A Transition to Advanced Mathematics

To the First Edition......Page 3
To the Fourth Edition......Page 5
CONTENTS......Page 7
1 Propositions and Connectives......Page 10
2 Conditionals and Biconditionals......Page 18
3 Quantifiers......Page 27
4 Mathematical Proofs......Page 35
5 Proofs Involving Quantifiers......Page 49
6 Additional Examples of Proofs......Page 58
1 Basic Notions of Set Theory......Page 68
2 Set Operations......Page 76
3 Extended Set Operations and Indexed Families of Sets......Page 83
4 Induction......Page 94
5 Equivalent Forms of Induction......Page 106
6 Principles of (ounting......Page 113
1 Cartesian Products and Relations......Page 124
2 Equivalence Relations......Page 137
3 Partitions......Page 145
4 Ordering Relations......Page 150
5 Graphs of Relations......Page 160
1 Functions as Relations......Page 170
2 Constructions of Functions......Page 180
3 Functions That Are Onto; One-to-One Functions......Page 188
4 Induced Set Functions......Page 197
1 Equivalent Sets; Finite Sets......Page 204
2 Infinite Sets......Page 212
3 Countable Sets......Page 218
4 The Ordering of Cardinal Numbers......Page 227
5 Comparability of Cardinal Numbers and the Axiom of Choice......Page 235
1 Algebraic Strudures......Page 242
2 Groups......Page 250
3 Examples of Groups......Page 255
4 Subgroups......Page 260
5 Cosels and Lagrange\'s Theorem......Page 267
6 Quotient Groups......Page 271
7 Isomorphism; The Fundamental Theorem of Group Homomorphisms......Page 275
1 Ordered Field Properties of the Real Numbers......Page 282
2 The Heine-Borel Theorem......Page 289
3 The Bolzano-Weierstrass Theorem......Page 299
4 The Bounded Monotone Sequence Theorem......Page 303
5 Equivalents of Completeness......Page 312
Exercises 1.1......Page 316
Exercises 1.2......Page 317
Exercises 1.4......Page 319
Exercises 1.5......Page 320
Exercises 1.6......Page 321
Exercises 2.1......Page 322
Exercises 2.2......Page 323
Exercises 2.3......Page 324
Exercises 2.5......Page 326
Exercises 2.6......Page 327
Exercises 3.1......Page 328
Exercises 3.2......Page 329
Exercises 3.4......Page 330
Exercises 4.1......Page 331
Exercises 4.2......Page 332
Exercises 4.3......Page 334
Exercises 4.4......Page 335
Exercises 5.2......Page 336
Exercises 5.3......Page 337
Exercises 6.1......Page 338
Exercises 6.3......Page 340
Exercises 6.4......Page 341
Exercises 6.7......Page 342
Exercises 7.1......Page 343
Exercises 7.3......Page 344
Exercises 7.5......Page 345
INDEX......Page 346