Transition to Higher Mathematics: Structure and Proof

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