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Mathematical Proofs: A Transition to Advanced Mathematics

by: Chartrand, Gary; Polimeni, Albert D.; Zhang, Ping

9780201710908

Mathematical Proofs: A Transition to Advanced Mathematics

by: Chartrand, Gary; Polimeni, Albert D.; Zhang, Ping

ISBN 13:
9780201710908
ISBN 10:
0201710900
Edition: 2nd
Format: Hardcover
Copyright: 01/01/2008
Publisher: Addison Wesley
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Summary

Mathematical Proofs is designed to prepare students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise providing solid introductions to relations, functions, and cardinalities of sets.

Communicating Mathematics

01

(12)

Learning Mathematics

01

(2)

What Others Have Said About Writing

03

(1)

Mathematical Writing

04

(1)

Using Symbols

05

(2)

Writing Mathematical Expressions

07

(1)

Common Words and Phrases in Mathematics

08

(3)

Some Closing Comments About Writing

11

(2)

Sets

13

(16)

Describing a Set

13

(2)

Special Sets

15

(1)

Subsets

16

(2)

Set Operations

18

(3)

Indexed Collections of Sets

21

(2)

Partitions of Sets

23

(1)

Cartesian Products of Sets

24

(5)

Exercises for Chapter 1

24

(5)

Logic

29

(22)

Statements

29

(2)

The Negation of a Statement

31

(1)

The Disjunction and Conjunction of Statements

32

(1)

The Implication

33

(2)

More On Implications

35

(1)

The Biconditional

36

(2)

Tautologies and Contradictions

38

(1)

Logical Equivalence

39

(2)

Some Fundamental Properties of Logical Equivalence

41

(1)

Characterizations of Statements

42

(2)

Quantified Statements and Their Negatiors

44

(7)

Exercises for Chapter 2

46

(5)

Direct Proof and Proof By Contrapositive

51

(16)

Trivial and Vacuous Proofs

51

(2)

Direct Proofs

53

(3)

Proof by Contrapositive

56

(4)

Proof by Cases

60

(3)

Proof Evaluations

63

(4)

Exercises for Chapter 3

64

(3)

More on Direct Proof and Proof by Contrapositive

67

(16)

Proofs Involving Divisibility of Integers

67

(3)

Proofs Involving Congruence of Integers

70

(3)

Proofs Involving Real Numbers

73

(1)

Proofs Involving Sets

74

(3)

Fundamental Properties of Set Operations

77

(2)

Proofs Involving Cartesian Products of Sets

79

(4)

Exercises for Chapter 4

80

(3)

Proof by Contradiction

83

(10)

Proof by Contradiction

83

(1)

Examples of Proof by Contradiction

84

(1)

The Three Prisoners Problem

85

(2)

Other Examples of Proof by Contradiction

87

(1)

The Irrationality of √2

87

(1)

A Review of the Three Proof Techniques

88

(5)

Exercises for Chapter S

90

(3)

Prove or Disprove

93

(20)

Conjectures in Mathematics

93

(3)

A Review of Quantifiers

96

(2)

Existence Proofs

98

(2)

A Review of Negations of Quantified Statements

100

(1)

Counterexamples

101

(2)

Disproving Statements

103

(2)

Testing Statements

105

(2)

A Quiz of ``Prove or Disprove'' Problems

107

(6)

Exercises for Chapter 6

108

(5)

Equivalence Relations

113

(22)

Relations

113

(1)

Reflexive, Symmetric, and Transitive Relations

114

(2)

Equivalence Relations

116

(3)

Properties of Equivalence Classes

119

(4)

Congruence Modulo n

123

(4)

The Integers Modulo n

127

(8)

Exercises for Chapter 7

130

(5)

Functions

135

(18)

The Definition of Function

135

(3)

The Set of All Functions From A to B

138

(1)

One-to-one and Onto Functions

138

(2)

Bijective Functions

140

(3)

Composition of Functions

143

(3)

Inverse Functions

146

(3)

Permutations

149

(4)

Exercises for Chapter 8

150

(3)

Mathematical Induction

153

(22)

The Well-Ordering Principle

153

(2)

The Principle of Mathematical Induction

155

(3)

Mathematical Induction and Sums of Numbers

158

(4)

Mathematical Induction and Inequalities

162

(1)

Mathematical Induction and Divisibility

163

(2)

Other Examples of Induction Proofs

165

(1)

Proof By Minimum Counterexample

166

(2)

The Strong Form of Induction

168

(7)

Exercises for Chapter 9

171

(4)

Cardinalities of Sets

175

(22)

Numerically Equivalent Sets

176

(1)

Denumerable Sets

177

(6)

Uncountable Sets

183

(5)

Comparing Cardinalities of Sets

188

(3)

The Schroder-Bernstein Theorem

191

(6)

Exercises for Chapter 10

194

(3)

Proofs in Number Theory

197

(18)

Divisibility Properties of Integers

197

(1)

The Division Algorithm

198

(4)

Greatest Common Divisors

202

(2)

The Euclidean Algorithm

204

(2)

Relatively Prime Integers

206

(2)

The Fundamental Theorem of Arithmetic

208

(2)

Concepts Involving Sums of Divisors

210

(5)

Exercises for Chapter 11

211

(4)

Proofs in Calculus

215

(28)

Limits of Sequences

215

(5)

Infinite Series

220

(4)

Limits of Functions

224

(6)

Fundamental Properties of Limits of Functions

230

(5)

Continuity

235

(2)

Differentiability

237

(6)

Exercises for Chapter 12

239

(4)

Proofs in Group Theory

243

(26)

Binary Operations

243

(4)

Groups

247

(5)

Permutation Groups

252

(3)

Fundamental Properties of Groups

255

(2)

Subgroups

257

(3)

Isomorphic Groups

260

(9)

Exercises for Chapter 13

263

(6)

Answers and Hints to Selected Odd-Numbered Exercises

269

(12)

References

281

(2)

Index of Symbols

283

(2)

Index of Mathematical Terms

285

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