An Introduction to Mathematical Reasoning: Numbers, Sets and Functions

by Peter J. Eccles

• ISBN13: 9780521592697
• ISBN10: 0521592690
• Format: Hardcover
• Copyright: 1998-01-13
• 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

Preface	ix
Part I: Mathematical statements and proofs	1	(58)
1 The language of mathematics	3	(7)
2 Implications	10	(11)
3 Proofs	21	(9)
4 Proof by contradiction	30	(9)
5 The induction principle	39	(14)
Problems I	53	(6)
Part II: Sets and functions	59	(62)
6 The language of set theory	61	(13)
7 Quantifiers	74	(15)
8 Functions	89	(12)
9 Injections, surjections and bijections	101	(14)
Problems II	115	(6)
Part III: Numbers and counting	121	(68)
10 Counting	123	(10)
11 Properties of finite sets	133	(11)
12 Counting functions and subsets	144	(13)
13 Number systems	157	(13)
14 Counting infinite sets	170	(12)
Problem III	182	(7)
Part IV: Arithmetic	189	(40)
15 The division theorem	191	(8)
16 The Euclidean algorithm	199	(8)
17 Consequences of the Euclidean algorithm	207	(9)
18 Linear diophantine equations	216	(9)
Problems IV	225	(4)
Part V: Modular arithmetic	229	(46)
19 Congruence of integers	231	(9)
20 Linear congruences	240	(10)
21 Congruence classes and the arithmetic of remainders	250	(12)
22 Partitions and equivalence relations	262	(9)
Problems V	271	(4)
Part VI: Prime numbers	275	(24)
23 The sequence of prime numbers	277	(12)
24 Congruence modulo a prime	289	(6)
Problems VI	295	(4)
Solutions to exercises	299	(46)
Bibliography	345	(1)
List of symbols	346	(1)
Index	347