An Introduction to Mathematical Reasoning: Numbers, Sets and Functions

by Peter J. Eccles

• ISBN13: 9780521597180
• ISBN10: 0521597188
• Format: Paperback
• Copyright: 1998-01-28
• Publisher: Cambridge University Press

Summary

This book eases students into the rigors of university mathematics. The emphasis is on understanding and constructing proofs and writing clear mathematics. The author achieves this by exploring set theory, combinatorics, and number theory, topics that include many fundamental ideas and may not be a part of a young mathematician's toolkit. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all-time-great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas.

Table of Contents

Part I. Mathematical Statements and Proofs: 1. The language of mathematics

2. Implications

3. Proofs

4. Proof by contradiction

5. The induction principle

Part II. Sets and Functions: 6. The language of set theory

7. Quantifiers

8. Functions

9. Injections, surjections and bijections

Part III. Numbers and Counting: 10. Counting

11. Properties of finite sets

12. Counting functions and subsets

13. Number systems

14. Counting infinite sets

Part IV. Arithmetic: 15. The division theorem

16. The Euclidean algorithm

17. Consequences of the Euclidean algorithm

18. Linear diophantine equations

Part V. Modular Arithmetic: 19. Congruences of integers

20. Linear congruences

21. Congruence classes and the arithmetic of remainders

22. Partitions and equivalence relations

Part VI. Prime Numbers: 23. The sequence of prime numbers

24. Congruence modulo a prime

Solutions to exercises.