Mathematical Reasoning- Writing and Proof, Version 2.1, 2014a

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6 Chapter 1. Introduction to Writing Proofs in Mathematics The conditional statement P ! Q means that Q is true whenever P is true. It says nothing about the truth value of Q when P is false. Using this as a guide, we define the conditional statement P ! Q to be false only when P is true and Q is false, that is, only when the hypothesis is true and the conclusion is false. In all other cases, P ! Q is true. This is summarized in Table 1.1, which is called a truth table for the conditional statement P ! Q. (In Table 1.1, T stands for “true” and F stands for “false.”) P Q P ! Q T T T T F F F T T F F T Table 1.1: Truth Table for P ! Q The important thing to remember is that the conditional statement P ! Q has its own truth value. It is either true or false (and not both). Its truth value depends on the truth values for P and Q, but some find it a bit puzzling that the conditional statement is considered to be true when the hypothesis P is false. We will provide a justification for this through the use of an example. Example 1.3 Suppose that I say “If it is not raining, then Daisy is riding her bike.” We can represent this conditional statement as P ! Q where P is the statement, “It is not raining” and Q is the statement, “Daisy is riding her bike.” Although it is not a perfect analogy, think of the statement P ! Q as being false to mean that I lied and think of the statement P ! Q as being true to mean that I did not lie. We will now check the truth value of P ! Q based on the truth values of P and Q. 1. Suppose that both P and Q are true. That is, it is not raining and Daisy is riding her bike. In this case, it seems reasonable to say that I told the truth and that P ! Q is true. 2. Suppose that P is true and Q is false or that it is not raining and Daisy is not riding her bike. It would appear that by making the statement, “If it is not
1.1. Statements and Conditional Statements 7 raining, then Daisy is riding her bike,” that I have not told the truth. So in this case, the statement P ! Q is false. 3. Now suppose that P is false and Q is true or that it is raining and Daisy is riding her bike. Did I make a false statement by stating that if it is not raining, then Daisy is riding her bike? The key is that I did not make any statement about what would happen if it was raining, and so I did not tell a lie. So we consider the conditional statement, “If it is not raining, then Daisy is riding her bike,” to be true in the case where it is raining and Daisy is riding her bike. 4. Finally, suppose that both P and Q are false. That is, it is raining and Daisy is not riding her bike. As in the previous situation, since my statement was P ! Q, I made no claim about what would happen if it was raining, and so I did not tell a lie. So the statement P ! Q cannot be false in this case and so we consider it to be true. Progress Check 1.4 (Explorations with Conditional Statements) 1. Consider the following sentence: If x is a positive real number, then x 2 C 8x is a positive real number. Although the hypothesis and conclusion of this conditional sentence are not statements, the conditional sentence itself can be considered to be a statement as long as we know what possible numbers may be used for the variable x. From the context of this sentence, it seems that we can substitute any positive real number for x. We can also substitute 0 for x or a negative real number for x provided that we are willing to work with a false hypothesis in the conditional statement. (In Chapter 2, we will learn how to be more careful and precise with these types of conditional statements.) (a) Notice that if x D 3, thenx 2 C 8x D 15, which is negative. Does this mean that the given conditional statement is false? (b) Notice that if x D 4, thenx 2 C 8x D 48, which is positive. Does this mean that the given conditional statement is true? (c) Do you think this conditional statement is true or false? Record the results for at least five different examples where the hypothesis of this conditional statement is true.
Page 3: Mathematical Reasoning Writing and
Page 7 and 8: Contents Note to Students Preface v
Page 9 and 10: Contents v 7.4 Modular Arithmetic .
Page 11 and 12: Note to Students vii Progress Chec
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Page 15 and 16: Preface xi With the exception of S
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2.3. Open Sentences and Sets 59 Exa
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2.3. Open Sentences and Sets 61 Whe
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2.4. Quantifiers and Negations 63
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2.4. Quantifiers and Negations 65 6
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2.4. Quantifiers and Negations 67 N
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2.4. Quantifiers and Negations 73 C
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2.4. Quantifiers and Negations 77 ?
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2.5. Chapter 2 Summary 81 De Morgan
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3.1. Direct Proofs 83 mathematics i
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3.1. Direct Proofs 85 3. This is a
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3.1. Direct Proofs 87 Please rememb
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3.1. Direct Proofs 89 Proof. We ass
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3.1. Direct Proofs 91 The problem i
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3.1. Direct Proofs 93 This means th
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3.1. Direct Proofs 95 4. Write a fi
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3.1. Direct Proofs 97 Let a be an i
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3.1. Direct Proofs 99 The point .a
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3.1. Direct Proofs 101 (a) Proposit
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3.2. More Methods of Proof 103 2. T
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3.2. More Methods of Proof 105 mean
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3.2. More Methods of Proof 107 Prop
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3.2. More Methods of Proof 109 belo
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3.2. More Methods of Proof 111 Nonc
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3.2. More Methods of Proof 113 (b)
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3.2. More Methods of Proof 115 19.
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3.3. Proof by Contradiction 117 A l
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3.3. Proof by Contradiction 119 Wri
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3.3. Proof by Contradiction 121 1.
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3.3. Proof by Contradiction 123 whi
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3.3. Proof by Contradiction 125 r i
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3.3. Proof by Contradiction 127 ? (
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3.3. Proof by Contradiction 129 18.
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3.4. Using Cases in Proofs 131 Proo
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3.4. Using Cases in Proofs 133 The
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3.4. Using Cases in Proofs 135 by 1
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3.4. Using Cases in Proofs 137 Theo
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3.4. Using Cases in Proofs 139 (b)
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3.5. The Division Algorithm and Con
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3.6. Review of Proof Methods 159 Pr
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3.6. Review of Proof Methods 161 Co
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3.6. Review of Proof Methods 163 8.
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3.6. Review of Proof Methods 165 ca
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3.7. Chapter 3 Summary 167 Exercis
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Chapter 4 Mathematical Induction 4.
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4.1. The Principle of Mathematical
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4.3. Induction and Recursion 201 Us
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4.3. Induction and Recursion 207 Ex
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4.3. Induction and Recursion 211 Pr
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4.4. Chapter 4 Summary 213 4.4 Chap
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Chapter 5 Set Theory 5.1 Sets and O
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5.1. Sets and Operations on Sets 21
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5.2. Proving Set Relationships 231
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5.3. Properties of Set Operations 2
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5.4. Cartesian Products 255 is an o
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5.4. Cartesian Products 257 6. For
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5.4. Cartesian Products 259 boundar
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5.4. Cartesian Products 261 Since u
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5.4. Cartesian Products 263 9. Is t
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5.5. Indexed Families of Sets 265 D
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5.5. Indexed Families of Sets 267 W
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5.5. Indexed Families of Sets 269 E
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5.5. Indexed Families of Sets 271 t
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5.5. Indexed Families of Sets 275 9
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5.6. Chapter 5 Summary 277 5.6 Chap
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5.6. Chapter 5 Summary 279 Theorem
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282 Chapter 6. Functions For this f
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284 Chapter 6. Functions 2. Let s b
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286 Chapter 6. Functions Nowhere do
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288 Chapter 6. Functions For each m
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290 Chapter 6. Functions Arrow Diag
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292 Chapter 6. Functions 4. Let f W
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294 Chapter 6. Functions Exploratio
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296 Chapter 6. Functions (a) f.x/ D
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298 Chapter 6. Functions verify tha
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300 Chapter 6. Functions Progress C
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302 Chapter 6. Functions and there
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304 Chapter 6. Functions 5. Let A a
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306 Chapter 6. Functions (a) Calcul
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308 Chapter 6. Functions A f B A g
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310 Chapter 6. Functions Definition
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312 Chapter 6. Functions Let f W A
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314 Chapter 6. Functions An Importa
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316 Chapter 6. Functions Notice tha
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318 Chapter 6. Functions (c) Draw a
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320 Chapter 6. Functions 13. Let A
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322 Chapter 6. Functions Hence, x a
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324 Chapter 6. Functions a A f B g
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326 Chapter 6. Functions Compositio
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330 Chapter 6. Functions Proof of T
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332 Chapter 6. Functions ? 6. Prove
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334 Chapter 6. Functions 6.5 Invers
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336 Chapter 6. Functions 3. If the
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338 Chapter 6. Functions Definition
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340 Chapter 6. Functions that f is
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342 Chapter 6. Functions Theorems a
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344 Chapter 6. Functions Now these
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346 Chapter 6. Functions ? (a) If g
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348 Chapter 6. Functions input and
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350 Chapter 6. Functions (a) fx 2 R
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354 Chapter 6. Functions 3. For eac
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356 Chapter 6. Functions Since x 2
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358 Chapter 6. Functions 3. Let gW
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360 Chapter 6. Functions Onto func
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Chapter 7 Equivalence Relations 7.1
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364 Chapter 7. Equivalence Relation
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Chapter 8 Topics in Number Theory 8
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416 Chapter 8. Topics in Number The
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Chapter 9 Finite and Infinite Sets
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454 Chapter 9. Finite and Infinite
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Appendix A Guidelines for Writing M
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494 Appendix A. Writing Guidelines
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496 Appendix A. Writing Guidelines
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498 Appendix B. Answers for Progres
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Appendix C Answers and Hints for Se
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538 Appendix C. Answers for Exercis
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Appendix D List of Symbols Symbol M
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584 Appendix D. List of Symbols Sym
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586 Index conditional, 33 condition
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588 Index idempotent laws for sets,
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590 Index open, 228 real function,
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Mathematical Reasoning- Writing and Proof, Version 2.1, 2014a

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