Mathematical Proofs.pdf

M

a t h e m a t ic a l

P

r o o f s

A Transition to Advanced Mathematics

T H I R D

E D I T I O N

T h i r cl E d i t i o n

Mathematical Proofs

A Transition to

Advanced Mathematics

Gary Chartrand

W estern M ichigan U niversity

Albert D. Polim eni

State U niversity o f N ew York at Fredonia

Ping Zhang

W estern M ichigan U niversity

PEARSON

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Library of Congress Cataloging-in-Publication Data

Chartrand, Gary.

Mathematical proofs : a transition to advanced mathematics / Gary Chartrand, Albert D. Polimeni, Ping Zhang. - 3rd ed.

p. cm.

Includes bibliographical references and index. ISBN-13: 978-0-321-79709-4

ISBN-10: 0-321-79709-4

1. Proof theory—Textbooks. I. Polimeni, Albert D.. 1938- II. Zhang, Ping, 1957- III. Title. QA9.54.C48 2013 51 1.3'6— dc23 . 2012012552 10 9 8 7 6 5 4 3 2 — CW— 16 15 14 13 12

PEARSON

To

the memory o f my mother and father G.C.

the memory o f my uncle Joe and my brothers John and Rocky A.D.P.

C ontents

0

Communicating Mathematics

1

L earning M athem atics 2

W hat O thers H ave Said A bout W riting 4

M athem atical W riting 5

U sing Sym bols 6

W riting M athem atical E xpressions 8

C om m on W ords and P hrases in M athem atics 10

Som e C losing C om m ents A bout W riting 12

1

Sets

l.i

1 .2 Subsets 18

1.3 Set O perations 21

1.4 Indexed C ollections o f Sets

1 .5 P artitions o f Sets 27

1.6

E xercises for C hapter 1 29

2

Logic

37

2.1 S tatem ents 37*

2 .2 The N egation o f a S tatem ent 39

2 .3 The D isjunction and C onjunction o f Statem ents 41

2 .4 The Im plication 42

2 .5 M ore on Im plications 44

2 .6 The B iconditional 47

2 .8 L ogical Equivalence 51

2 .9 Som e F undam ental P roperties o f L ogical E quivalence 53

2 .1 0 Q uantified S tatem ents 55

2 .1 1 C haracterizations o f S tatem ents 63

E xercises for C hapter 2 64

3

Direct Proof and Proof by Contrapositive

3 .1 Trivial and Vacuous P roofs 78

3 .2 D irect P roofs 80

3 .3 P ro o f by C ontrapositive 84

3 .4 P ro o f by C ases 89

3 .5 P ro o f E valuations 92

E xercises for C hapter 3 93

4

More on Direct Proof and Proof by Contrapositive

4 .1 P roofs Involving D ivisibility o f Integers 99

4 .2 P roofs Involving C ongruence o f Integers 103

4 .3 P roofs Involving R eal N um bers 105

4 .4 P roofs Involving Sets 108

4 .5 F undam ental P roperties o f Set O perations 111

4 .6 P roofs Involving C artesian Products o f Sets 113

E xercises fo r C hapter 4 114

5

Existence and Proof by Contradiction

5 .1 C ounterexam ples 120

5 .2 P ro o f by C ontradiction 124

5 .3 A R eview o f T hree P ro o f T echniques 130

5 .4 E xistence Proofs 132

5 .5 D isproving E xistence S tatem ents 136

E xercises for C hapter 5 137

6

Mathematical Induction

6 .1 The P rinciple o f M athem atical Induction 142

6 .2 A M ore G eneral P rinciple o f M athem atical Induction 151

6 .3 P ro o f by M inim um C ounterexam ple 158

6 .4 The S trong Principle o f M athem atical Induction 161

[7

Prove or Disprove

7.1 C onjectures in M athem atics 170

7 .2 R evisiting Q uantified Statem ents 173

7 .3 Testing S tatem ents 178

E xercises for C hapter 7 185

8

Equivalence Relations

8 .1 R elations 192

8 .2 P roperties o f R elations 193

8 .3 Equivalence R elations 196

8 .4 P roperties o f E quivalence C lasses

8 .5 C ongruence M odulo n 202

8 .6 The Integers M odulo n 207

E xercises fo r C hapter 8 210

9

Functions

9.1 The D efinition o f F unction 216

9 .2 The Set o f A ll F unctions from A to fi 219

9 .3 O ne-to-O ne and O nto Functions 220

9 .4 Bijective Functions 222

9 .5 C om position o f Functions 225

9 .6 Inverse Functions 229

9 .7 Perm utations 232

E xercises for C hapter 9 234

flo

Cardinalities of Sets

1 0 .1 N um erically E quivalent Sets 243

1 0 .2 D enum erable Sets 244

1 0 .3 U ncountable Sets 250

1 0 .4 C om paring C ardinalities o f Sets 255

1 0 .5 T he S ch ro d er-B ern stein T heorem 258

E xercises for C hapter 10 262

Proofs in Number Theory

11.1 D ivisibility P roperties o f Integers 266

Contents

vii

1 1 .3 G reatest C om m on D ivisors 271

1 1 .4 The E uclidean A lgorithm 272

1 1 .5 R elatively Prim e Integers 275

1 1 .6 The F undam ental T heorem o f A rithm etic 277

1 1 .7 C oncepts Involving Sum s o f D ivisors 280

E xercises for C hapter 11 281

12

Proofs in Calculus

12.1 L im its o f S equences 288

1 2 .2 Infinite Series 295

1 2 .3 L im its o f Functions 300

1 2 .4 F undam ental P roperties o f L im its o f Functions 307

1 2 .5 C ontinuity 312

12.6 D ifferentiability 314

E xercises for C hapter 12 317

13

Proofs in Group Theory

1 3 .1 B inary O perations 322

1 3 .2 G roups 326

1 3 .3 Perm utation G roups 330

1 3 .4 F undam ental P roperties o f G roups 333

1 3 .5 S ubgroups 336

1 3 .6 Isom orphic G roups 340

E xercises for C hapter 13 344

14

Proofs in Ring Theory (Online)

1 4 .1 Rings

1 4 .2 E lem entary P roperties o f R ings

1 4 .3 Subrings

1 4 .4 Integral D om ains

1 4 .5 Fields

E xercises for C hapter 14

115

Proofs in Linear Algebra (Online)

1 5 .1 P roperties o f Vectors in 3-Space

1 5 .2 Vector Spaces

1 5 .3 M atrices

1 5 .4 Som e P roperties o f Vector Spaces

288

1 5 .5 Subspaces

1 5 .6 Spans o f Vectors

1 5 .7 L inear D ependence and Independence

1 5 .8 L inear T ransform ations

1 5 .9 P roperties o f L inear T ransform ations

E xercises for C hapter 15

viii

1 6 .1 M etric Spaces

1 6 .2 O pen Sets in M etric Spaces

1 6 .3 C ontinuity in M etric Spaces

1 6 .4 T opological Spaces

1 6 .5 C ontinuity in T opological Spaces

E xercises for C hapter 16

Answers and Hints to Selected Odd-Numbered

Exercises in Chapters 14-16 (online)

351

394

395

396

Answers and Hints to Odd-Numbered Section Exercises

References

Index of Symbols

Index

PREFACE TO THE THIRD

EDITION

As w e m entioned in the prefaces o f the first tw o editions, because the teaching o f calculus in m any colleges and universities has becom e m ore problem -oriented w ith added em ­ phasis on the use o f calculators and com puters, the theoretical gap betw een the m aterial presented in calculus and the m athem atical background expected (or at least hoped for) in m ore advanced courses such as abstract algebra and advanced calculus has w idened. In an attem pt to narrow this gap and to b etter p repare students for the m ore abstract m athem atics courses to follow, m any colleges and universities have introduced courses that are now com m only called transition courses. In these courses, students are in tro ­ duced to problem s w hose solution involves m athem atical reasoning and a know ledge o f p ro o f techniques and w riting clear proofs. Topics such as relations, functions and cardi­ nalities o f sets are encountered throughout theoretical m athem atics courses. In addition, transition courses often include theoretical aspects o f num ber theory, abstract algebra, and calculus. This textbook has been w ritten for such a course.

The idea for this textbook originated in the early 1980s, long before transition courses becam e fashionable, during the supervising o f undergraduate m athem atics re ­ search projects. We cam e to realize that even advanced undergraduates lack a sound understanding o f p ro o f techniques and have difficulty w riting correct and clear proofs. A t that tim e, a set o f notes w as developed for these students. This w as follow ed by the introduction o f a transition course, for w hich a m ore detailed set o f notes was w ritten. The first edition o f this book em anated from these notes, w hich in turn has led to a second edition and now this third edition.

W hile understanding proofs and p ro o f techniques and w riting good proofs are m ajor goals here, these are not things that can be accom plished to any great degree in a single course during a single sem ester. T hese m ust continue to be em phasized and p racticed in succeeding m athem atics courses.

Our A pproach

Since this textbook originated from notes that w ere w ritten exclusively for undergradu­ ates to help them understand p ro o f techniques and to w rite good proofs, this is the tone

Preface to the T hird Edition

in w hich all editions o f this book have been w ritten: to be student-friendly. N um erous exam ples o f proofs are presented in the text. F ollow ing com m on practice, w e indicate the end o f a p ro o f w ith the square sym bol ■. O ften we precede a p ro o f by a discussion, referred to as a p r o o f strategy, w here we think through w hat is n eeded to present a pro o f o f the result in question. O ther tim es, w e find it useful to reflect on a p ro o f we have ju st presented to point out certain key details. We refer to a discussion o f this type as a p r o o f

analysis. Periodically, problem s are p resented and solved, and we m ay find it convenient

to discuss som e features o f the solution, w hich w e refer to sim ply as an analysis. For clarity, we indicate the end o f a discussion o f a p ro o f strategy, p ro o f analysis, analysis or solution o f an exam ple w ith the diam ond sym bol # .

A m ajor goal o f this textbook is to help students learn to construct proofs o f their ow n that are not only m athem atically correct but clearly w ritten. M ore advanced m athe­ m atics students should strive to present proofs that are convincing, readable, notationally consistent, and gram m atically correct. A secondary goal is to have students gain suffi­ cient know ledge o f and confidence w ith proofs so that they will recognize, understand, and appreciate a p ro o f that is properly w ritten.

As w ith the first tw o editions, the third edition o f this book is intended to assist the student in m aking the transition to courses that rely m ore on m athem atical p ro o f and reasoning. We envision students w ould take a course based on this book after they have had a y ear o f calculus (and possibly another course, such as elem entary linear algebra). It is likely that, prio r to taking this course, a stu d e n t’s training in m athem atics consisted prim arily o f doing p atterned problem s; that is, students have been taught m ethods for solving problem s, likely including som e explanation as to w hy these m ethods worked. Students m ay very w ell have had exposure to som e proofs in earlier courses but, m ore than likely, w ere unaw are o f the logic involved and o f the m ethod o f p ro o f being used. There m ay have even been tim es w hen the students w ere not certain w hat w as being proved.

O utline of the Contents

Since w riting good proofs requires a certain degree o f com petence in w riting, w e have devoted C hapter 0 to w riting m athem atics. The em phasis o f this chapter is on effective and clear exposition, correct usage o f sym bols, w riting and displaying m athem atical expressions, and using key w ords and phrases. A lthough every instructor w ill em phasize w riting in his or h er ow n way, we feel that it is useful to read C hapter 0 periodically throughout the course. It w ill m ean m ore as the student progresses through the course.

A m ong the additions to and changes in the second edition that resulted in this third edition are the follow ing.

® M ore than 250 exercises have been added, m any o f w hich require m ore thought to solve.

® N ew exercises have been added dealing w ith conjectures to give students practice w ith this im portant aspect o f m ore advanced m athem atics.

® A dditional exam ples have been provided to assist in understanding and solving new exercises.

® In a num ber o f instances, expanded discussions o f a topic have been given to provide added clarity. In particular, the im portant topic o f quantified statem ents is introduced in S ection 2.10 and then review ed in Section 7.2 to enhance o n e ’s understanding o f this.

Preface to the T hird Edition

xi

• A discussion o f cosets and L ag ran g e’s theorem has been added to C hapter 13 (Proofs in G roup Theory).

E ach chapter is divided into sections and the exercises for each chapter occur at the end o f the chapter, divided into sections in the sam e way. T here is also a final section of exercises for the entire chapter.

C hapter 1 contains a gentle introduction to sets, so that everyone has the sam e back­ ground and is using the sam e notation as w e prepare for w hat lies ahead. N o proofs in­ volving sets occur until C hapter 4. M uch o f C hapter 1 m ay very w ell be a review fo r many. C hapter 2 deals exclusively w ith logic. The goal here is to p resent w hat is needed to get into proofs as quickly as possible. M uch o f the em phasis in C hapter 2 is on statem ents, im plications, and quantified statem ents, including a discussion o f m ixed quantifiers. Sets are introduced before logic so that the stu d en t's first encounter w ith m athem atics here is a fam iliar one and because sets are needed to discuss quantified statem ents properly in C hapter 2.

The tw o p ro o f techniques o f direct p ro o f and p ro o f by contrapositive are introduced in C hapter 3 in the fam iliar setting o f even and odd integers. P ro o f by cases is discussed in this chapter as well as proofs o f “ if and only i f ” statem ents. C hapter 4 continues this discussion in other settings, nam ely divisibility o f integers, congruence, real num bers, and sets.

The technique o f p ro o f by contradiction is introduced in C hapter 5. Since existence proofs and counterexam ples have a connection w ith p ro o f by contradiction, these also occur in C hapter 5. The topic o f uniqueness (of an elem ent w ith specified properties) is also addressed in C hapter 5.

P roof by m athem atical induction occurs in C hapter 6 . In addition to the P rinciple o f M athem atical Induction and the S trong P rinciple o f M athem atical Induction, this chapter includes p ro o f by m inim um counterexam ple. The m ain goal o f C hapter 7 (Prove or D isprove) concerns the testing o f statem ents o f unknow n truth value, w here it is to be determ ined, w ith justification, w hether a given statem ent is true or false. In addition to the challenge o f determ ining w hether a statem ent is true or false, such problem s provide added practice w ith counterexam ples and the various p ro o f techniques. Testing statem ents is preceded in this chapter w ith an historical discussion o f conjectures in m athem atics and a review o f quantifiers.

C hapter 8 deals w ith relations, especially equivalence relations. M any exam ples in­ volving congruence are presented and the set o f integers m odulo n is described. C hapter 9 involves functions, w ith em phasis on the properties o f one-to-one and onto. This gives rise to a d iscussion o f bijective functions and inverses o f functions. The w ell-defined property o f functions is discussed in m ore detail in this edition. In addition, there is a discussion o f im ages and inverse im ages o f sets w ith regard to functions and a num ber o f added exercises involving these concepts.

C hapter 10 deals w ith infinite sets and a discussion o f cardinalities o f sets. This chapter includes an historical discussion o f infinite sets, beginning w ith C antor and his fascination and difficulties with the S ch ro d er-B ern stein Theorem , then to Z erm elo and the A xiom o f C hoice, and ending w ith a p ro o f o f the S ch ro d er-B ern stein Theorem .

A ll o f the p ro o f techniques are used in C hapter 1 1 w here num erous results in the area o f num ber theory are introduced and proved. C hapter 12 deals w ith proofs that occur in calculus. B ecause these proofs are quite different than those previously encountered but are often m ore predictable in nature, m any illustrations are given that involve lim its o f

x ii Preface to the T hird Edition

sequences and lim its o f functions and their connections w ith infinite series, continuity, and differentiability. The final C hapter 13 deals with m odem algebra, beginning with binary operations and m oving into proofs that are encountered in the area o f group theory.

Web Site for M athem atical Proofs

T hree additional chapters, C hapters 14 -1 6 (dealing w ith proofs in ring theory, linear algebra, and topology), can be found on the W eb site: http://w w w .aw .com /info/chartrand.

T eaching a C ourse from T his Text

A lthough a course using this textbook could be designed in m any w ays, here are our view s on such a course. As we noted earlier, w e think it is useful for students to reread (at least portions of) C hapter 0 throughout the course as we feel that w ith each reading, the chapter becom es m ore m eaningful. T he first part o f C hapter 1 (Sets) w ill likely be fam iliar to m ost students, although the last part m ay not. C hapters 2 - 6 w ill probably be part o f any such course, although certain topics could receive varying degrees o f em phasis (w ith perhaps p ro o f by m inim um counterexam ple in C hapter 6 possibly even om itted). O ne could spend little or m uch tim e on C hapter 7, depending on how m uch tim e is used to discuss the large num ber o f “prove or disprove” exercises. We think that m ost o f C hapters 8 and 9 w ould be covered in such a course. It w ould be useful to cover som e o f the fundam ental ideas addressed in C hapter 10 (C ardinalities o f Sets). A s tim e perm its, portions o f the later chapters could be covered, especially those o f interest to the instructor, including the possibility o f going to the W eb site for even m ore variety in the three online chapters.

E xercises

T here are num erous exercises for C hapters 1-13 (as w ell as for C hapters 14 -1 6 on the W eb site). The degree o f difficulty o f the exercises ranges from routine to m edium difficulty to m oderately challenging. As m entioned earlier, the third edition contains m ore exercises in the m oderately challenging category. T here are exercises that present students w ith statem ents, asking them to decide w hether they are true or false (w ith justification). T here are proposed proofs o f statem ents, asking if the argum ent is valid. T here are proofs w ithout a statem ent given, asking students to supply a statem ent o f w hat has been proved. A lso, there are exercises that call upon students to m ake conjectures o f their ow n and possibly to provide proofs o f these conjectures.

C hapter 3 is the first chapter in w hich students w ill be called upon to w rite proofs. A t such an early stage, we feel that students need to (1) concentrate on constructing a valid p ro o f and not be distracted by unfam iliarity w ith the m athem atics, (2 ) develop som e self-confidence w ith this process, and (3) learn how to w rite a p ro o f properly. W ith this in m ind, m any o f the exercises in C hapter 3 have been intentionally structured so as to be sim ilar to the exam ples in that chapter.

In general, there are exercises for each section at the end o f a chapter (section exercises) and additional exercises for the entire chapter (chapter exercises). A nsw ers or

Preface lo the Third Edition

xiii

hints to the odd-num bered section exercises appear at the end o f text. O ne should also keep in m ind, however, that proofs o f results are not unique in general.

A ck n ow led gm en t s

It is a pleasure to thank the review ers o f the third edition: D aniel A costa, S outheastern L ouisiana U niversity Scott A nnin, C alifornia State U niversity, F ullerton J. M arshall A sh, D eP aul U niversity

A ra B asm ajian, H unter C ollege o f CU N Y M atthias B eck, San F rancisco State U niversity R ichard B elshoff, M issouri State U niversity

Jam es Braw ner, A rm strong A tlantic State U niversity M anav D as, U niversity o f L ouisville

D avid D em psey, Jacksonville State U niversity

C ristina D om okos, C alifornia State U niversity, Sacram ento Jose D. F lores, U niversity o f South D akota

Eric G ottlieb, Rhodes College

R ichard H am m ack, V irginia C om m onw ealth U niversity A lan K och. A gnes S cott College

M. H arper L angston, C ourant Institute o f M athem atical Sciences, N ew York U niversity M aria N ogin, C alifornia State U niversity, F resno

D aniel N ucinkis, U niversity o f S outham pton T hom as P olaski, W inthrop U niversity John R andall, R utgers U niversity E ileen T. Shugart, V irginia Tech

B rian A. Snyder, Lake S uperior State U niversity M elissa S utherland, SU N Y G eneseo

M .B. U lm er, U niversity o f South C arolina U pstate M ike W inders, W orcester State U niversity

We also thank R enato M irollo, B oston C ollege and Tom W eglaitner for giving a final reading o f portions o f the third edition.

We have been m ost fortunate to receive the enthusiastic support from m any at Pearson. F irst, w e w ish to thank the editorial team , as w ell as others at P earson who have been so helpful and supportive: G reg Tobin, Publisher, M athem atics and S tatis­ tics; W illiam H offm an, S enior A cquisitions Editor; Jeff W eidenaar, E xecutive M arket­ ing M anager, M athem atics; and B randon Raw nsley, A ssociate Editor, A rts & Sciences, H igher E ducation. O ur thanks to all o f you. Finally, thank you as w ell to B eth H ouston, S enior P roduction P roject M anager; K ailash Jadli, P roject M anager, A ptara, Inc.; and M ercedes H eston, C opy Editor, for guiding us through the final stages o f the third edition.

G ary C hartrand A lbert D. P olim eni P ing Z hang

'

0

C om m unicating M athem atics

uite likely, the m athem atics you have already encountered consists o f doing pro b ­ lem s using a specific approach or procedure. These m ay include solving equations

sim plifying algebraic expressions, verifying trigonom etric identities, using certain rules to find and sim plify the derivatives o f functions and setting up and evaluat­ ing a definite integral that give the area o f a region o r the volum e o f a solid. A ccom plishing all o f these is often a m atter o f practice.

M any o f the m ethods that one uses to solve problem s in m athem atics are b ased on results in m athem atics that w ere discovered by people and show n to be true. This kind o f m athem atics m ay very w ell be new to you and, as w ith anything th a t’s new, there are things to be learned. B ut learning som ething new can be (in fact should be) fun. There are several steps involved here. The first step is discovering som ething in m athem atics that w e believe to be true. H ow does one discover new m athem atics? T his usually com es about by considering exam ples and observing that a pattern seem s to be occurring w ith the exam ples. T his m ay lead to a guess on our p art as to w hat appears to be happening. W e then have to convince ourselves that our guess is correct. In m athem atics this involves constructing a p ro o f o f w hat we believe to be true is, in fact, true. B ut this is not enough. We need to convince others that w e are right. So w e need to w rite a p ro o f that is w ritten so clearly and so logically that people w ho know the m ethods o f m athem atics w ill be convinced. W here m athem atics differs from all other scholarly fields is that once a pro o f has been given o f a certain m athem atical statem ent, there is no longer any doubt. This statem ent is true. Period. T here is no other alternative.

O ur m ain em phasis here w ill be in learning how to construct m athem atical proofs and learning to w rite the p ro o f in such a m anner that it w ill be clear to and understood by others. E ven though learning to guess new m athem atics is im portant and can be fun, w e w ill spend only a little tim e on this as it often requires an understanding o f m ore m athem atics than can be discussed at this tim e. B ut w hy w ould we w ant to discover new m athem atics? W hile one possible answ er is that it com es from the curiosity that m ost m athem aticians possess, a m ore com m on explanation is that we have a p roblem to solve that requires know ing that som e m athem atical statem ent is true.

2

Chapter 0

L earning M athem atics

O ne o f the m ajor goals o f this book is to assist you as you progress from an individual w ho uses m athem atics to an individual w ho understands m athem atics. P erhaps this w ill m ark the beginning o f you becom ing som eone w ho actually develops m athem atics o f your own. This is an attainable goal if y ou have the desire.

T he fact that y o u 'v e gone this far in your study o f m athem atics suggests that you have ability in m athem atics. This is a real opportunity for you. M uch o f the m athem atics that you w ill encounter in the future is based on w hat you are about to learn here. The b etter you learn the m aterial and the m athem atical thought process now, the m ore you w ill u nderstand later. To be sure, any area o f study is considerably m ore enjoyable w hen you understand it. B ut getting to that point will require effort on your part.

T here are probably as m any excuses for doing poorly in m athem atics as there are strategies for doing w ell in m athem atics. We have all heard students say (som etim es, rem arkably, even w ith pride) that they are not good at m athem atics. T h a t’s only an alibi. M athem atics can be learned like any other subject. E ven som e students w ho have done w ell in m athem atics say that they are not good w ith proofs. T his, too, is unacceptable. W hat is required is determ ination and effort. To have done w ell on an exam w ith little o r no studying is nothing to be proud of. Confidence based on being w ell-prepared is good, however.

H ere is som e advice that has w orked for several students. F irst, it is im portant to understand w hat goes on in class each day. This m eans being p resent and being prepared for every class. A fter each class, recopy any lecture notes. W hen recopying the notes, express sentences in your ow n w ords and add details so that everything is as clear as possible. If you run into snags (and you w ill), talk them over w ith a classm ate or your instructor. In fact, it’s a good idea (at least in our opinion) to have som eone w ith w hom to discuss the m aterial on a regular basis. N ot only does it often clarify ideas, it gets you into the habit o f using correct term inology and notation.

In addition to learning m athem atics from your instructor, solidifying your under­ standing by careful note-taking and by talking w ith classm ates, yo u r text is (or at least should be) an excellent source as well. R ead your text carefully w ith pen (or pencil) and p aper in hand. M ake a serious effort to do every hom ew ork problem assigned and, eventually, be certain th at y ou know how to solve them . I f there are exercises in the text that have not been assigned, you m ight even try to solve these as well. A nother g ood idea is to try to create your ow n problem s. In fact, w hen studying for an exam , try creating your ow n exam . If you start doing this for all o f y our classes, you m ight be surprised at how good you becom e. C reativity is a m ajor part o f m athem atics. D iscovering m ath e­ m atics not only contributes to your understanding o f the subject but has the potential to contribute to m athem atics itself. C reativity can com e in all form s. The follow ing quote is due to the w ell-know n w riter J. K. R ow ling (author o f the H arry P otter novels).

Som etim es ideas ju s t com e to me. O ther tim es I have to sw eat a n d alm ost bleed to m ake ideas come. I t ’s a m ysterious p ro c e ss, but I hope I never fin d o ut exactly how it works.

Learning M athem atics 3

In the book D efying G ravity: The Creative C areer o f Stephen Schw artz fro m

G odspell to Wicked, the author C arol de G iere w rites a biography o f S tephen Schw artz,

one o f the m ost successful com poser-lyricists, in w hich she discusses not only creativ­ ity in m usic but how an idea can lead to som ething special and interesting and how creative people m ay have to deal w ith disappointm ent. Indeed, de G iere dedicates her book to the creative spirit w ithin each o f us. W hile he w rote the m usic for such fam ous show s as G odspell and W icked, S chw artz discusses creativity head-on in the song “The S park o f C reatio n ” he w rote for the m usical C hildren o f E den. In h er book, de G iere writes:

In m any w ays, this song expresses the them e o f Stephen S ch w a rtz’s life— the naturalness a n d im portance o f the creative urge w ithin us. A t the sam e tim e he created an anthem f o r artists.

In m athem atics our goal is to seek the truth. Finding answ ers to m athem atical questions is im portant, but we cannot be satisfied w ith this alone. We m ust be certain that w e are right and that our explanation for w hy we believe w e are correct is convincing to others. T he reasoning w e use as w e proceed from w hat we know to w hat we w ish to show m ust be logical. It m ust m ake sense to others, not ju st to ourselves.

There is jo in t responsibility here. As w riters, it is ou r responsibility to give an accurate clear argum ent w ith enough details provided to allow the reader to understand w hat w e have w ritten and to be convinced. It is the re a d e r’s responsibility to know the basics o f logic and to study the concepts involved so that a w ell-presented argum ent w ill be understood. C onsequently, in m athem atics w riting is im portant, very im portant. Is it really im portant to w rite m athem atics w ell? A fter all, is n ’t m athem atics m ainly equations and sym bols? N ot at all. It is not only im portant to w rite m athem atics well, it is im portant to w rite w ell. You will be w riting the rest o f your life, at least reports, letters and e-m ail. M any people w ho never m eet you w ill know you only by w hat you w rite and how you w rite.

M athem atics is a sufficiently com plicated subject that we d o n 't need vague, hazy and boring w riting to add to it. A teacher has a very positive im pression o f a student w ho hands in w ell-w ritten and w ell-organized assignm ents and exam inations. You w ant people to enjoy reading w hat y o u ’ve w ritten. It is im portant to have a good reputation as a writer. I t ’s part o f being an educated person. E specially with the large num ber o f e-m ail letters that so m any o f us w rite, it has becom e com m onplace for w riting to be m ore casual. A lthough all people w ould probably subscribe to this (since it is m ore efficient), w e should know how to w rite w ell form ally and professionally w hen the situation requires it.

You m ight think that considering how long y o u ’ve been w riting and that y o u 're set in your w ays, it w ill be very difficult to im prove yo u r w riting. N ot really. If you w ant to im prove, you can and w ill. E ven if you are a good w riter, your w riting can alw ays be im proved. O rdinarily, people d o n ’t think m uch about th eir w riting. O ften ju s t thinking about your w riting is the first step to w riting better.

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Chapter 0

W hat Others H ave S aid ab ou t

Writing-M any people w ho are w ell know n in their areas o f expertise have expressed their thoughts about w riting. H ere are quotes by som e o f these individuals.

A nything that helps com m unication is good. A nything that hurts it is bad. I like words m ore than num bers, an d I alw ays did— conceptual more than co m ­ putational.

Paul H alm os, m athem atician

W riting is easy. A ll you have to do is cross out all the w rong words.

M ark Twain, author (The A dventures o f H uckleberry F inn)

You d o n ’t w rite because you w a n t to say som ething; yo u w rite because yo u ’ve g o t som ething to say.

F. Scott Fitzgerald, author (The G reat G atsby)

W riting com es m ore easily i f yo u have som ething to say.

S cholem A sch, author

E ith e r w rite som ething w orth reading or do som ething w orth writing.

B enjam in F ranklin, statesm an, writer, inventor

W hat is w ritten w ithout effort is in general read w ithout pleasure.

S am uel Johnson, w riter

E a sy reading is d a m n ed hard w riting.

N athaniel H aw thorne, novelist (The S carlet Letter)

E verything th a t is w ritten m erely to p le a se the author is w orthless. The last thing one know s w hen w riting a b o o k is w hat to p u t first. I have m ade this letter longer because I lack the tim e to m ake it short.

B laise Pascal, m athem atician and physicist

The best w ay to becom e a cquainted w ith a subject is to w rite a book a bout it.

B enjam in D israeli, prim e m inister o f E ngland

In a very real sense, the w riter w rites in order to teach him self, to understand h im ­ self, to satisfy him self; the p u b lish in g o f his ideas, though it brings gratification, is a curious anticlim ax.

A lfred K azin, literary critic

The skill o f w riting is to create a context in w hich other p eo p le can think.

E dw in S chlossberg, exhibit designer

A w riter needs three things, experience, observation, an d im agination, any two o f w hich, a t tim es any one o f w h ich , can supply the lack o f the other.

W illiam Faulkner, w riter (The So u n d an d the F ury)

I f confusion runs ram pant in the p a ssa g e ju s t read, It m ay very w ell be that too m uch has been said.

M athem atical Writing 5

So th a t's w hat he m eant! Then w hy d id n ’t he say so?

F rank Harary, m athem atician A m athem atical theory is not to be considered com plete until yo u have m ade it

so clear that you can explain it to the fir st m an w hom you m eet on the street.

D avid H ilbert, m athem atician

E verything sh o u ld be m ade as sim ple as possible, but not simpler.

A lbert E instein, physicist

N e ve r let anything you w rite be p u b lish ed w ithout having ha d others critique it.

D onald E. K nuth, com puter scientist and w riter

Som e books are to be tasted, others to be swcdlowed, an d som e fe w to be chew ed an d digested.

R eading m aketh a f u l l m an, conference a ready m an, an d w riting an exact mcm.

Francis B acon, w riter and philosopher

fu d g e an article not by the quality o f w hat is fr a m e d a n d hanging on the wall, but by the quality o f w hat 's in the w astebasket.

A nonym ous (Q uote by L eslie L am port)

We are a ll apprentices in a craft where no-one ever becom es a master.

E rnest H em ingw ay, author (For W hom the B ell Tolls)

There are three rules f o r w riting a novel. U nfortunately, no one know s w hat they are.

W. Som erset M augham , author (O f H um an Bondage)

M athem atical W riting

M ost o f the quotes given above p ertain to w riting in general, not to m athem atical w riting in particular. H ow ever these suggestions for w riting apply as w ell to w riting m athem at­ ics. F or us, m athem atical w riting m eans w riting assignm ents for a m athem atics course (particularly a course w ith proofs). Such an assignm ent m ight consist o f w riting a single proof, w riting solutions to a n um ber o f problem s or perhaps w riting a term p ap er w hich, m ore than likely, includes definitions, exam ples, background an d proofs. W e’ll refer to any o f these as an assignm ent. Your goal should be to w rite correctly, clearly and in an interesting m anner.

B efore you even begin to w rite, you should have already thought about a num ber o f things. F irst, you should know w hat exam ples and proofs you p lan to include if this is appropriate for your assignm ent. You should not be overly concerned about w riting good proofs on your first attem pt— but be certain that you do have proofs.

As y o u ’re w riting your assignm ent, you m ust be aw are o f your audience. W hat is the target group for your assignm ent? O f course, it should be w ritten for yo u r instructor. B ut it should be w ritten so that a classm ate w ould understand it. A s you grow m athem atically, your audience w ill grow w ith you and you w ill adapt your w riting to this new audience.

G ive y ourself enough tim e to w rite your assignm ent. D o n ’t try to put it together ju st a few m inutes before it’s due. The disappointing result w ill be obvious to your

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instructor. A nd to you! F ind a place to w rite that is com fortable for you: your room , an office, a study room , the library and sitting at a desk, at a table, in a chair. D o w hat w orks best for you. Perhaps you w rite best w hen it ’s quiet or w hen there is background m usic.

N ow that y o u ’re com fortably settled and have allow ed enough tim e to do a good job, le t’s p u t a plan together. If the assignm ent is fairly lengthy, you m ay need an outline,

w hich, m ost likely, w ill include one or m ore o f the follow ing:

1. B ackground and m otivation

2. The definitions to be presented and possibly the notation to be used

3. The exam ples to include

4. The results to be presented (w hose proofs have already been w ritten, probably

in rough form )

5. R eferences to other results you intend to use

6 . The order o f everything m entioned above.

I f the assignm ent is a term paper, it m ay include extensive background m aterial and m ay need to be carefully m otivated. The subject o f the pap er should be placed in perspective. W here does it fit in w ith w hat w e already know ?

M any w riters w rite in spirals. E ven though you have a plan for your assignm ent w hich includes an ordered list o f things you w ant to say, it is likely that you w ill reach som e poin t (perhaps sooner than you think) w hen you realize that you should have in ­ cluded som ething earlier— perhaps a definition, a theorem , an exam ple, som e notation. (This happened to us m any tim es w hile w riting this textbook.) Insert the m issing m aterial, start over again and w rite until once again y ou realize that som ething is m issing. It is im ­ portant, as you reread, that you start at the beginning each tim e. T hen repeat the steps listed above.

We are about to give you som e advice, som e pointers, about w riting m athem atics. S uch advice is necessarily subjective. N ot everyone subscribes to these suggestions on w riting. Indeed, w riting experts d o n ’t agree on all issues. F or the present, yo u r instructor will be your best guide. B ut w riting does not follow a list o f rules. As you m ature m athem atically, perhaps the best advice about your w riting is the sam e advice given by Jim iny C ricket to D isn e y ’s Pinocchio: A lw a y s let yo u r conscience be y o u r guide. You m ust be yourself. A nd one additional piece o f advice: B e careful about accepting advice on w riting. O riginality and creativity d o n ’t follow rules. U ntil you reach the stage o f being com fortable and confident w ith your ow n w riting, however, w e believe that it is useful to consider a few w riting tips.

Since a num ber o f these w riting tips m ay not m ake sense (since, after all, w e d o n ’t even have anything to w rite as yet), it w ill probably be m ost useful to return to this chapter p eriodically as you proceed through the chapters that follow.

U sing Sym bols

Since m athem atics is a sym bol-oriented subject, m athem atical w riting involves a m ixture o f w ords and sym bols. H ere are several guidelines to w hich a num ber o f m athem aticians subscribe.

Using Symbols 7

1. N e ve r sta rt a sentence w ith a symbol.

W riting m athem atics follow s the sam e practice as w riting all sentences, nam ely that the first w ord should be capitalized. This is confusing if the sentence w ere to begin w ith a sym bol since the sentence appears to be incom plete. A lso, in general, a sentence sounds better if it starts w ith a word. Instead o f w riting:

x 2 — 6x + 8 = 0 has tw o distinct roots. write:

The equation x 2 — 6x + 8 = 0 has tw o distinct roots.

2. Separate sym bols n o t in a list by words i f possible.

S eparating sym bols by w ords m akes the sentence easier to read and therefore easier to understand. The sentence:

E xcept for a, b is the only root o f (x — a ) (x — b) = 0. w ould be clearer if it w ere w ritten as:

E xcept for a, the num ber b is the only ro o t o f (x — a ) (x — b) — 0. 3. E xcep t w hen discussing logic, avo id w riting the fo llo w in g sym bols in yo u r

assignm ent:

—>. V, 3, 9 , e tc .

The first four sym bols stand for "im p lies” , “for every” , “there ex ists” and “ such th a t” , respectively. You m ay have already seen these sym bols and know w hat they m ean. If so, this is good. It is useful w hen taking notes or w riting early drafts o f an assignm ent to use shorthand sym bols but m any m athem aticians avoid such sym bols in their professional w riting. (We will visit these sym bols later.)

4. B e careful a bout using i.e. an d e.g.

T hese stand for that is a n d /o r exam ple, respectively. T here are situations w hen w riting the w ords is preferable to w riting the abbreviations as there m ay be confusion w ith nearby sym bols. F or exam ple, y — 1 and

( 1 Y

lim 1 H— are not rational num bers, that is, i and e are not rational

n—>oo \ n J

num bers.

5. W rite out integers as w ords w hen they are used as adjectives an d when the num bers are relatively sm a ll or are easy to describe in words. W rite out num bers num erically w hen they specify the value o f som ething.

There are exactly tw o groups o f o rder 4. Fifty m illion F renchm en c a n ’t be w rong.

There are one m illion positive integers less than 1,000,001. 6 . D o n 't m ix w ords a n d sym bols improperly.

Instead o f w riting:

8

Chapter 0

Every integer exceeding 1 is a prim e or is com posite. or

If n > 2 is an integer, then n is prim e or com posite. A lthough

Since (.v — 2)(x — 3) = 0, it follow s that x = 2 or 3.

sounds correct, it is n o t w ritten correctly. It should be:

Since (x — 2)(.v — 3) = 0, it follow s that x = 2 or x = 3. 7. A vo id using a sym bol in the statem ent o f a theorem w hen it's not needed.

D o n 't w rite:

T heorem E very bijective fu n c tio n f has an inverse.

D elete “ / ” . It serves no useful purpose. The theorem does not depend on w hat the function is called. A sym bol should not be used in the statem ent o f a theorem (or in its proof) exactly once. If it is useful to have a nam e for an arbitrary bijective function in the p ro o f (as it probably w ill be), then “ / ” can be introduced there.

8 . E xplain the m eaning o f every sym bol that you introduce.

A lthough w hat you intended m ay seem clear, d o n ’t assum e this. F or exam ple, if you w rite n = 2k + \ and k has never appeared before, then say

that k is an integer (if indeed k is an integer). 9. Use “fro ze n sym bols" properly.

I f m and n are typically used fo r integers (as they probably are), then d o n ’t use them for real num bers. I f A and B are used fo r sets, then d o n ’t use these as typical elem ents o f a set. If / is used for a function, then d o n ’t use this as an integer. W rite sym bols that the read er w ould expect. To do otherw ise could very w ell confuse the reader.

10. Use consistent sym bols.

U nless there is som e special reason to the contrary, use sym bols that “fit” together. O therw ise, it is distracting to the reader.

Instead o f w riting

I f x and y are even integers, then x = 2a and y = 2r for som e integers a and r.

replace 2r by 2b (w here then, o f course, we w rite “for som e integers a and

b ”). O n the other hand, you m ight p refer to w rite x — 2r and y = 2s.

W riting M ath em atical E x p ressio n s

T here w ill be num erous occasions w hen you w ill w ant to w rite m athem atical expres­ sions in your assignm ent, such as algebraic equations, inequalities, and form ulas. If

Writing M athem atical Expressions 9

these expressions are relatively short, then they should probably be w ritten w ithin the text o f the p ro o f or discussion. (W e’ll explain this in a m om ent.) If the expressions are rath er lengthy, then it is probably preferred for these expressions to be w ritten as

displays.

F or exam ple, suppose that we are discussing the B inom ial Theorem . (It’s not im ­ portant if you d o n ’t recall w hat this theorem is.) I t ’s possible that w hat w e are w riting includes the follow ing passage:

For exam ple, if we expand (,a + b)4 , then we obtain (a + b)4 = a 4 + 4 a 3b + 6 a2b2 + 4 a b3 + b4.

It w ould probably be b etter to w rite the expansion o f (a + b) 4 as a d isp la y , w here

the m athem atical expression is placed on a line or lines by itself and is centered. This is illustrated below.

F or exam ple, if we expand (a + b )4 , then w e obtain

(a + b )4 = a4 + 4 a 3b + 6 a2b2 + 4 a b3 + b4.

If there are several m athem atical expressions that are linked by equal signs and inequality sym bols, then we w ould alm ost certainly w rite this as a display. For exam ple, suppose that we w anted to w rite n3 + 3 n2 — n + 4 in term s o f k , w here n = 2 k + 1. A

possible display is given next: Since n = 2 k + 1, it follow s that

n3 + 3 n~ - n + 4 = (2k + l )3 + 3(2 k + i f - (2k + 1) + 4

= (8£3 + 12k2 + 6k + 1) + 3(4k2 + 4 k + 1) - 2 k - 1 + 4

= 8/r3 + 2 4 k2 + 16k + 7 = 8 k3 + 2 4 k 2 + 16 k + 6 + 1

= 2(4 k3 + 12k1 + 8Ar + 3) + 1.

N otice how the equal signs are lined up. (We w rote tw o equal signs on one line since that line w ould have contained very little m aterial otherw ise, as well as to balance the lengths o f the lines better.)

L e t’s return to the expression (a + b) 4 — a4 + 4 a 3b + 6a2b2 + 4a b3 + b4 for the

m om ent. I f we w ere to w rite this expression in the text o f a paragraph (as we are doing) and if we find it necessary to w rite portions o f this expression on tw o separate lines, then this expression should be broken so that the first line ends w ith an operation or com parative sym bol s u c h a s + , —, < , > o r = . I n other w ords, the second line should not begin w ith one o f these sym bols. The reason for doing this is that ending the line w ith one o f these sym bols alerts the read er that m ore w ill follow ; otherw ise, the reader m ight conclude (incorrectly) that the portion o f the expression appearing on the first line is the entire expression. C onsequently, write

F or exam ple, if we expand (a + b)4, then we obtain (a + b)4 = a4 + 4a 3b + 6a2b2 + 4a b3 + b4.

and not

F or exam ple, if we expand (a + b)4, then we obtain (a + b)4 = a4 + 4a 3b

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Chapter 0

If there is an occasion to refer to an expression that has already appeared, then this expression should have been w ritten as a display and labeled as below:

(ia + b) 4 = a4 + 4 a 3b + 6 a2b2 + 6 a b2 + b 4. (1)

Then w e can sim ply refer to expression (1) rather than w riting it out each tim e.

C om m on W ords an d P h rases in M athem atics

T here are som e w ords and phrases that appear so often in m athem atical w riting that it is useful to discuss them.

1. I We One L e t’s

I w ill now show that n is even. We w ill now show that n is even. O ne now show s that n is even. L et's now show that n is even.

T hese are fo u r ways that w e m ight w rite a sentence in a proof. W hich o f these sounds the best to you? It is not considered good practice to use “I” unless you are w riting a personal account o f som ething. O therw ise, “I ” sounds egotistical and can be annoying. U sing “ one” is often aw kw ard. U sing “w e” is standard practice in m athem atics. T his word also brings the reader into the discussion w ith the author and gives the im pression o f a team effort. T he w ord “le t’s” accom plishes this as w ell but is m uch less form al. T here is a danger o f being too casual, however. In general, your w riting should be balanced, m aintaining a professional style. O f course, there is the possibility o f avoiding all o f these words:

The integer n is now show n to be even. 2. C learly O bviously O f course C ertainly

T hese and sim ilar w ords can turn a reader off if w h at’s w ritten is n o t clear to the reader. It can give the im pression that the author is putting the reader down. T hese w ords should be used sparingly and w ith caution. If they are used, then at least be certain that w hat you say is true. T here is also the possibility that the w riter (a student?) has a lack o f understanding o f the m athem atics or is not being careful and is using these w ords as a cover-up. T his gives us even m ore reasons to avoid these words.

3. A ny E ach E very

This statem ent is true for any integer n.

D oes this m ean that the statem ent is true for som e integer n or all integers «? Since the w ord any can be vague, perhaps it is best to avoid it. If by any, we m ean each or every, then use one o f these two w ords instead. W hen the w ord

Common Words and Phrases in Mathematics

11

4. S in ce • • • , then ■ ■ ■

A num ber o f people connect these tw o w ords. You should use either “If ■ ■ •, then • ■ •” (should this be the intended m eaning) or “ Since ■ • •, it follow s that • • •” or, possibly, “ Since • • w e have • • For exam ple, it is correct to w rite

I f n2 is even, then n is even.

or

Since n2 is even, it follow s that n is even.

or perhaps

Since n2 is even, n is even.

but avoid

Since n2 is even, then n is even.

In this context, the w ord since can be replaced by because.

5. Therefore Thus H ence C onsequently So It fo llo w s that This im plies that

This is tricky. M athem aticians cannot survive w ithout these w ords. Often w ithin a proof, we proceed from som ething w e ’ve ju st learned to som ething else that can be concluded from it. There are m any (m any!) openings to sentences w hich attem pt to say this. A lthough each o f the w ords or phrases

Therefore Thus H en ce C onsequently So It fo llo w s that This im plies that

is suitable, it is good to introduce som e variety into your w riting and not use the sam e w ords or phrases any m ore often than necessary.

6. T hat W hich

These w ords are often confused w ith each other. S om etim es they are interchangeable; m ore often they are not.

The solution to the equation is the num ber less than 5 that is positive. (2)

The solution to the equation is the num ber less than 5 w hich is positive. (3)

W hich o f these tw o sentences is correct? The sim ple answ er is: B oth are correct— or, at least, both m ight be correct.

For exam ple, sentence (2) could be the response to the question: W hich o f the num bers 2, 3, and 5 is the solution o f the equation? S entence (3) could be the response to the question: W hich o f the num bers 4.9 and 5.0 is the solution o f the equation?

The w ord that introduces a restrictive clause and, as such, the clause is essential to the m eaning o f the sentence. T hat is, if sentence (2) w ere w ritten only as “The solution to the equation is the num ber less than 5” then the entire m eaning is changed. Indeed, we no longer know w hat the solution o f the equation is.

O n the other hand, the w ord w hich does n o t introduce a restrictive clause. It intro­ duces a nonrestrictive (or parenthetical) clause. A nonrestrictive clause only provides additional inform ation that is not essential to the m eaning o f the sentence. In sentence (3)

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Chapter 0

the p hrase “w hich is p ositive" sim ply provides m ore inform ation about the solution. This clause m ay have been added because the solution to an earlier equation is negative. In fact, it w ould be m ore appropriate to add a com m a:

T he solution to the equation is the num ber less than 5, w hich is positive. For another illustration, consider the follow ing tw o statem ents:

I alw ays keep the m ath text that I like w ith m e. (4)

I alw ays keep the m ath text w hich I like w ith me. (5 )

W hat is the difference betw een these tw o sentences? In (4), the w riter o f the sentence clearly has m ore than one m ath text and is referring to the one that he/she likes. In (5), the w riter has only one m ath text and is providing the added inform ation that he/she likes it. The nonrestrictive clause in (5) should be set off by com m as:

I alw ays keep the m ath text, w hich I like, w ith me.

A possible guideline to follow as you seek to determ ine w hether that or which is the p roper w ord to use is to ask yourself: D oes it sound right if it reads “w hich, by the w ay”? In general, that is norm ally used considerably m ore often than which. H ence the advice here is: B ew are o f w icked w h ich ’s!

W hile we are discussing the w ord that, w e m ention that the w ords assum e and

suppose often precede restrictive clauses and, as such, the w ord that should im m ed i­

ately follow one o f these w ords. O m itting that leaves us w ith an im plied that. M any m athem aticians prefer to include it rather than om it it.

In other w ords, instead o f w riting:

A ssum e N is a norm al subgroup. m any w ould write

A ssum e that N is a norm al subgroup.

Som e C losing C om m ents abou t W riting

1. U se good English. W rite in com plete sentences, ending each sentence w ith a

period (or a question m ark w hen appropriate) and capitalize the first w ord o f each sentence. (Rem em ber: N o sentence begins w ith a sym bol!)

2. C apitalize theorem and lem m a as in T heorem 1 and L em m a 4.

3. M any m athem aticians do not hyphenate w ords containing the prefix non, such

as

nonem pty, nonnegative, nondecreasing, nonzero.

4. M any w ords that occur often in m athem atical w riting are com m only

m isspelled. A m ong these are:

com m utative (independent o f order)

com plem ent (supplem ent, balance, rem ainder) consistent (conform ing, agreeing)

Some Closing Comments about Writing

feasible (suitable, attainable) its (possessive, not “ it is”) occurrence (incident) p arallel (non-intersecting) preceding (foregoing, form er) principle (postulate, regulation, rule) proceed (continue, m ove on)

and, o f course,

corollary, lem m a, theorem .

T here are m any pairs o f w ords that fit to gether in m athem atics (w hile interchanging w ords am ong the pairs do not). F or exam ple,

We ask questions. We pose problem s. We present solutions. We prove theorem s. We solve problem s, and

u

Sets

I

fundam ental id ea that occurs throughout m athem atics: sets. Indeed, a set is an object from w hich every m athem atical structure is constructed (as w e w ill often see in the succeeding chapters). A lthough there is a form al subject called se t theory in w hich the properties o f sets follow from a num ber o f axiom s, this is n either our interest nor our need. It is our desire to keep the discussion o f sets inform al w ithout sacrificing clarity. It is alm ost a certainty that portions o f this chapter w ill be fam iliar to you. N evertheless, it is im portant th at w e understand w hat is m eant by a set, how m athem aticians describe sets, the notation used w ith sets, and several concepts that involve sets.

Y ou’ve b een experiencing sets all y our life. In fact, all o f the follow ing are exam ples o f sets: the students in a particular class w ho have an iPod, the item s on a shopping list, the integers. As a sm all child, you learned to say the alphabet. W hen you did this, you w ere actually listing the letters that m ake up the set w e call the alphabet. A set is a collection o f objects. The objects that m ake up a set are called its elem en ts (or

m em bers). The elem ents o f a softball team are the players: w hile the elem ents o f the

alphabet are letters.

It is custom ary to use capital (upper case) letters (such as A, B, C , S, X , Y ) to designate sets and low er case letters (for exam ple, a, b, c, s, x , y ) to represent elem ents o f sets. If a is an elem ent o f the set A, then w e w rite a e A; if a does not belong to A, then w e w rite a £ A.

1.1 D escrib in g a Set

T here w ill be m any occasions w hen w e (or you) w ill need to describe a set. The m ost im ­ portant requirem ent w hen describing a set is that the description m akes it clear precisely w hich elem ents belong to the set.

If a set consists o f a sm all num ber o f elem ents, then this set can be described by explicitly listing its elem ents betw een braces (curly brackets) w here the elem ents are separated by com m as. Thus .S' — {I. 2, 3} is a set, consisting o f the num bers 1, 2 and 3. The order in w hich the elem ents are listed d o esn ’t matter. Thus the set S ju s t m entioned could be w ritten as S = {3, 2, 1} or S — {2, 1, 3}, for exam ple. T hey describe the sam e

1.1 Describing a Set

15

Exam ple 1.1

set. I f a set T consists o f the first five letters o f the alphabet, then it is not essential that w e w rite T = {a, b, c, d , e}\ that is, the elem ents o f T need not be listed in alphabet­ ical order. O n the other hand, listing the elem ents o f T in any other o rder m ay create unnecessary confusion.

The set A o f all p eople w ho signed the D eclaration o f Independence and later becam e president o f the U nited States is A = {John A dam s, T hom as Jefferson} and the set B o f all positive even integers less than 20 is B = {2, 4, 6 , 8 , 10, 12, 14, 16, 18}. Som e sets contain too m any elem ents to be listed this way. Perhaps even the set B ju s t given contains too m any elem ents to describe in this m anner. In such cases, the ellipsis or “three dot notation” is often helpful. F or exam ple, X = { 1 ,3 , 5 ...49} is the set o f all positive odd integers less than 50, w hile Y = {2, 4 , 6 , . . . } is the set o f all positive even integers. The three dots m ean “ and so on ” for Y and “and so on up to ” for X .

A set need n ot contain any elem ents. A lthough it m ay seem p eculiar to consider sets w ithout elem ents, these kinds o f sets occur surprisingly often and in a variety o f settings. For exam ple, if S is the set o f real num ber solutions o f the equation A'2 + 1 = 0 , then S contains no elem ents. T here is only one set that contains no elem ents, and it is called the

em p ty set (or som etim es the n ull set or void set). T he em pty set is denoted by 0. We

also w rite 0 = { }. In addition to the exam ple given above, the set o f all real num bers x such that .v" < 0 is also empty.

The elem ents o f a set m ay in fact be sets them selves. The sym bol # below indicates the conclusion o f an exam ple.

The set S = {1, 2, {1, 2}, 0} consists o f fo u r elem ents, tw o o f w hich are sets, namely,

{1, 2} an d 0. I f w e w rite C = {1, 2}, then w e can also w rite S = {1, 2, C, 0}.

The set T = {0, {1, 2, 3}, 4, 5} also has fo u r elem ents, namely, the three integers

0, 4 an d 5 a n d the set { 1, 2, 3}. E ven though 2 € {1 ,2 , 3}, the num ber 2 is n o ta n elem ent

i t f f i t h a t is, 2 f T . ♦

O ften sets consist o f those elem ents satisfying som e condition or possessing som e speci­ fied property. In this case, we can define such a set as S — {a : p ( x )} , w here, by this, we m ean that S consists o f all those elem ents a satisfying som e condition p ( x ) concerning a . Som e m athem aticians w rite S — [x | p(x)}; that is, som e prefer to w rite a vertical line rather than a colon (w hich, by itself here, is understood to m ean “ such th at” ). For exam ple, if we are studying real num ber solutions o f equations, then

S = {a : (a - I )i'.v + 2 )(a + 3) = 0}

is the set o f all real num bers x such that (a — 1)(a + 2)(.v + 3) = 0; that is, S is the

solution set o f the equation (x — 1 )(.v + 2)(x + 3) = 0. We could have w ritten S = {1, —2, —3}; however, even though this w ay o f expressing S is apparently sim pler, it does not tell us th at w e are interested in the solutions o f a p articular equation. The

ab solute value |a | o f a re a l num ber a is a if a > 0; w hile \x\ = —a if a < 0. Therefore,

T = {x : |a | = 2}

is the set o f all real num bers having absolute value 2, that is, T = {2, —2}. In the sets S and T that w e have ju st described, w e understand that “x ” refers to a real num ber x . If