How to Think Like a Mathematician: A Companion to Undergraduate Mathematics

by Kevin Houston

• ISBN13: 9780521719780
• ISBN10: 052171978X
• Format: Paperback
• Copyright: 2009-02-23
• Publisher: Cambridge University Press

Summary

Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many courses such as divisors, Euclidean algorithms, modular arithmetic, equivalence relations, and injectivity and surjectivity of functions. The material has been tested by real students over many years so all the essentials are covered. With over 300 exercises to help you test your progress, you'll soon learn how to think like a mathematician.

Table of Contents

Preface	p. ix
Study skills for mathematicians	p. 1
Sets and functions	p. 3
Reading mathematics	p. 14
Writing mathematics I	p. 21
Writing mathematics II	p. 35
How to solve problems	p. 41
How to think logically	p. 51
Making a statement	p. 53
Implications	p. 63
Finer points concerning implications	p. 69
Converse and equivalence	p. 75
Quantifiers-For all and There exists	p. 80
Complexity and negation of quantifiers	p. 84
Examples and counterexamples	p. 90
Summary of logic	p. 96
Definitions, theorems and proofs	p. 97
Definitions, theorems and proofs	p. 99
How to read a definition	p. 103
How to read a theorem	p. 109
Proof	p. 116
How to read a proof	p. 119
A study of Pythagoras' Theorem	p. 126
Techniques of proof	p. 137
Techniques of proof I: Direct method	p. 139
Some common mistakes	p. 149
Techniques of proof II: Proof by cases	p. 155
Techniques of proof III: Contradiction	p. 161
Techniques of proof IV: Induction	p. 166
More sophisticated induction techniques	p. 175
Techniques of proof V: Contrapositive method	p. 180
Mathematics that all good mathematicians need	p. 185
Divisors	p. 187
The Euclidean Algorithm	p. 196
Modular arithmetic	p. 208
Injective, surjective, bijective-and a bit about infinity	p. 218
Equivalence relations	p. 230
Closing remarks	p. 241
Putting it all together	p. 243
Generalization and specialization	p. 248
True understanding	p. 252
The biggest secret	p. 255
Appendices	p. 257
Greek alphabet	p. 257
Commonly used symbols and notation	p. 258
How to prove that	p. 260
Index	p. 263
Table of Contents provided by Ingram. All Rights Reserved.