p</m></cell><cell><m>q \wedge (\negate p)</m></cell><cell><m>(q \wedge p) \vee [q \wedge (\negate p)]</m></cell>
                    </row>
                    <row>
                        <cell>T</cell><cell>T</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>F</cell><cell>F</cell><cell>F</cell>
                    </row>                                        
                    <row>
                        <cell>F</cell><cell>T</cell><cell>T</cell><cell>F</cell><cell>T</cell><cell>T</cell>
                    </row>                
                    <row bottom="medium">
                        <cell>F</cell><cell>F</cell><cell>T</cell><cell>F</cell><cell>F</cell><cell>F</cell>
                    </row>
               </tabular>
            </table>
        </example>

        <p>We now give a final definition that relates conditional statements and negation.</p>

        <definition xml:id="logic-definition-converse-inverse-contrapositive">
            <p>Consider two statements <m>p</m> and <m>q</m>. The statement <m>q \rightarrow p</m> is the <em><term>converse</term></em> of <m>p \rightarrow q</m>. The statement <m>\negate q \rightarrow \negate p</m> is the <em><term>contrapositive</term></em> of <m>p \rightarrow q</m>. The statement <m>\negate p \rightarrow \negate q</m> is the <em><term>inverse</term></em> of <m>p \rightarrow q</m>.</p>
        </definition><idx><h>Converse</h></idx><idx><h>Inverse</h></idx><idx><h>Contrapositive</h></idx>

        <p>It is worth noting that the converse of the inverse is the contrapositive. You should also note that the terms <q>inverse</q> and <q>negation</q> are not interchangeable.</p>

        <remark>
            <title>About Notation</title>
            <p>You should be aware that there are conventions governing the use or lack of use of parentheses in logical statements that are similar to those used to interpret algebraic expressions. Although we sometimes use grouping symbols for emphasis, such grouping symbols are often unnecessary for clarity of meaning.  For example, the expression <m>[ (\negate p) \wedge q] \rightarrow [(\negate r) \vee (\negate s)]</m> could have been written  <m>\negate p \wedge q \rightarrow \negate r \vee \negate s</m>. It is important for you to realize that the negation symbol preceding the <m>p</m> statement applies only to <m>p</m> unless indicated otherwise. However, the grouping symbols in the expressions <m>[ \negate (p \wedge q)] \rightarrow [(\negate r) \rightarrow (\negate s)]</m> and <m>\negate \{(p \wedge q) \rightarrow [(\negate r) \rightarrow (\negate s)]\}</m> produce statements with entirely different meanings.</p>
        </remark>

        <fact xml:id="logic-fact-equivalency-of-contrapositive">
            <p>Consider two statements <m>p</m> and <m>q</m>.
                <ol>

                    <li><m>(p \rightarrow q) \leftrightarrow [(\negate q) \rightarrow (\negate p)]</m>; that is, the conditional is equivalent to its contrapositive.</li>

                    <li><m>[(\negate p) \rightarrow (\negate q)] \leftrightarrow (q \rightarrow p) </m>; that is, the inverse is equivalent to the converse.</li>

                </ol></p>
        </fact>

        <proof>
            <p>To demonstrate the proof of (1) in <xref ref="logic-fact-equivalency-of-contrapositive" />, we need only examine the corresponding truth table <xref ref="logic-table-equivalency-of-contrapositive" />.  Since the last two columns are the same, the conditional statement and its contrapostive are equivalent.</p>

            <table xml:id="logic-table-equivalency-of-contrapositive">
                <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>\negate q \rightarrow \negate p</m></cell>
                    </row>
                    <row>
                        <cell>T</cell><cell>T</cell><cell>F</cell><cell>F</cell><cell>T</cell><cell>T</cell>
                    </row>                            <row>
                        <cell>T</cell><cell>F</cell><cell>F</cell><cell>T</cell><cell>F</cell><cell>F</cell>
                    </row>                                        
                    <row>
                        <cell>F</cell><cell>T</cell><cell>T</cell><cell>F</cell><cell>T</cell><cell>T</cell>
                    </row>                
                    <row bottom="medium">
                        <cell>F</cell><cell>F</cell><cell>T</cell><cell>T</cell><cell>T</cell><cell>T</cell>
                    </row>
               </tabular>
                <title>Truth table for <m>(p \rightarrow q) \leftrightarrow (\negate q \rightarrow \negate p)</m></title>
            </table>

            <p>We will leave the proof of (2) as an exercise.</p>
        </proof>

        <exercises>

            <exercise>
                <statement>
                    <p>Translate the following English statements using propositional notation.
                        <ol>
                            
                            <li>An integer is odd if and only if its square is odd.</li>

                            <li>If I do not study, then I will fail this class.</li>

                            <li>Either I will go shopping or I will go to a movie.</li>

                            <li>I was well qualified, but I did not get the job.</li>

                            <li>If <m>n</m> is an integer, then <m>n</m> is even or <m>n</m> is odd.</li>

                            <li>The square of an even integer is an even integer.</li>

                        </ol></p>
                </statement>
            </exercise>
            
            <exercise>
                <statement>
                    <p>Negate each of the following statements. (Refer to the definition of negation.)
                        <ol>

                            <li>A positive number is larger than zero.</li>

                            <li>If today is Saturday, then I do not have to go to work.</li>

                            <li>Dogs can bark and cats can climb trees.</li>

                            <li>If <m>x^2 - 9 = 0</m>,then either <m>x = 3</m> or <m>x = -3</m>.</li>

                        </ol>
                    Note: The difficulties of negating compound statements will be vastly simplified by the tautologies studied in the next section.</p>
                </statement>
            </exercise>  

            <exercise>
                <statement>
                    <p>For the conditional statements given below, give the converse, the inverse, and the contrapositive.
                        <ol>

                            <li>If I teach third grade, then I am an elementary school teacher.</li>

                            <li>If I do not get to class on time, then I will not be allowed to take the exam.</li>

                            <li>I will return the calls and dictate the letter when I arrive at the office.</li>

                            <li>If <m>(x + 1)(x - 4) = 0</m>, then <m>x = -1</m> or <m>x = 4</m>.</li>

                            <li>If a number has a factor of <m>4</m>, then it has a factor of <m>2</m>.</li>

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

            <exercise>
                <statement>
                    <p>Restate the following in a logically equivalent form.
                        <ol>

                            <li>It is not true that both today is Wednesday and the month is June.</li>

                            <li>It is not true that yesterday I both ate breakfast and watched television.</li>

                            <li>It is not raining, or it is not July.</li>

                        </ol></p>
                </statement>
            </exercise>            
          
            <exercise>
                <statement>
                    <p>In the following statements, remove those grouping symbols which are unnecessary for clarity of meaning.
                        <ol>

                            <li><m>p \vee [(\negate p) \wedge q]</m></li>

                            <li><m>[\negate(p \rightarrow q)] \wedge q</m></li>

                            <li><m>[p \wedge (\negate q)] \vee (p \wedge q)</m></li>

                            <li><m>\{  \negate[ p \vee (\negate r) ] \vee (q \wedge p)\} \rightarrow p</m></li>

                        </ol></p>
                </statement>
            </exercise>            
          
            <exercise>
                <statement>
                    <p>Construct truth tables for the following compound statements.
                    	<ol>

                    		<li><m>p \vee (\negate p \wedge q)</m></li>

                    		<li><m>\negate(p \rightarrow q) \wedge q</m></li>

                    		<li><m>(p \, \wedge \negate q) \vee (p \wedge q)</m></li>

                    		<li><m>[ \negate(p \, \vee \negate r) \wedge (p \vee q)] \rightarrow p</m></li>


                    	</ol></p>
                </statement>
            </exercise>            
          
            <exercise>
                <statement>
                    <p>For integers <m>x</m> and <m>y</m>, find the inverse, the converse, the contrapositive, and the negation of each of the following statements.
                    	<ol>

                    		<li>If <m>x = 3</m>, then <m>x^4 = 81</m>.</li>

                    		<li>If <m>x \gt 0</m>, then <m>x \neq -4</m>.</li>

                    		<li>If <m>x</m> is odd and <m>y</m> is even, then <m>xy</m> is even.</li>

                    		<li>If <m>x^2 = x</m>, then either <m>x = 0</m> or <m>x = 1</m>.</li>

                    		<li>If <m>xy \neq 0</m>, then <m>x \neq 0</m> and <m>y \neq 0</m>.</li>

                    	</ol></p>
                </statement>
            </exercise>            
          
            <exercise>
                <statement>
                    <p>Give two examples from mathematics which satisfy the given conditions.
                    	<ol>

                    		<li>A statement and its converse that are both true.</li>

                    		<li>A statement that is true, but its converse is false.</li>

                    		<li>A biconditional statement that is true.</li>

                    		<li>A biconditional statement that is false.</li>

                    	</ol></p>
                </statement>
            </exercise>            
          
            <exercise>
                <statement>
                    <p>Decide if the conditional statements are true or false.
                    	<ol>

                    		<li>If <m>n</m> is a natural number, then the last digit of <m>n^4</m> is <m>0</m>, <m>1</m>, <m>5</m>, or <m>6</m>.</li>

                    		<li>If the last digit of a natural number is <m>0</m>, <m>1</m>, <m>5</m>, or <m>6</m>, then it is a fourth power of some natural number.</li>

                    		<li><m>n</m> is a natural number only if <m>n + 1</m> is a whole number.</li>

                    		<li><m>n + 1</m> is a whole number if <m>n</m> is a natural number.</li>

                    	</ol></p>
                </statement>
            </exercise>            
          
            <exercise>
                <statement>
                    <p>Let <m>m</m> and <m>n</m> be integers and consider the statement <m>p \rightarrow q</m> given by, <q>If <m>m +
                    n</m> is even, then <m>m</m> and <m>n</m> are even.</q>
                		<ol>

                			<li>Express the contrapositive, converse, and inverse of the given conditional.</li>

                			<li>For the given conditional or any statements in part (a) that are false, give a counterexample.</li>

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

    </section>

    <section xml:id="logic-section-tautologies">
    	<title> Tautologies, Contradictions, &amp; Quantifiers</title>
    	
    	<p>By definition, a simple statement is either true or false. In mathematics/logic, statements which are always true or always false are of great value, but the greatest benefit occurs when dealing with compound statements fitting this description. We give the formal definitions below.</p>

        <definition xml:id="logic-definition-tautology">
        	<p>A compound statement which is always true is called a <em><term>tautology</term></em>, while a compound statement which is always false is called a <em><term>contradiction</term></em>.</p>
        </definition><idx><h>Tautology</h></idx>

        <example xml:id="logic-example-contradiction">
            <p>The statement <m>p \leftrightarrow \negate p</m> is a contradiction since its truth table indicates this statement is always false (<xref ref="logic-table-contradiction" />).  That is, a statement and its negation can never have the same truth value.</p>

            <table xml:id="logic-table-contradiction">
                           <tabular halign="center" top="medium">
                <row bottom="medium">
                    <cell><m>p</m></cell><cell><m>\negate p</m></cell><cell><m>p \leftrightarrow \negate p</m></cell>
                </row>
                            
                <row>
                    <cell>T</cell><cell>F</cell><cell>F</cell>
                </row>                
                <row bottom="medium">
                    <cell>F</cell><cell>T</cell><cell>F</cell>
                </row>
           </tabular>
                <title>Truth table for <m>p \leftrightarrow \negate p</m></title>
            </table>
        </example>

        <example xml:id="logic-example-tautology">
            <p>The statement <m>p \leftrightarrow \negate(\negate p)</m> is a tautology since its truth table indicates this statement is always true (<xref ref="logic-table-tautology" />).  Thus, the double negation of a statement is equivalent to the original statement.</p>

            <table xml:id="logic-table-tautology">
                           <tabular halign="center" top="medium">
                <row bottom="medium">
                    <cell><m>p</m></cell><cell><m>\negate p</m></cell><cell><m>\negate( \negate p)</m></cell><cell><m>p \leftrightarrow \negate(\negate p)</m></cell>
                </row>
                            
                <row>
                    <cell>T</cell><cell>F</cell><cell>T</cell><cell>T</cell>
                </row>                
                <row bottom="medium">
                    <cell>F</cell><cell>T</cell><cell>F</cell><cell>T</cell>
                </row>
           </tabular>
                <title>Truth table for <m>p \leftrightarrow \negate(\negate p)</m></title>
            </table>

        </example>

        <p>The following theorem enumerates a list of tautologies which will be useful to us. The proofs will be left as exercises.</p>

        <theorem xml:id="logic-theorem-tautology">
        	<p>The following are tautologies. Statements (1)<ndash/>(13) are basic properties, while (14)<ndash/>(22) can be considered additional laws.
        		<ol>

        			<li><m>p \leftrightarrow p</m></li>

        			<li><m>p \leftrightarrow \negate(\negate p)</m></li>

        			<li><m>[\negate(p \vee q)] \leftrightarrow [(\negate p) \wedge (\negate q)]</m></li>

        			<li><m>[\negate(p \wedge q)] \leftrightarrow [(\negate p) \vee (\negate q)]</m></li>


        			<li><m>[\negate(p \rightarrow q)]  \leftrightarrow [p \, \wedge (\negate q)] </m></li>

        			<li><m>[\negate(p \leftrightarrow q)]  \leftrightarrow \{ [p \, \wedge \negate q)] \vee [q \, \wedge \negate p)] \} </m></li>

        			<li><m>( p \vee q ) \leftrightarrow (\negate p \rightarrow q)</m></li>


        			<li><m>(p \rightarrow q)  \leftrightarrow ( \negate q \rightarrow \negate p ) </m></li>

        			<li><m>( \negate p \rightarrow \negate q )  \leftrightarrow  (q \rightarrow p) </m></li>

        			<li><m>[(p \rightarrow q) \wedge (q \rightarrow p)] \leftrightarrow (p \leftrightarrow q)</m></li>

        			<li><m>\{(\negate p) \rightarrow [ q \wedge (\negate q)] \} \rightarrow p</m></li>

        			<li><m>(p \leftrightarrow q) \rightarrow [(r \wedge p) \rightarrow (r \wedge q)]</m></li>

        			<li><m>(p \leftrightarrow q) \rightarrow [(r \vee p) \leftrightarrow (r \vee q)]</m></li>

        			<li><m>(p \leftrightarrow q) \leftrightarrow (q \leftrightarrow p)</m></li>

        			<li><m>(p \wedge q) \leftrightarrow (q \wedge p)</m></li>

        			<li><m>(p \vee q) \leftrightarrow (q \vee p)</m></li>

        			<li><m>[(p \rightarrow p) \wedge (q \rightarrow r)] \rightarrow (p \rightarrow r)</m></li>

        			<li><m>[(p \leftrightarrow p) \wedge (q \leftrightarrow r)] \rightarrow (p \rightarrow r)</m></li>

        			<li><m>[p \vee (q \wedge r)] \leftrightarrow [(p \vee q) \wedge (p \vee r)]</m></li>

        			<li><m>[p \wedge (q \vee r)] \leftrightarrow [(p \wedge q) \vee (p \wedge r)]</m></li>


        			<li><m>[p \vee (q \vee r)] \leftrightarrow [(p \vee q) \vee r]</m></li>

        			<li><m>[p \wedge (q \wedge r)] \leftrightarrow [(p \wedge q) \wedge r]</m></li>
        		</ol></p>
        </theorem>

        <paragraphs>
            <title>Historical Note</title>
            <p>Augustus De Morgan (27 June 1806<ndash/>18 March 1871) was a British mathematician and logician. Use internet and/or library resources to research his major contributions to the fields of mathematics and logic, specifically De Morgan's Laws.</p><idx><h>Augustus De Morgan</h></idx>
        </paragraphs>
        
        <p>The following theorem lists some useful contradictions, and again, the proof requires construction of the appropriate truth tables and is left to the exercises.</p>

        <theorem xml:id="logic-theorem-contradiction">
        	<p>The following statements are contradictions.
        		<ol>

        			<li><m>(p \rightarrow q) \wedge (p \wedge \negate q)</m></li>

        			<li><m>[(p \vee q)  \wedge \negate p] \wedge \negate q</m></li>

        			<li><m>(p \wedge q) \wedge \negate p</m></li>

        		</ol></p>
        </theorem>

        <p>You should be aware that the list of possible tautologies and contradictions we could have chosen is virtually endless. We have simply chosen those which will be of most benefit to us later.</p>

        <p>As a final example we will provide the following example of a truth table involving three statements.</p>

        <example xml:id="logic-example-three-statements">
            <p>The following truth table (<xref ref="logic-table-three-statements" />) can be used to verify the statement
            	<me>[( p \rightarrow q ) \vee r] \leftrightarrow [(p \; \wedge \negate q) \rightarrow r].</me>
            	Since the columns for <m>( p \rightarrow q ) \vee r</m> and <m>(p \; \wedge \negate q) \rightarrow r</m> match, we have a tautology.</p>

            <table xml:id="logic-table-three-statements">
                           <tabular halign="center" top="medium">
                <row bottom="medium">
                    <cell><m>p</m></cell>
                    <cell><m>q</m></cell>
                    <cell><m>r</m></cell>
                    <cell><m>p \rightarrow q</m></cell>
                    <cell><m>( p \rightarrow q ) \vee r</m></cell>
                    <cell><m>\negate q</m></cell>
                    <cell><m>p \; \wedge \negate q</m></cell>
                    <cell><m>(p \; \wedge \negate q) \rightarrow r</m></cell>
                </row>             
                <row>
                    <cell>T</cell><cell>T</cell><cell>T</cell><cell>T</cell><cell>T</cell><cell>F</cell><cell>F</cell><cell>T</cell>
                </row>                
                <row>
                    <cell>T</cell><cell>T</cell><cell>F</cell><cell>T</cell><cell>T</cell><cell>F</cell><cell>F</cell><cell>T</cell>
                </row>                           
                <row>
                    <cell>T</cell><cell>F</cell><cell>T</cell><cell>F</cell><cell>T</cell><cell>T</cell><cell>T</cell><cell>T</cell>
                </row>                           
                <row>
                    <cell>T</cell><cell>F</cell><cell>F</cell><cell>F</cell><cell>F</cell><cell>T</cell><cell>T</cell><cell>F</cell>
                </row>                           
                <row>
                    <cell>F</cell><cell>T</cell><cell>T</cell><cell>T</cell><cell>T</cell><cell>F</cell><cell>F</cell><cell>T</cell>
                </row>                           
                <row>
                    <cell>F</cell><cell>T</cell><cell>F</cell><cell>T</cell><cell>T</cell><cell>F</cell><cell>F</cell><cell>T</cell>
                </row>                           
                <row>
                    <cell>F</cell><cell>F</cell><cell>T</cell><cell>T</cell><cell>T</cell><cell>T</cell><cell>F</cell><cell>T</cell>
                </row>                           
                <row bottom="medium">
                    <cell>F</cell><cell>F</cell><cell>F</cell><cell>T</cell><cell>T</cell><cell>T</cell><cell>F</cell><cell>T</cell>
                </row>
           </tabular>
                <title>Truth table for <m>[( p \rightarrow q ) \vee r] \leftrightarrow [(p \; \wedge \negate q) \rightarrow r]</m></title>
            </table>

        </example>

        <exercises>

            <exercise>
                <statement>
                    <p>Verify each of the following using truth tables.
                        <ol>
                            
                            <li>Statements (1)<mdash/>(13) of <xref ref="logic-theorem-tautology" />.</li>

                            <li>Statements (14)<mdash/>(22) of <xref ref="logic-theorem-tautology" />.</li>

                        </ol></p>
                </statement>
            </exercise>
            
            <exercise>
                <statement>
                    <p>Verify each of the following using truth tables.
                        <ol>

                            <li>Part (1) of <xref ref="logic-theorem-contradiction" />.</li>

                            <li>Part (2) of <xref ref="logic-theorem-contradiction" />.</li>

                            <li>Part (3) of <xref ref="logic-theorem-contradiction" />.</li>

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

            <exercise xml:id="exercise-logic-negation">
                <statement>
                    <p>Using the appropriate tautologies from <xref ref="logic-theorem-tautology" />, negate the following statements.
                        <ol>

                            <li>A foot has 12 inches and a yard has three feet.</li>

                            <li>Either I will get a job or I will not be able to pay my bills.</li>

                            <li>If you study logic one hour per day, then you will make an A in the course.</li>

                            <li>If <m>x^2 - 5x + 6 = 0</m>, then <m>x - 3 = 0</m> or <m>x - 2 = 0</m>.</li>

                            <li>An integer <m>m</m> is odd if and only if <m>m^2</m> is odd.</li>

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

                            <li>I will call home if I win the game.</li>

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

            <exercise>
                <statement>
                    <p>State the converse, inverse, and contrapositive of the statements indicated below.
                        <ol>

                            <li>The statement in (c) of <xref ref="exercise-logic-negation" />.</li>

                            <li>The statement in (d) of <xref ref="exercise-logic-negation" />.</li>

                            <li>The statement in (f) of <xref ref="exercise-logic-negation" />.</li>

                            <li>The statement in (g) of <xref ref="exercise-logic-negation" />.</li>

                        </ol></p>
                </statement>
            </exercise>            
          
            <exercise>
                <statement>
                    <p>Justify why each of the following are true by way of a truth table and a brief paragraph explaining what the statement means.
                        <ol>

                            <li><m>[\negate p \wedge (p \vee q)] \rightarrow q</m></li>

                            <li><m>[(p \rightarrow q) \wedge \negate q] \rightarrow \negate p</m></li>

                            <li><m>[\negate p \rightarrow (q \wedge \negate q)] \rightarrow p</m></li>

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

    </section>

    <section xml:id="logic-section-quantifiers">
        <title>Propositional Functions and Quantifiers</title>

        <p>In mathematics we frequently wish to consider sentences (propositions) which involve variables. Since for different values of the variables (called propositional variables) we get different propositions with possibly different truth values, we call such sentences <em><term>propositional functions</term></em> or <em><term>open sentences</term></em>.</p><idx><h>Propositional functions</h></idx><idx><h>Open sentences</h></idx>

         <example xml:id="logic-example-variable-x">
            <p>For each real number <m>x</m> consider the sentence <m>x^2 + x = 1</m>.  Thus, <m>x^2 + x = 1</m> is a propositional function which has different truth values. The proposition is true for <m>x = 1</m> and <m>x = -2</m> and false for all other values of the propositional variable.</p>
        </example>

        <p>We can limit propositional functions by prefixing various expressions we call quantifiers, the most important of which are existential quantifiers and universal quantifiers. Phrases such as
            <ul>

                <li><q>there exists a value <m>x</m></q></li>

                <li><q>there are <m>x</m>, <m>y</m>, and <m>z</m></q></li>

                <li><q>for some values of <m>x</m></q></li>

                <li><q>at least one value of <m>x</m></q></li>

            </ul>
        make use of <em><term>existential quantifiers</term></em>.  On the other hand, phrases such as
            <ul>
                <li><q>for each value of <m>x</m></q></li>

                <li><q>for every value of <m>x</m></q></li>

                <li><q>for all values of <m>x</m></q></li>

                <li><q>no value of <m>x</m></q></li>
                
            </ul>
        are called <em><term>universal quantifiers</term></em>.</p><idx><h>Universal quantifiers</h></idx><idx><h>Existential quantifiers</h></idx>

         <example xml:id="logic-example-quatifiers">
            <p>For each real number <m>x</m> consider the propositional function <m>p(x)</m> that states <m>x^2 + x = 1</m>. We can alter that propositional function using the two types of quantifiers.
                <ol>

                    <li>There exists <m>x</m> such that <m>p(x)</m> is true.</li>

                    <li>For all <m>x</m>, <m>p(x)</m> is true.</li>

                    </ol>
            Clearly, (1) is true and (2) is false.</p>
        </example>

        <p>Notationally, we will let <m>p(x)</m> be a propositional function which states <m>p</m> is true for each <m>x</m>. Using the existential quantifier, we change <m>p(x)</m> into a proposition, namely <q>There exists <m>x</m> such that <m>p(x)</m>.</q>  In mathematics, <q>there exists</q> is replaced by the symbol <m>\exists</m>, and we replace the statement above by <m>(\exists x)(p(x))</m>, which is read <q>There exists <m>x</m> such that <m>p(x)</m> is true.</q>Similarly, we use the notation for the universal quantifier, <m>\forall</m>, and we have the proposition <m>(\forall x)(p(x))</m>, which is read <q>For all <m>x</m>, <m>p(x)</m> is true.</q></p>

        <example xml:id="logic-example-quantifier-notation">
            <p>Consider all states in the USA and the propositional function <m>p(x)</m>, which states that <m>x</m> is a state which borders on the Pacific Ocean. The proposition <m>(\forall x)(p(x))</m> is false, while the proposition <m>(\exists x)(p(x))</m>, is true.</p>
        </example>

        <remark>
            <title>Caution!</title>
            <p>Be careful when using the symbols <m>\exists</m> and <m>\forall</m>.  While their use is quite common in logic, it is very easy to write confusing sentences.  You will rarely see these symbols used in an algebra or calculus textbook.  You may wish to avoid using these symbols for the time being.</p>
        </remark>

        <p>General forms of qualified statements with their negations can be found in <xref ref="logic-table-quantifier-negation" />.</p>

        <table xml:id="logic-table-quantifier-negation">
            <title>Truth table for negation</title>
            <tabular halign="center" top="medium">
                <row bottom="medium">
                    <cell>Statement</cell><cell>Negation</cell>
                </row>    
                <row>
                    <cell>Some <m>a</m> are <m>b</m>.</cell><cell>No <m>a</m> is <m>b</m>.</cell>
                </row>                
                <row>
                    <cell>Some <m>a</m> are not <m>b</m>.</cell><cell>All <m>a</m> are <m>b</m>.</cell>
                </row>           
                <row>
                    <cell>All <m>a</m> are <m>b</m>.</cell><cell>Some <m>a</m> are not <m>b</m>.</cell>
                </row>                           <row bottom="medium">
                    <cell>No <m>a</m> is <m>b</m>.</cell><cell>Some <m>a</m> are <m>b</m>.</cell>
                </row>
           </tabular>
        </table>

        <exercises>

            <exercise>
                <statement>
                    <p>Write each of the following statements in <q>if-then</q> form.
                        <ol>
                            
                            <li>Every figure that is a square is a rectangle.</li>

                            <li>All integers are rational numbers.</li>

                            <li>Figures with exactly <m>3</m> sides may be triangles.</li>

                            <li>It rains only if it is cloudy.</li>

                        </ol></p>
                </statement>
            </exercise>
            
            <exercise>
                <statement>
                     <p>The open sentence <q><m>x^2 + 8 = 6x</m>,</q> can be made either true or false by using different quantifiers. For example, <q>For some whole number <m>x</m>, <m>x^2 + 8 = 6x</m></q> is true, since <m>x = 4</m> or <m>x = 2</m> make the equation true; however, <q>For all whole numbers <m>x</m>. <m>x^2 + 8 = 6x</m>,</q> is false since the equation is false for the whole number <m>x = 0</m> (and for countless other values of <m>x</m>).</p>

                    <p>Use an appropriate quantifier to make each of the following open sentences true, where <m>x</m> is a whole number. Then use quantifiers to make each statement false.
                        <ol>
                            
                            <li><m>x + 5 = 8</m></li>

                            <li><m>x + x^2 = x(x + 1)</m></li>

                            <li><m>x \cdot 1 = x \cdot 3</m></li>

                            <li><m>x^2 + 1 = 0</m></li>

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

            <exercise xml:id="exercise-logic-negation-quantifier">
                <statement>
                    <p>Negate the following statements.
                        <ol>

                            <li>There exists at least one real number <m>x</m> such that <m>x^2 = 9</m>.</li>

                            <li>There is no real number <m>x</m> that makes the sentence <m>x^2 = -1</m> true.</li>

                            <li>Some students attend night school.</li>

                            <li>No children are allowed in this building.</li>

                            <li>There is some number that is both odd and even.</li>

                            <li>All college students are math or engineering majors.</li>

                            <li>For all real numbers <m>x</m>, if <m>x</m> is positive, then <m>-x</m> is negative.</li>

                            <li>Some cars are red, and all students take math.</li>

                            <li>There are some people who go to school in the morning and work in the afternoons.</li>

                            <li>Not all numbers are rational and positive.</li>

                            <li>All dogs have <m>4</m> legs.</li>

                            <li>Not all rectangles are squares.</li>

                        </ol></p>
                </statement>
            </exercise>                      
          
            <exercise>
                <statement>
                    <p>Find the negation of each of the following.
                        <ol>

                            <li><m>p \wedge (q \vee r)</m></li>

                            <li><m>\negate p \wedge (q \rightarrow p)</m></li>

                            <li><m>[p \wedge (q \rightarrow r)] \vee (\negate q \wedge p)</m></li>

                            <li><m>x^2</m> is even only if <m>x</m> is even.</li>

                            <li>There is an integer <m>x</m> such that <m>x/2</m> is an integer, and for every integer <m>y</m>, <m>x/(2y)</m> is not an integer.</li>

                            <li>For every postive integer <m>x</m>, either <m>x</m> is prime or <m>x^2 + 1</m> is prime.</li>

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

    </section>

</chapter>