An Introduction to Mathematical Reasoning: Numbers, Sets and Functions
Cambridge University Press, 12/11/1997
EAN 9780521597180, ISBN10: 0521597188
Paperback, 362 pages, 22.9 x 15.2 x 2 cm
Language: English
The purpose of this book is to introduce the basic ideas of mathematical proof to students embarking on university mathematics. The emphasis is on helping the reader in understanding and constructing proofs and writing clear mathematics. This is achieved by exploring set theory, combinatorics and number theory, topics which include many fundamental ideas which are part of the tool kit of any mathematician. 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 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. Over 250 problems include questions to interest and challenge the most able student as well as plenty of routine exercises to help familiarize the reader with the basic ideas.
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.
'The book is written with understanding of the needs of students ...' European Mathematical Society