Mathematical proofs: a transition to advanced mathematics

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Contents......Page 5
0 Communicating Mathematics......Page 16
Learning Mathematics......Page 17
What Others Have Said About Writing......Page 19
Mathematical Writing......Page 20
Using Symbols......Page 21
Writing Mathematical Expressions......Page 23
Common Words and Phrases in Mathematics......Page 25
Some Closing Comments About Writing......Page 27
1.1 Describing a Set......Page 29
1.2 Subsets......Page 33
1.3 Set Operations......Page 36
1.4 Indexed Collections of Sets......Page 39
1.5 Partitions of Sets......Page 42
1.6 Cartesian Products of Sets......Page 43
Exercises for Chapter 1......Page 44
2.1 Statements......Page 52
2.2 The Negation of a Statement......Page 54
2.3 The Disjunction and Conjunction of Statements......Page 56
2.4 The Implication......Page 57
2.5 More on Implications......Page 59
2.6 The Biconditional......Page 62
2.7 Tautologies and Contradictions......Page 64
2.8 Logical Equivalence......Page 66
2.9 Some Fundamental Properties of Logical Equivalence......Page 68
2.10 Quantified Statements......Page 70
2.11 Characterizations of Statements......Page 78
Exercises for Chapter 2......Page 79
3 Direct Proof and Proof by Contrapositive......Page 92
3.1 Trivial and Vacuous Proofs......Page 93
3.2 Direct Proofs......Page 95
3.3 Proof by Contrapositive......Page 99
3.4 Proof by Cases......Page 104
3.5 Proof Evaluations......Page 107
Exercises for Chapter 3......Page 108
4.1 Proofs Involving Divisibility of Integers......Page 114
4.2 Proofs Involving Congruence of Integers......Page 118
4.3 Proofs Involving Real Numbers......Page 120
4.4 Proofs Involving Sets......Page 123
4.5 Fundamental Properties of Set Operations......Page 126
4.6 Proofs Involving Cartesian Products of Sets......Page 128
Exercises for Chapter 4......Page 129
5.1 Counterexamples......Page 135
5.2 Proof by Contradiction......Page 139
5.3 A Review of Three Proof Techniques......Page 145
5.4 Existence Proofs......Page 147
5.5 Disproving Existence Statements......Page 151
Exercises for Chapter 5......Page 152
6.1 The Principle of Mathematical Induction......Page 157
6.2 A More General Principle of Mathematical Induction......Page 166
6.3 Proof by Minimum Counterexample......Page 173
6.4 The Strong Principle of Mathematical Induction......Page 176
Exercises for Chapter 6......Page 180
7.1 Conjectures in Mathematics......Page 185
7.2 Revisiting Quantified Statements......Page 188
7.3 Testing Statements......Page 193
Exercises for Chapter 7......Page 200
8.1 Relations......Page 207
8.2 Properties of Relations......Page 208
8.3 Equivalence Relations......Page 211
8.4 Properties of Equivalence Classes......Page 213
8.5 Congruence Modulo n......Page 217
8.6 The Integers Modulo n......Page 222
Exercises for Chapter 8......Page 225
9.1 The Definition of Function......Page 231
9.2 The Set of All Functions from A to B......Page 234
9.3 One-to-One and Onto Functions......Page 235
9.4 Bijective Functions......Page 237
9.5 Composition of Functions......Page 240
9.6 Inverse Functions......Page 244
9.7 Permutations......Page 247
Exercises for Chapter 9......Page 249
10 Cardinalities of Sets......Page 257
10.1 Numerically Equivalent Sets......Page 258
10.2 Denumerable Sets......Page 259
10.3 Uncountable Sets......Page 265
10.4 Comparing Cardinalities of Sets......Page 270
10.5 The Schröder–Bernstein Theorem......Page 273
Exercises for Chapter 10......Page 277
11.1 Divisibility Properties of Integers......Page 281
11.2 The Division Algorithm......Page 282
11.3 Greatest Common Divisors......Page 286
11.4 The Euclidean Algorithm......Page 287
11.5 Relatively Prime Integers......Page 290
11.6 The Fundamental Theorem of Arithmetic......Page 292
11.7 Concepts Involving Sums of Divisors......Page 295
Exercises for Chapter 11......Page 296
12.1 Limits of Sequences......Page 303
12.2 Infinite Series......Page 310
12.3 Limits of Functions......Page 315
12.4 Fundamental Properties of Limits of Functions......Page 322
12.5 Continuity......Page 327
12.6 Differentiability......Page 329
Exercises for Chapter 12......Page 332
13.1 Binary Operations......Page 337
13.2 Groups......Page 341
13.3 Permutation Groups......Page 345
13.4 Fundamental Properties of Groups......Page 348
13.5 Subgroups......Page 351
13.6 Isomorphic Groups......Page 355
Exercises for Chapter 13......Page 359
14.1 Rings......Page 366
14.2 Elementary Properties of Rings......Page 371
14.3 Subrings......Page 374
14.4 Integral Domains......Page 376
14.5 Fields......Page 378
Exercises for Chapter 14......Page 381
15.1 Properties of Vectors in 3-Space......Page 384
15.2 Vector Spaces......Page 386
15.3 Matrices......Page 389
15.4 Some Properties of Vector Spaces......Page 392
15.5 Subspaces......Page 394
15.6 Spans of Vectors......Page 397
15.7 Linear Dependence and Independence......Page 399
15.8 Linear Transformations......Page 403
15.9 Properties of Linear Transformations......Page 407
Exercises for Chapter 15......Page 410
16.1 Metric Spaces......Page 415
16.2 Open Sets in Metric Spaces......Page 418
16.3 Continuity in Metric Spaces......Page 423
16.4 Topological Spaces......Page 426
16.5 Continuity in Topological Spaces......Page 428
Exercises for Chapter 16......Page 430
Answers and Hints to Selected Odd-Numbered\n\nExercises in Chapters 14–16 (online)......Page 9
Answers and Hints to Odd-Numbered Section Exercises......Page 440
References......Page 483
Index of Symbols......Page 484
C......Page 485
F......Page 486
N......Page 487
S......Page 488
Z......Page 489