How to think like a mathematician : a companion to undergraduate mathematics

279 Pages·2009·1.073 MB·English

by  Kevin Houston

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HowtoThinkLikeaMathematician
Lookingforaheadstartinyourundergraduatedegreeinmathematics?Maybeyou’ve
alreadystartedyourdegreeandfeelbewilderedbythesubjectyoupreviouslyloved?
Don’tpanic!Thisfriendlycompanionwilleaseyourtransitiontorealmathematical
thinking.
Workingthroughthebookyouwilldevelopanarsenaloftechniquestohelpyou
unlockthemeaningofdefinitions,theoremsandproofs,solveproblems,andwrite
mathematicseffectively.Allthemajormethodsofproof–directmethod,cases,
induction,contradictionandcontrapositive–arefeatured.Concreteexamplesareused
throughout,andyou’llgetplentyofpracticeontopicscommontomanycoursessuchas
divisors,EuclideanAlgorithm,modulararithmetic,equivalencerelations,andinjectivity
andsurjectivityoffunctions.
Thematerialhasbeentestedbyrealstudentsovermanyyearssoalltheessentialsare
covered.Withover300exercisestohelpyoutestyourprogress,you’llsoonlearnhowto
thinklikeamathematician.
Essentialforanystartingundergraduateinmathematics,thisbookcanalsohelp
ifyou’restudyingengineeringorphysicsandneedaccesstoundergraduatemathematics
topics,orifyou’retakingasubjectthatrequireslogicsuchascomputerscience,
philosophyorlinguistics.

How to Think Like a
Mathematician
A Companion to Undergraduate Mathematics
KEVIN HOUSTON
UniversityofLeeds
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.
ToMumandDad–Thanksforeverything.

Contents
Preface Page ix
I Studyskillsformathematicians 1
1 Setsandfunctions 3
2 Readingmathematics 14
3 WritingmathematicsI 21
4 WritingmathematicsII 35
5 Howtosolveproblems 41
II Howtothinklogically 51
6 Makingastatement 53
7 Implications 63
8 Finerpointsconcerningimplications 69
9 Converseandequivalence 75
10 Quantifiers–ForallandThereexists 80
11 Complexityandnegationofquantifiers 84
12 Examplesandcounterexamples 90
13 Summaryoflogic 96
III Definitions,theoremsandproofs 97
14 Definitions,theoremsandproofs 99
15 Howtoreadadefinition 103
16 Howtoreadatheorem 109
17 Proof 116
18 Howtoreadaproof 119
19 AstudyofPythagoras’Theorem 126
IV Techniquesofproof 137
20 TechniquesofproofI:Directmethod 139
21 Somecommonmistakes 149
22 TechniquesofproofII:Proofbycases 155
23 TechniquesofproofIII:Contradiction 161
24 TechniquesofproofIV:Induction 166
vii
viii Contents
25 Moresophisticatedinductiontechniques 175
26 TechniquesofproofV:Contrapositivemethod 180
V Mathematicsthatallgoodmathematiciansneed 185
27 Divisors 187
28 TheEuclideanAlgorithm 196
29 Modulararithmetic 208
30 Injective,surjective,bijective–andabitaboutinfinity 218
31 Equivalencerelations 230
VI Closingremarks 241
32 Puttingitalltogether 243
33 Generalizationandspecialization 248
34 Trueunderstanding 252
35 Thebiggestsecret 255
Appendices 257
A Greekalphabet 257
B Commonlyusedsymbolsandnotation 258
C Howtoprovethat… 260
Index 263