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

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156 Chapter 3. Constructing and Writing Proofs in Mathematics 21. (a) Determine several pairs of integers a and b such that a b .mod 5/. For each such pair, calculate 4a C b, 3a C 2b,and7a C 3b. Are each of the resulting integers congruent to 0 modulo 5? (b) Prove or disprove the following proposition: Let m and n be integers such that .m C n/ 0 .mod 5/ and let a; b 2 Z. Ifa b.mod 5/, then.ma C nb/ 0.mod 5/. 22. Evaluation of proofs See the instructions for Exercise (19) onpage100 from Section 3.1. (a) Proposition. For all integers a and b, if.a C 2b/ 0.mod 3/, then .2a C b/ 0.mod 3/. Proof. We assume a; b 2 Z and .a C 2b/ 0.mod 3/. This means that 3 divides a C 2b and, hence, there exists an integer m such that a C 2b D 3m. Hence, a D 3m 2b. For.2a C b/ 0.mod 3/,there exists an integer x such that 2a C b D 3x. Hence, 2.3m 2b/ C b D 3x 6m 3b D 3x 3.2m b/ D 3x 2m b D x: Since .2m b/ is an integer, this proves that 3 divides .2a C b/ and hence, .2a C b/ 0.mod 3/. (b) Proposition. For each integer m, 5 divides m 5 m . Proof. Let m 2 Z. We will prove that 5 divides m 5 m by proving that m 5 m 0.mod 5/. We will use cases. For the first case, if m 0.mod 5/,thenm 5 0.mod 5/ and, hence, m 5 m 0.mod 5/. For the second case, if m 1 .mod 5/, thenm 5 1 .mod 5/ and, hence, m 5 m .1 1/ .mod 5/, which means that m 5 m 0.mod 5/. For the third case, if m 2 .mod 5/, thenm 5 32 .mod 5/ and, hence, m 5 m .32 2/ .mod 5/, which means that m 5 m 0.mod 5/.
3.5. The Division Algorithm and Congruence 157 Explorations and Activities 23. Using a Contradiction to Prove a Case Is Not Possible. Explore the statements in Parts (a) and (b) by considering several examples where the hypothesis is true. (a) If anintegera is divisible by both 4 and 6, then it divisible by 24. (b) If anintegera is divisible by both 2 and 3, then it divisible by 6. (c) What can you conclude from the examples in Part (a)? (d) What can you conclude from the examples in Part (b)? The proof of the following proposition based on Part (b) uses cases. In this proof, however, we use cases and a proof by contradiction to prove that a certain integer cannot be odd. Hence, it must be even. Complete the proof of the proposition. Proposition. Let a 2 Z. If 2 divides a and 3 divides a, then 6 divides a. Proof: Leta 2 Z and assume that 2 divides a and 3 divides a. We will prove that 6 divides a. Since 3 divides a, there exists an integer n such that a D 3n: The integer n is either even or it is odd. We will show that it must be even by obtaining a contradiction if it assumed to be odd. So, assume that n is odd. (Now complete the proof.) 24. The Last Two Digits of a Large Integer. Notice that 7; 381; 272 72 .mod 100/ since 7; 381; 272 72 D 7; 381; 200, which is divisible by 100. In general, if we start with an integer whose decimal representation has more than two digits and subtract the integer formed by the last two digits, the result will be an integer whose last two digits are 00. This result will be divisible by 100. Hence, any integer with more than 2 digits is congruent modulo 100 to the integer formed by its last two digits. (a) Start by squaring both sides of the congruence 3 4 81 .mod 100/ to prove that 3 8 61 .mod 100/ andthenprovethat3 16 21 .mod 100/. What does this tell you about the last two digits in the decimal representation of 3 16 ? (b) Use the two congruences in Part (24a) and laws of exponents to determine r where 3 20 r .mod 100/ and r 2 Z with 0 r < 100 .What does this tell you about the last two digits in the decimal representation of 3 20 ?
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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 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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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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