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

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400 Chapter 7. Equivalence Relations What we did for the specific partition in Part (12b) can be done for any partition of a set. So to generalize Part (12b), we let A be a nonempty set and let C be a partition of A. We then define a relation on A as follows: For x; y 2 A, x y if and only if there exists a set U in C such that x 2 U and y 2 U. (c) Prove that is an equivalence relation on the set A. (d) Leta 2 A and let U 2 C such that a 2 U . Prove that Œa D U . 13. Equivalence Relations on a Set of Matrices. The following exercises require a knowledge of elementary linear algebra. We let M n;n .R/ be the set of all n by n matrices with real number entries. (a) Define a relation on M n;n .R/ as follows: For all A; B 2 M n;n .R/, A B if and only if there exists an invertible matrix P in M n;n .R/ such that B D PAP 1 . Is an equivalence relation on M n;n .R/? Justify your conclusion. (b) Define a relation R on M n;n .R/ as follows: For all A; B 2 M n;n .R/, ARB if and only if det.A/ D det.B/. IsR an equivalence relation on M n;n .R/? Justify your conclusion. (c) Let be an equivalence relation on R. Define a relation on M n;n .R/ as follows: For all A; B 2 M n;n .R/, A B if and only if det.A/ det.B/. Is an equivalence relation on M n;n .R/? Justify your conclusion. 7.4 Modular Arithmetic Preview Activity 1 (Congruence Modulo 6) For this activity, we will only use the relation of congruence modulo 6 on the set of integers. 1. Find five different integers a such that a 3.mod 6/ and find five different integers b such that b 4 .mod 6/. That is, find five different integers in Œ3, the congruence class of 3 modulo 6 and five different integers in Œ4, the congruence class of 4 modulo 6. 2. Calculate s D aCb using several values of a in Œ3 and several values of b in Œ4 from Part (1). For each sum s that is calculated, find r so that 0 r
7.4. Modular Arithmetic 401 3. Calculate p D a b using several values of a in Œ3 and several values of b in Œ4 from Part (1). For each product p that is calculated, find r so that 0 r
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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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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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328 Chapter 6. Functions Progress C
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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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450 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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