Introductory Mathematics: Algebra and Analysis

226 Pages·1998·8.906 MB·English

by  Geoff Smith

Springer Undergraduate Mathematics Series
Springer-Verlag London Ltd.
Advisory Board
P.J. Cameron Queen Mary and Westfield College
M.A.J. Chaplain University ofDundee
K. Erdmann Oxford University
L.C.G. Rogers University ofC ambridge
E. SOli Oxford University
J.F. Toland University ofBath
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Basic Stochastic Processes Z. Brzefniak and T. Zastawniak
Elementary Differential Geometry A. Pressley
Elementary Number Theory G.A. ]ones and ].M. ]ones
Elements of Abstract Analysis M. 6 Searc6id
Elements of Logic via Numbers and Sets D.L. ]ohnson
Further Linear Algebra T.S. Blyth and E.F. Robertson
Geometry R. Fenn
Groups, Rings and Fields D.A.R. Wallace
Hyperbolic Geometry ]. W. Anderson
Information and Coding Theory G.A. ]ones and ]M. ]ones
Introduction to Laplace Transforms and Fourier Series P.P.G. Dyke
Introduction to Ring Theory P.M. Cohn
Introductory Mathematics: Algebra and Analysis G. Smith
Linear Functional Analysis B.P. Rynne and M.A. Youngson
Matrix Groups: An Introduction to Lie Group Theory A. Baker
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Numerical Methods for Partial Differential Equations G. Evans, J. Blackledge, P. Yardley
Probability Models J. Haigh
Real Analysis JM. Howie
Sets, Logic and Categories P. Cameron
Special Relativity N.M.J. Woodhouse
Symmetries D.L. ]ohnson
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GeoffSmith
Introductory
Mathematics:
Algebra and Analysis
With 19 Figures
, Springer
Geoff Smith, MA, MSc, PhD
Department of Mathematical Sciences, University of Bath, Claverton Down,
Bath BAl 7AY, UK
Cover illustration elements reproduced by kind permission of
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American Statistica1 Association: Chance Voi 8 No 1,1995 article by KS and KW Heiner 'Tree Rings ofthe Northern Shawangunks' page 32 fig 2
Springer-Verlag: Mathematica in Education and Research Voi 4 Issue 3 1995 article by Roman E Maeder, Beatrice Amrhein and Oliver Gloor
'llinstrated Mathematica: Visualization ofMathematical Objects' page 9 fig Il, originally pnblished as a CD ROM 'llinstrated Mathernatics' by
TELOS: ISBN 0-387-14222-3, German edition by Birkhanser: ISBN 3-7643-5100-4.
Mathematica in Education and Research VoI 4 Issue 3 1995 article by Richard J Gaylord and Kazume Nishidate 'Traffic Engineering with CeUular
Automata' page 35 fig 2. Mathernatica in Education and Research VoI 5 Issue 2 1996 article by Michael Trott 'The Implicitization of a Trefoil
Knot' page 14.
Mathematica in Education and Research VoI 5 Issue 2 1996 article by l.ee de Cola 'Coins, Trees, Bars and BeUs: Simu1ation of the Binontia1 Pro
cess' page 19 fig 3. Mathematica in Education and Research VoI 5 Issue 2 1996 article by Richard Gaylord and Kazume Nishidate 'Contagions
Spreading' page 33 fig 1. Mathematicain Education andResearch Voi 5 Issue 21996 article by Joe Buhler and Stan Wagon 'Secrets ofthe
Madelung Constant' page 50 fig 1.
British Library Cataloguing in Publication Data
Smith, Geoffrey Charles
Introductory mathematics: algebra and analysis. -(Springer
undergraduate mathematics series)
1. Mathematics 2. Algebra 3. Mathematical analysis
1. Title
510
ISBN 978-3-540-76178-5 ISBN 978-1-4471-0619-7 (eBook)
DOI 10.1007/978-1-4471-0619-7
Library of Congress Cataloging-in-Publication Data
Smith, Geoff, 1953-
Introductory mathematics: algebra and analysis / Geoff Smith
p. cm. --(Springer undergraduate mathematics series)
Includes index.
1. Algebra 2. Mathematical analysis 1. Title. II. Series.
QAI54.2.S546 1998 97-41196
510-dc21 CIP
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted
under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or
transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of
reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency.
Enquiries conceming reproduction outside those terms should be sent to the publishers.
Springer Undergraduate Mathematics Series ISSN 1615-2085
© Springer-Verlag London 1998
Originally published by Springer-Verlag London Limited in 1998
The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a specific
statement, that such names are exempt from the relevant laws and regulations and therefore free for general use.
The publisher makes no representation, express or implied, with regard to the accuracy of the information
contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may
be made.
Typesetting: Camera ready by author
12/3830-54 Printed on acid-free paper SPIN 11012368
To my current wife
Preface
This book is a gentle and relaxed introduction to the two branches of pure
mathematics which dominate the early stages ofthe subject as it is taught to
undergraduates in many countries. It is not a substitute for more advanced
texts, and has no pretensions to comprehensiveness. There are several places
where I would have liked to press on further, but you have to stop somewhere.
It should, I hope, be easy to read, and to this end the style is decidedly more
colloquial than is traditional in text books. I only hope that the language will
not date too quickly. Thus this is not a book in the relentless theorem-proof
style; it contains discursive commentary.
The ways in which pure mathematicians express themselves, and the cu
mulative nature ofthe subject, may make pure mathematics seem daunting to
a beginner. The mathematical mode ofexpression and the deductive method
are vital to pure mathematics. We wish to explore strange geometries, new
algebraic systems, and infinite dimensional spaces. There is no point in em
barkingon thisenterpriseunless we are prepared to be ruthlessly precise, since
otherwise, no-one will have any idea what we are talking about (ifanything).
Theseexoticspaces and objectsarenot partofeverydayexperience, unlike,
for example a dog. Ifwe mention that "thereis a dog in the garden",wedo not
expect the response "what is a dog, what isa garden, what does ismean in this
sentence, why have you used the indefinite article, and whatis the contribution
ofthe word there?" We know a lot about dogs and gardens, and do not need to
putthesentenceunderscrutinyinorderto understandthemeaning. However,if
insteadsomeonesays "everylineargroupiseithervirtuallysolvable,orcontains
a free subgroup ofrank 2", then either you have to live in a world where these
termsareasfamiliar asdogs andgardens,oryouhavetotakethe remarkapart,
and analyze every part of it until you understand what it asserts. Of course
there is little point in doing this unless you happen to know that linear groups
VIII Introductory Mathematics: Algebra and Analysis
are very interesting - which, incidentally, they are.
There is a web site which supports this book.
http://www.maths.bath.ac.uk/~masgcs/bookl/
Ifthat ever changes, a link to the newsite will be put in place. At the website
you will find additional exercises and solutions, and corrections to any errors
that are discovered after publication.
The material in this book is not always arranged in a logically perfect
sequence. This is deliberate, and is a consequence oftrying to make the book
accessible. The ideal way to read the book is from cover to cover. Chapter 1
establishes notation and sets the scene, and Chapter 2concerns mathematical
proof- many readers will want to read that before proceeding with the main
topics. I have tried to make subsequent chapters (fairly) independent, though
Chapter 6 should definitely be read before either Chapter 7 or Chapter 8. In
consequenceofthe partialindependence, somematerialis repeated in different
parts ofthe book, though the treatments vary.
I alsofelt that this bookshouldcontainearlychaptersoncomplex numbers
and matrices. These topics are basic to university mathematics, engineering
and science, and are rarely or barely taught at secondary level.
It is standard practice to thank everyone who had anything to do with
the creation ofa book. I therefore wish to thank and congratulate my parents
Eileen and Roy. This volume is being published in the year of their golden
wedding anniversary, and Springer-Verlaghave kindly agreed to celebrate this
event by supplying a cover ofappropriate colour.
Inductively, ifI thank anyone, I thank their parents too, thereby acknowl
edging the assistance of a vast number of people, creatures, single-celled or
ganisms and amino acids. Please note that this (thinly spread) gratitude was
expressed with considerable economy (see Section 2.1).
Despite the millions of generations who have already been thanked, there
are some deserving cases who have not yet qualified. I also acknowledge the
help and support ofvarious colleagues at the University ofBath. My 'lEX and
U,TFfC guru is Fran Burstall, and Teck Sin How provided figures at amazing
speed. James Davenport and Aaron Wilson helped to weed out errors, and
suggested improvements. I must also thank John Tolandwho persuaded me to
write this bookwith his usualcombinationofcharm,flattery and threats- and
supplied the beautiful question that constitutes Exercise 8.4. Any remaining
errors are mine, and copyright.
I would also like to thank my bright, enthusiastic and frustrated students,
without whom this book would not have been necessary, and my wife Olga
MarkovnaTabachnikova, without whom my salary would have been sufficient.
GCS, Bath, ll-xi-1997.
Introductory Mathematics: Algebra and Analysis ix
Added at second printing I thank the following people who have reported ty
pographical errors in the first printing of this book: Verity Jeffery (Meridian
School), ProfCharles F. Miller III (Melbourne University), Martyn Partridge
(Intertype), Carrie Rutherford (Q.M.W., London) and Aaron Wilson (Univer
sity of Bath). These errors have been eliminated. I also wish to thank Prof
Edward Fraenkel FRS (University ofBath) for his tactful attempts to improve
my grammar.
In addition to solutions of problems and amplifications on material in the
book, the web site now contains supplementary materialon many topics, some
ofwhich weresuggestedby GregorySankaranand WafaaShabana. This mate
rial includes Cardinality and Countability, Functions, Preimages, Unions and
Intersections,theInclusion-ExclusionPrinciple,InjectionsandSurjections,Fer
mat's Two Squares Theorem, Group Actions (and exercises) and the Integers
modulo N.
Added at third printing A few more errors have been dealt with thanks to Dr
VictoriaGould (University ofYork) and ProfDave Johnson (University ofthe
West Indies). The proof of Proposition 6.9 replaces the garbled mush which
disgraced the first two printings.
Inthis latestiterationwegiveDedekindand von Neumann propercreditfor
creating the natural numbers. However, the provenance ofthe whole numbers
is slightly controversial since according to Kronecker the integers were made
by God.
Contents
Preface ......................................................... vii
1. Sets, Functions and Relations. .............................. 1
1.1 Sets.................................................... 1
1.2 Subsets................................................. 2
1.3 Well-known Sets. ......................................... 3
1.4 Rationals, Reals and Pictures " 6
1.5 Set Operations ........................................... 8
1.6 Sets ofSets. ............................................. 11
1.7 Paradox................................................. 14
1.8 Set-theoretic Constructions. ............................... 15
1.9 Notation................................................ 16
1.10 Venn Diagrams. ......... .... .................... ......... 17
1.11 Quantifiers and Negation. ........................ ......... 19
1.12 Informal Description ofMaps 21
1.13 Injective, Surjective and Bijective Maps. .................... 22
1.14 Composition ofMaps. ... ............ ...... ...... ......... 23
1.15 Graphs and Respectability Reclaimed 29
1.16 Characterizing Bijections. ................................. 30
1.17 Sets ofMaps. ............................................ 31
1.18 Relations. ............................................... 31
1.19 Intervals 37
2. Proof 39
2.1 Induction................................................ 39
2.2 Complete Induction. ...................................... 43
XII Contents
2.3 Counter-examples and Contradictions 47
2.4 Method ofDescent 50
2.5 Style.................................................... 53
2.6 Implication.............................................. 54
2.7 Double Implication .. ..................................... 54
2.8 The Master Plan ......................................... 56
3. Complex Numbers and Related Functions 57
3.1 Motivation.............................................. 57
3.2 Creating the Complex Numbers ............................ 62
3.3 A Geometric Interpretation. ............................... 70
3.4 Sine, Cosine and Polar Form. .............................. 76
3.5 e 80
3.6 Hyperbolic Sine and Hyperbolic Cosine. ..................... 85
3.7 Integration Tricks 88
3.8 Extracting Roots and Raising to Powers. .................... 89
3.9 Logarithm............................................... 90
3.10 Power Series 92
4. Vectors and Matrices 95
4.1 Row Vectors ............................................. 95
4.2 Higher Dimensions 97
4.3 Vector Laws ........... 98
4.4 Lengths and Angles 98
4.5 Position Vectors 103
4.6 Matrix Operations 104
4.7 Laws ofMatrix Algebra 106
4.8 Identity Matrices and Inverses 108
4.9 Determinants 110
4.10 Geometry ofDeterminants 119
4.11 Linear Independence 120
4.12 Vector Spaces 121
4.13 Transposition 123
5. Group Theory 125
5.1 Permutations 125
5.2 Inverse Permutations 130
5.3 The AlgebraofPermutations 131
5.4 The Order ofa Permutation 133
5.5 Permutation Groups 135
5.6 Abstract Groups 136
5.7 Subgroups 142