This page intentionally left blank How to Think Like a Mathematician 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 Algorithm, 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 Essential for any starting undergraduate in mathematics, this book can also help if you’re studying engineering or physics and need access to undergraduate mathematics topics, or if you’re taking a subject that requires logic such as computer science, philosophy or linguistics How to Think Like a Mathematician A Companion to Undergraduate Mathematics KE VIN H OUSTON University of Leeds CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521895460 © K Houston 2009 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2009 ISBN-13 978-0-511-50645-1 eBook (EBL) ISBN-13 978-0-521-89546-0 hardback ISBN-13 978-0-521-71978-0 paperback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate To Mum and Dad – Thanks for everything Contents Preface Page ix I Study skills for mathematicians Sets and functions Reading mathematics Writing mathematics I Writing mathematics II How to solve problems 14 21 35 41 II How to think logically Making a statement Implications Finer points concerning implications Converse and equivalence 10 Quantifiers – For all and There exists 11 Complexity and negation of quantifiers 12 Examples and counterexamples 13 Summary of logic 51 53 63 69 75 80 84 90 96 III Definitions, theorems and proofs 14 Definitions, theorems and proofs 15 How to read a definition 16 How to read a theorem 17 Proof 18 How to read a proof 19 A study of Pythagoras’ Theorem 97 99 103 109 116 119 126 IV Techniques of proof 20 Techniques of proof I: Direct method 21 Some common mistakes 22 Techniques of proof II: Proof by cases 23 Techniques of proof III: Contradiction 24 Techniques of proof IV: Induction 137 139 149 155 161 166 vii viii Contents 25 More sophisticated induction techniques 26 Techniques of proof V: Contrapositive method 175 180 V Mathematics that all good mathematicians need 27 Divisors 28 The Euclidean Algorithm 29 Modular arithmetic 30 Injective, surjective, bijective – and a bit about infinity 31 Equivalence relations 185 187 196 208 218 230 VI Closing remarks 32 Putting it all together 33 Generalization and specialization 34 True understanding 35 The biggest secret 241 243 248 252 255 Appendices A Greek alphabet B Commonly used symbols and notation C How to prove that … Index 257 257 258 260 263 ...This page intentionally left blank How to Think Like a Mathematician Looking for a head start in your undergraduate degree in mathematics? Maybe you’ve already started your degree and feel... able to think like a mathematician and so my aim is to give you a book jam-packed with practical advice and helpful hints on how to acquire skills specific to Add your own favourite term of abuse... you’re taking a subject that requires logic such as computer science, philosophy or linguistics How to Think Like a Mathematician A Companion to Undergraduate Mathematics KE VIN H OUSTON University