An Introduction to Mathematical Reasoning: Numbers, Sets and Functions

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Cambridge University Press, Dec 11, 1997 - Mathematics - 350 pages
3 Reviews
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.
 

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Contents

The language of mathematics
3
Implications
10
Proofs
21
Proof by contradiction
30
The induction principle
39
Mathematical statements and proofs
53
Part II
59
The language of set theory
61
The division theorem
191
The Euclidean algorithm
199
Consequences of the Euclidean algorithm
207
Linear diophantine equations
216
Problems IV
225
Part V
229
Congruence of integers
231
Linear congruences
240

Quantifiers
74
Functions
89
Injections surjections and bijections
101
Sets and functions
115
Part III
121
Counting
123
Properties of finite sets
133
Counting functions and subsets
144
Number systems
157
Counting infinite sets
170
Numbers and counting
182
Part IV
189
Congruence classes and the arithmetic of remainders
250
Partitions and equivalence relations
262
Problems V
271
Part VI
275
The sequence of prime numbers
277
Congruence modulo a prime
289
Problems VI
295
Solutions to exercises
299
Bibliography
345
List of symbols
346
Index
347
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