# An Introduction to Mathematical Reasoning: Numbers, Sets and Functions

Cambridge University Press, Dec 11, 1997 - Mathematics - 350 pages
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 List of symbols Index Copyright

### References to this book

 A Concise Introduction to Pure Mathematics, Second EditionMartin LiebeckLimited preview - 2005
 A Concise Introduction to Pure Mathematics, Second EditionMartin LiebeckLimited preview - 2000
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