includes index

Chapter 0. Introduction 0.1. Why this book is 0.2. What this book is 0.3. What this book is not 0.4. Advice to the Student 0.5. Advice to the Instructor 0.6. Acknowledgements Chapter 1. Preliminaries 1.1. “And” “Or” 1.2. Sets 1.3. Functions 1.4. Injections, Surjections, Bijections 1.5. Images and Inverses 1.6. Sequences 1.7. Russell’s Paradox 1.8. Exercises 1.9. Hints to Get Started on Some Exercises Chapter 2. Relations 2.1. Definitions 2.2. Orderings 2.3. Equivalence Relations 2.4. Constructing Bijections 2.5. Modular Arithmetic 2.6. Exercises Chapter 3. Proofs 3.1. Mathematics and Proofs 3.2. Propositional Logic 3.3. Formulas 3.4. Quantifiers 3.5. Proof Strategies 3.6. Exercises Chapter 4. Principle of Induction 4.1. Well-Orderings 4.2. Principle of Induction 4.3. Polynomials 4.4. Arithmetic-Geometric Inequality 4.5. Exercises Chapter 5. Limits 5.1. Limits 5.2. Continuity 5.3. Sequences of Functions 5.4. Exercises Chapter 6. Cardinality 6.1. Cardinality 6.2. Infinite Sets 6.3. Uncountable Sets 6.4. Countable Sets 6.5. Functions and Computability 6.6. Exercises Chapter 7. Divisibility 7.1. Fundamental Theorem of Arithmetic 7.2. The Division Algorithm 7.3. Euclidean Algorithm 7.4. Fermat’s Little Theorem 7.5. Divisibility and Polynomials 7.6. Exercises Chapter 8. The Real Numbers 8.1. The Natural Numbers 8.2. The Integers 8.3. The Rational Numbers 8.4. The Real Numbers 8.5. The Least Upper Bound Principle 8.6. Real Sequences 8.7. Ratio Test 8.8. Real Functions 8.9. Cardinality of the Real Numbers 8.10. Order-Completeness 8.11. Exercises Chapter 9. Complex Numbers 9.1. Cubics 9.2. Complex Numbers 9.3. Tartaglia-Cardano Revisited 9.4. Fundamental Theorem of Algebra 9.5. Application to Real Polynomials 9.6. Further Remarks 9.7. Exercises Appendix A. The Greek Alphabet Appendix B. Axioms of Zermelo-Fraenkel with the Axiom of Choice Bibliography Index