This text is intended for the Foundations of Higher Math bridge course taken by prospective math majors following completion of the mainstream Calculus sequence, and is designed to help students develop the abstract mathematical thinking skills necessary for success in later upper-level majors math courses. As lower-level courses such as Calculus rely more exclusively on computational problems to service students in the sciences and engineering, math majors increasingly need clearer guidance and more rigorous practice in proof technique to adequately prepare themselves for the advanced math curriculum. With their friendly writing style Bob Dumas and John McCarthy teach students how to organize and structure their mathematical thoughts, how to read and manipulate abstract definitions, and how to prove or refute proofs by effectively evaluating them. Its wealth of exercises give students the practice they need, and its rich array of topics give instructors the flexibility they desire to cater coverage to the needs of their school’s majors curriculum.
This text is part of the Walter Rudin Student Series in Advanced Mathematics.
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
