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

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36 Chapter 2. Logical Reasoning P :P T F F T P Q P ^ Q T T T T F F F T F F F F P Q P _ Q T T T T F T F T T F F F P Q P ! Q T T T T F F F T T F F T Rather than memorizing the truth tables, for many people it is easier to remember the rules summarized in Table 2.1. Operator Symbolic Form Summary of Truth Values Conjunction P ^ Q True only when both P and Q are true Disjunction P _ Q False only when both P and Q are false Negation :P Opposite truth value of P Conditional P ! Q False only when P is true and Q is false Table 2.1: Truth Values for Common Connectives Other Forms of Conditional Statements Conditional statements are extremely important in mathematics because almost all mathematical theorems are (or can be) stated as a conditional statement in the following form: If “certain conditions are met,” then “something happens.” It is imperative that all students studying mathematics thoroughly understand the meaning of a conditional statement and the truth table for a conditional statement.
2.1. Statements and Logical Operators 37 We also need to be aware that in the English language, there are other ways for expressing the conditional statement P ! Q other than “If P ,thenQ.” Following are some common ways to express the conditional statement P ! Q in the English language: If P ,thenQ. P implies Q. P only if Q. Q if P . Whenever P is true, Q is true. Q is true whenever P is true. Q is necessary for P . (This means that if P is true, then Q is necessarily true.) P is sufficient for Q. (This means that if you want Q to be true, it is sufficient to show that P is true.) In all of these cases, P is the hypothesis of the conditional statement and Q is the conclusion of the conditional statement. Progress Check 2.1 (The “Only If” Statement) Recall that a quadrilateral is a four-sided polygon. Let S represent the following true conditional statement: If a quadrilateral is a square, then it is a rectangle. Write this conditional statement in English using 1. the word “whenever” 2. the phrase “only if” 3. the phrase “is necessary for” 4. the phrase “is sufficient for” Constructing Truth Tables Truth tables for compound statements can be constructed by using the truth tables for the basic connectives. To illustrate this, we will construct a truth table for .P ^:Q/ ! R. The first step is to determine the number of rows needed. For a truth table with two different simple statements, four rows are needed since there are four different combinations of truth values for the two statements. We should be consistent with how we set up the rows. The way we
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
Page 13 and 14: Preface ix the emphasis is on the f
Page 15 and 16: Preface xi With the exception of S
Page 17 and 18: Preface xiii students with a thorou
Page 19 and 20: Chapter 1 Introduction to Writing P
Page 21 and 22: 1.1. Statements and Conditional Sta
Page 33 and 34: 1.2. Constructing Direct Proofs 15
Page 49 and 50: 1.3. Chapter 1 Summary 31 1.3 Chapt
Page 51 and 52: Chapter 2 Logical Reasoning 2.1 Sta
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Page 99 and 100: 2.5. Chapter 2 Summary 81 De Morgan
Page 101 and 102: 3.1. Direct Proofs 83 mathematics i
Page 103 and 104: 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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4.2. Other Forms of Mathematical In
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4.3. Induction and Recursion 201 Us
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4.3. Induction and Recursion 203 Th
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4.3. Induction and Recursion 205 if
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4.3. Induction and Recursion 207 Ex
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4.3. Induction and Recursion 209 11
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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 273 E
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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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