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Copyright, ii
Preface, xi
0. Communicating Mathematics, 1
Learning Mathematics, 1
What Others Have Said about Writing, 3
Mathematical Writing, 5
Using Symbols, 6
Writing Mathematical Expressions, 8
Common Words and Phrases in Mathematics, 9
Some Closing Comments about Writing, 12
1. Sets, 13
1.1. Describing a Set, 13
1.2. Subsets, 16
1.3. Set Operations, 19
1.4. Indexed Collections of Sets, 22
1.5. Partitions of Sets, 25
1.6. Cartesian Products of Sets, 26
Exercises for Chapter 1, 27
Additional Exercises for Chapter 1, 31
2. Logic, 33
2.1. Statements, 33
2.2. The Negation of a Statement, 35
2.3. The Disjunction and Conjunction of Statements, 37
2.4. The Implication, 38
2.5. More on Implications, 40
2.6. The Biconditional, 42
2.7. Tautologies and Contradictions, 45
2.8. Logical Equivalence, 47
2.9. Some Fundamental Properties of Logical Equivalence, 48
2.10. Quantified Statements, 50
2.11. Characterizations of Statements, 56
Exercises for Chapter 2, 57
Additional Exercises for Chapter 2, 64
3. Direct Proof and Proof by Contrapositive, 67
3.1. Trivial and Vacuous Proofs, 68
3.2. Direct Proofs, 70
3.3. Proof by Contrapositive, 74
3.4. Proof by Cases, 78
3.5. Proof Evaluations, 81
Exercises for Chapter 3, 83
Additional Exercises for Chapter 3, 85
4. More on Direct Proof and Proof by Contrapositive, 87
4.1. Proofs Involving Divisibility of Integers, 87
4.2. Proofs Involving Congruence of Integers, 91
4.3. Proofs Involving Real Numbers, 93
4.4. Proofs Involving Sets, 96
4.5. Fundamental Properties of Set Operations, 99
4.6. Proofs Involving Cartesian Products of Sets, 100
Exercises for Chapter 4, 101
Additional Exercises for Chapter 4, 104
5. Existence and Proof by Contradiction, 107
5.1. Counterexamples, 107
5.2. Proof by Contradiction, 111
5.3. A Review of Three Proof Techniques, 116
5.4. Existence Proofs, 118
5.5. Disproving Existence Statements, 122
Exercises for Chapter 5, 124
Additional Exercises for Chapter 5, 125
6. Mathematical Induction, 129
6.1. The Principle of Mathematical Induction, 129
6.2. A More General Principle of Mathematical Induction, 138
6.3. Proof by Minimum Counterexample, 144
6.4. The Strong Principle of Mathematical Induction, 146
Exercises for Chapter 6, 150
Additional Exercises for Chapter 6, 152
7. Prove or Disprove, 155
7.1. Conjectures in Mathematics, 155
7.2. Revisiting Quantified Statements, 158
7.3. Testing Statements, 163
7.4. A Quiz of “Prove or Disprove” Problems, 167
Exercises for Chapter 7, 169
Additional Exercises for Chapter 7, 172
8. Equivalence Relations, 175
8.1. Relations, 175
8.2. Properties of Relations, 176
8.3. Equivalence Relations, 178
8.4. Properties of Equivalence Classes, 181
8.5. Congruence Modulo n, 185
8.6. The Integers Modulo n, 189
Exercises for Chapter 8, 192
Additional Exercises for Chapter 8, 195
9. Functions, 197
9.1. The Definition of Function, 197
9.2. The Set of All Functions from A to B, 200
9.3. One-to-One and Onto Functions, 200
9.4. Bijective Functions, 203
9.5. Composition of Functions, 205
9.6. Inverse Functions, 209
9.7. Permutations, 212
Exercises for Chapter 9, 213
Additional Exercises for Chapter 9, 216
10. Cardinalities of Sets, 221
10.1. Numerically Equivalent Sets, 222
10.2. Denumerable Sets, 223
10.3. Uncountable Sets, 229
10.4. Comparing Cardinalities of Sets, 234
10.5. The Schröder–Bernstein Theorem, 237
Exercises for Chapter 10, 241
Additional Exercises for Chapter 10, 243
11. Proofs in Number Theory, 245
11.1. Divisibility Properties of Integers, 245
11.2. The Division Algorithm, 246
11.3. Greatest Common Divisors, 250
11.4. The Euclidean Algorithm, 252
11.5. Relatively Prime Integers, 254
11.6. The Fundamental Theorem of Arithmetic, 256
11.7. Concepts Involving Sums of Divisors, 259
Exercises for Chapter 11, 260
Additional Exercises for Chapter 11, 263
12. Proofs in Calculus, 267
12.1. Limits of Sequences, 267
12.2. Infinite Series, 273
12.3. Limits of Functions, 277
12.4. Fundamental Properties of Limits of Functions, 284
12.5. Continuity, 289
12.6. Differentiability, 291
Exercises for Chapter 12, 293
Additional Exercises for Chapter 12, 295
13. Proofs in Group Theory, 297
13.1. Binary Operations, 297
13.2. Groups, 301
13.3. Permutation Groups, 305
13.4. Fundamental Properties of Groups, 308
13.5. Subgroups, 311
13.6. Isomorphic Groups, 313
Exercises for Chapter 13, 317
Additional Exercises for Chapter 13, 321
Solutions to Odd-Numbered Section Exercises, 323
References, 357
Index of Symbols, 359
Index, 361

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