An Introduction to Mathematical Reasoning [NOOK Book]

Overview

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...
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An Introduction to Mathematical Reasoning

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Overview

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.
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Editorial Reviews

From the Publisher
"A student planning to study advanced mathematics would be well served by first mastering the material in this book...a rigorous study of sevearl fundamental topics pervasive in mathematics, including sets, functions, cardinality, combinatorics, and modular arithmetic." D.S. Larson, Choice
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Product Details

  • ISBN-13: 9781139636315
  • Publisher: Cambridge University Press
  • Publication date: 12/5/2012
  • Sold by: Barnes & Noble
  • Format: eBook
  • Edition number: 1
  • Sales rank: 1,226,343
  • File size: 10 MB

Table of Contents

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