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

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228 Chapter 5. Set Theory (b) Use set builder notation or the roster method to specify the set of integers that are the sum of four consecutive integers. (c) Specify the set of all natural numbers that can be written as the sum of four consecutive natural numbers. (d) Prove that for each n 2 Z, n is the sum of eight consecutive integers if and only if n 4.mod 8/. (e) Use set builder notation or the roster method to specify the set of integers that are the sum of eight consecutive integers. (f) Specify the set of all natural numbers can be written as the sum of eight consecutive natural numbers. 14. One of the properties of real numbers is the so-called Law of Trichotomy, which states that if a; b 2 R, then exactly one of the following is true: ab. Is the following proposition concerning sets true or false? Either provide a proof that it is true or a counterexample showing it is false. If A and B are subsets of some universal set, then exactly one of the following is true: A B; A D B; B A. Explorations and Activities 15. Intervals of Real Numbers. In previous mathematics courses, we have frequently used subsets of the real numbers called intervals. There are some common names and notations for intervals. These are given in the following table, where it is assumed that a and b are real numbers and a
5.1. Sets and Operations on Sets 229 (a) Is.a; b/ a proper subset of .a; b? Explain. (b) IsŒa; b a subset of .a; C1/? Explain. (c) Use interval notation to describe i. the intersection of the interval Œ 3; 7 with the interval .5; 9; ii. the union of the interval Œ 3; 7 with the interval .5; 9; iii. the set difference Œ 3; 7 .5; 9. (d) Write the set fx 2 R jjxj 0:01g using interval notation. (e) Write the set fx 2 R jjxj >2g as the union of two intervals. 16. More Work with Intervals. For this exercise, use the interval notation described in Exercise 15. (a) Determine the intersection and union of Œ2; 5 and Œ 1; C1/. (b) Determine the intersection and union of Œ2; 5 and Œ3:4; C1/. (c) Determine the intersection and union of Œ2; 5 and Œ7; C1/. Now let a, b, andc be real numbers with a
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Mathematical Reasoning Writing and
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Contents Note to Students Preface v
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Contents v 7.4 Modular Arithmetic .
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Note to Students vii Progress Chec
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Preface ix the emphasis is on the f
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Preface xi With the exception of S
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Preface xiii students with a thorou
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Chapter 1 Introduction to Writing P
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1.1. Statements and Conditional Sta
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1.2. Constructing Direct Proofs 15
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1.3. Chapter 1 Summary 31 1.3 Chapt
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Chapter 2 Logical Reasoning 2.1 Sta
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2.1. Statements and Logical Operato
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2.2. Logically Equivalent Statement
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2.3. Open Sentences and Sets 53 The
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2.3. Open Sentences and Sets 55 5.
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2.3. Open Sentences and Sets 57 in
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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 71 :
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2.4. Quantifiers and Negations 73 C
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2.4. Quantifiers and Negations 75
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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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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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