Author(s): Bob A. Dumas, John E. McCarthy
Edition: 2
Publisher: Bob A. Dumas, John E. McCarthy
Year: 2015
Language: English
Commentary: Downloaded from https://www.math.wustl.edu/~mccarthy/SandP2.html
Pages: 290
Contents ix
Chapter 0. Introduction 1
0.1. Why this book is 1
0.2. What this book is 1
0.3. What this book is not 3
0.4. Advice to the Student 3
0.5. Advice to the Instructor 6
0.6. Acknowledgements 9
Chapter 1. Preliminaries 11
1.1. “And” “Or” 11
1.2. Sets 12
1.3. Functions 23
1.4. Injections, Surjections, Bijections 29
1.5. Images and Inverses 31
1.6. Sequences 37
1.7. Russell’s Paradox 40
1.8. Exercises 41
1.9. Hints to get started on some exercises 46
Chapter 2. Relations 49
2.1. Definitions 49
2.2. Orderings 51
2.3. Equivalence Relations 53
2.4. Constructing Bijections 57
2.5. Modular Arithmetic 60
2.6. Exercises 65
Chapter 3. Proofs 69
3.1. Mathematics and Proofs 69
3.2. Propositional Logic 73
3.3. Formulas 80
3.4. Quantifiers 82
3.5. Proof Strategies 87
3.6. Exercises 93
Chapter 4. Principle of Induction 99
4.1. Well-orderings 99
4.2. Principle of Induction 100
4.3. Polynomials 109
4.4. Arithmetic-Geometric Inequality 116
4.5. Exercises 121
Chapter 5. Limits 127
5.1. Limits 127
5.2. Continuity 136
5.3. Sequences of Functions 139
5.4. Exercises 146
Chapter 6. Cardinality 151
6.1. Cardinality 151
6.2. Infinite Sets 155
6.3. Uncountable Sets 162
6.4. Countable Sets 169
6.5. Functions and Computability 175
6.6. Exercises 177
Chapter 7. Divisibility 181
7.1. Fundamental Theorem of Arithmetic 181
7.2. The Division Algorithm 186
7.3. Euclidean Algorithm 190
7.4. Fermat’s Little Theorem 193
7.5. Divisibility and Polynomials 198
7.6. Exercises 204
Chapter 8. The Real Numbers 207
8.1. The Natural Numbers 208
8.2. The Integers 211
8.3. The Rational Numbers 213
8.4. The Real Numbers 214
8.5. The Least Upper Bound Property 217
8.6. Real Sequences 218
8.7. Ratio Test 223
8.8. Real Functions 225
8.9. Cardinality of the Real Numbers 230
8.10. Order-Completeness 233
8.11. Exercises 236
Chapter 9. Complex Numbers 243
9.1. Cubics 243
9.2. Complex Numbers 246
9.3. Tartaglia-Cardano Revisited 252
9.4. Fundamental Theorem of Algebra 255
9.5. Application to Real Polynomials 261
9.6. Further remarks 262
9.7. Exercises 262
Appendix A. The Greek Alphabet 265
Appendix B. Axioms of Zermelo-Fraenkel with the Axiom of Choice 267
Appendix C. Hints to get started on early exercises 271
Bibliography 273
Index 275
