An Invitation to Abstract Mathematics

406 Pages·2013·2.42 MB·English

by  Béla Bajnok (auth.)

Undergraduate Texts in Mathematics
Béla Bajnok
An Invitation
to Abstract
Mathematics
Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics
SeriesEditors:
SheldonAxler
SanFranciscoStateUniversity,SanFrancisco,CA,USA
KennethRibet
UniversityofCalifornia,Berkeley,CA,USA
AdvisoryBoard:
ColinAdams,WilliamsCollege,Williamstown,MA,USA
AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada
RuthCharney,BrandeisUniversity,Waltham,MA,USA
IreneM.Gamba,TheUniversityofTexasatAustin,Austin,TX,USA
RogerE.Howe,YaleUniversity,NewHaven,CT,USA
DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA
JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA
JillPipher,BrownUniversity,Providence,RI,USA
FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA
AmieWilkinson,UniversityofChicago,Chicago,IL,USA
Undergraduate Texts in Mathematics are generally aimed at third- and fourth-
year undergraduate mathematics students at North American universities. These
texts strive to provide students and teachers with new perspectives and novel
approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation
ofinterrelationsamongdifferentaspectsofthesubject.Theyfeatureexamplesthat
illustratekeyconceptsaswellasexercisesthatstrengthenunderstanding.
Forfurthervolumes:
http://www.springer.com/series/666
Be´la Bajnok
An Invitation to Abstract
Mathematics
123
Be´laBajnok
DepartmentofMathematics
GettysburgCollege
Gettysburg,PA,USA
ISSN0172-6056
ISBN978-1-4614-6635-2 ISBN978-1-4614-6636-9(eBook)
DOI10.1007/978-1-4614-6636-9
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013935711
MathematicsSubjectClassification:00-01,01-01,03-01
©Be´laBajnok2013
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Preface to Instructors
WhatKindofaBookIs This?
Ithasbeenmorethanthreedecadessincethefirstso-calledtransitionbookappeared
on the mathematics shelves of college bookstores, and there are currently several
dozen such booksavailable. The aim of these booksis to bridge the gap between
the traditionallower-levelcourses, primarilycalculus, and the upper-levelcourses
that require deeper understanding and maturity, such as modern algebra and real
analysis.Thus,themainfocusoftransitionbooksisonthefoundationsofabstract
mathematics, giving a thorough treatment of elementary logic and set theory and
introducingstudentstotheartandcraftofproofwriting.
While this book certainly hopes to provide students with a firm foundationfor
theupper-levelcoursesofanundergraduatemathematicsprogram,itisnotgeared
solelytowardstudentswhointendtomajorinmathematics.Itisthedisappointing
realityatmanyinstitutionsthatsomeofthemostablestudentsarenotconsidering
mathematics as a possible major; in fact, coming out of a standard calculus
sequence, most students are not familiar with the true nature of this beautiful
subject. Therefore, an important mission of the book is to provide students with
anunderstandingandanappreciationof(abstract)mathematics,withthehopethat
theychoosetostudythesetopicsfurther.
Recognizingthatnotallourstudentswillhavetheopportunitytotakeadditional
courses in mathematics, this textbook attempts to give a broad view of the field.
Evenstudentsmajoringinmathematicsusedto complainthattheywere notgiven
an opportunity to take a course on “mathematics” without an artificial division
of subjects. In this textbook, we make an attempt to remedy these concerns
by providing a unified approach to a diverse collection of topics, by revisiting
concepts and questions repeatedly from differing viewpoints, and by pointing
out connections, similarities, and differences among subjects whenever possible.
If during or after reading this book students choose to take further courses in
mathematics,thenwehaveachievedourmostimportantgoal.
v
vi PrefacetoInstructors
In order to provide students with a broad exposure to mathematics, we have
includedanunusuallydiversearrayoftopics.Beyondathoroughstudyofconcepts
that are expectedto be foundin similar books,we briefly discuss importantmile-
stonesinthehistoryofmathematicsandfeaturesomeofthemostinterestingrecent
accomplishmentsinthefield.Thisbookaimstoshowstudentsthatmathematicsisa
vibrantanddynamichumanenterprisebyincludinghistoricalperspectivesandnotes
on the giants of mathematics and their achievements; by mentioning more recent
resultsandupdatesonavarietyofquestionsofcurrentactivityinthemathematical
community; and by discussing many famous and less well-known questions that
have not yet been resolved and that remain open for the mathematicians of the
future.
We alsointendedtogobeyondthetypicalelementarytextbyprovidingamore
thoroughanddeepertreatmentwheneverfeasible.Whilewefinditimportantnotto
assumeanyprerequisitesforthebook,weattempttotravelfurtheronsomeofthe
mostenchantingpathsthaniscustomaryatthebeginninglevel.Althoughwerealize
thatperhapsnotallstudentsare willing tojoin usonthese excursions,we believe
thatthereareagreatmanystudentsforwhomtherewardsareworththeeffort.
Another important objective—and here is where the author’s Hungarian roots
are truly revealed—isto center much of the learning on problem solving. George
Po´lya’s famous book How to Solve It1 introduced students around the world to
mathematicalproblemsolving,and,asitseditorialreviewsays,“show[ed]anyonein
anyfieldhowtothinkstraight.”PaulHalmos—anothermathematicianofHungarian
origin2—isoftenquoted3abouttheimportanceofproblems:
Themajorpartofeverymeaningfullifeisthesolutionofproblems;aconsiderable
partoftheprofessionallifeoftechnicians,engineers,scientists,etc.,isthesolution
of mathematical problems. It is the duty of all teachers, and of teachers of
mathematics in particular, to expose their students to problems much more than
tofacts.4
Ourtexttakestheserecommendationstoheartbyofferingasetofcarefullychosen,
instructive,andchallengingproblemsineachchapter.
Itistheauthor’shopethatthisbookwillconvincestudentsthatmathematicsis
a wonderful and important achievement of humankind and will generate enough
enthusiasmtoconvincethemtotake morecoursesinmathematics.Intheprocess,
1OriginallypublishedinhardcoverbyPrincetonUniversityPressin1945.Availableinpaperback
fromPrincetonUniversityPress(2004).
2Halmos,authorofnumerousprize-winningbooksandarticlesonmathematicsanditsteaching,is
alsoknownastheinventorofthe(cid:2)symbol,usedtomarktheendofproofs,andtheword“iff,”a
now-standardabbreviationforthephrase“if,andonlyif.”
3Forexample,byareportoftheMathematicalAssociationofAmericaCommitteeontheTeaching
of Undergraduate Mathematics (Washington, D.C., 1983) and by the Notices of the American
MathematicalSociety(October,2007,page1141)
4“TheHeartofMathematics,”AmericanMathematicalMonthly87(1980),519–524
PrefacetoInstructors vii
studentsshouldlearnhowtothink,write,andtalk abstractlyandprecisely—skills
thatwillproveimmeasurablyusefulintheirfuture.
HowCan OneTeach from ThisBook?
Can abstract mathematical reasoning be taught? In my view, it certainly can be.
However, an honest answer would probably qualify this by saying that not all
students will be able (or willing) to acquire this skill to the maximal degree. I
often tell people that, when teaching this course, I feel like a ski instructor; I can
show them how the pros do it and be there for them when they need my advice,
praise, or criticism, but how well they will learn it ultimately depends on their
abilities, dedication, and enthusiasm. Some students will become able to handle
thesteepestslopesandthemostdangerouscurves,whileotherswillmostlyremain
on friendlier hills. A few might become Olympic champions, but most will not;
however, everyone who gives it an honest effort will at least learn how to move
forward without falling. And, perhaps most importantly, I hope that, even though
someoccasionallyfindthetrainingfrighteninganddifficult,theywillallenjoythe
process.
The book contains 24 chapters. Each chapter consists of a lecture followed
by about a dozen problems. I am a strong believer in the “spiral” method:
topics are often discussed repeatedly throughout the book, each time with more
depth, additional insights, or different viewpoints. The chapters are written in an
increasinglyadvancedfashion;thelasteightchapters(andespeciallythelastthree
orfour)areparticularlychallenginginbothcontentandlanguage.Thelecturesand
theproblemsbuildononeanother;theconceptsofthelecturesareoftenintroduced
byproblemsinpreviouschaptersorareextendedanddiscussedagaininproblems
insubsequentchapters.(TheLATEXcommand“ref”appearsmorethanonethousand
timesinthesourcefile.)Therefore,ifanypartofalectureoranyproblemisskipped,
this should be done with caution. The material can be covered in a one-semester
course or in a two-semester sequence;the latter choice will obviouslyallow for a
moreleisurelypacewithopportunitiesfordeeperdiscussionsandadditionalstudent
interactions.
The heart and soul of this book is in its approximately 280 problems (some
with multiple parts); the lectures are intended to be as brief as possible and yet
provideenoughinformationforstudentstoattacktheproblems.Iputconsiderable
effort into keeping the number of problems relatively small. Each problem was
carefully chosen to clarify a concept, to demonstrate a technique, or to enthuse.
Thereareveryfewroutineproblems;mostproblemswillrequirerelativelyextensive
arguments,creativeapproaches,orboth.Particularlyinlaterchapters,theproblems
aimforstudentstodevelopsubstantialinsight.Tomakeeventhemostchallenging
problems accessible to all students, hints are provided liberally. An Instructor’s
Guide, containingsolutions to all problemsin the book,is available on request at
thebook’sproductpageonwww.springer.com.
viii PrefacetoInstructors
Manyoftheproblemsarefollowedbyremarksaimedatconnectingtheproblems
to areasof currentresearchwith the hopethatsomeof these noteswill invitestu-
dentstocarryoutfurtherinvestigations.Thebookalsocontainsseveralappendices
withadditionalmaterialandquestionsforpossiblefurtherresearch.Someofthese
questions are not difficult, but others require a substantial amount of ingenuity—
thereareevenknownopenconjecturesamongthem;anyprogressonthesequestions
would indeed be considered significant and certainly publishable. I feel strongly
thateveryundergraduatestudentshouldengagein aresearchexperience.Whether
they will go on to graduate school, enroll in professional studies, or take jobs in
education,government,orbusiness,studentswillbenefitfromtheopportunitiesfor
perfectingavarietyofskillsthataresearchexperienceprovides.
Allowmetoaddafewnotesonmypersonalexperienceswiththisbook.Itaught
courses using this text more than 20 times but find it challenging each time. The
approach I find best suited for this course is one that maximizes active learning
andclass interaction(amongstudentsandbetweenstudentsandmyself).Students
are asked to carefully read the lecture before class and to generate solutions to
the assigned problems. Our organized and regular out-of-class “Exploratorium”
sessions—wherestudentsworkalone orwith otherstudentsin the class underthe
supervisionofteachingassociates—seemparticularlybeneficialinhelpingstudents
prepareforclass.Ispendnearlyeveryclassbyaskingstudentstopresenttheresults
oftheirworktotheclass.AsItellthem,itisnotnecessarythattheyhavecompletely
correctsolutions, but I expect them to have worked on all of the problemsbefore
classtothebestoftheirability.Itrytobegenerouswithencouragement,praise,and
constructivecriticism,butIamnotsatisfieduntilathoroughandcompletesolution
ispresentedforeachproblem.Itisnotunusualforaproblemtobediscussedseveral
timesbeforeitgetsmyfinalPFB(“PerfectforBe´la”)approval.
Without a doubt, teaching this course has been one of the most satisfying
experiencesthat I have had in this profession.Watching my students develop and
succeed,perhapsmoresothaninanyothercourse,isalwaysasuperblyrewarding
adventure.
Preface to Students
For WhomIs ThisBookWritten?
This bookis intendedfor a broadaudience.Any student who wishes to learn and
perfect his or her ability to think and reason at an advanced level will benefit
from taking a course based on this textbook. The skills of understanding and
communicating abstract ideas will prove useful in every professional career: law,
medicine,engineering,business,education,politics,science,economics,andothers.
Theabilitytoexpressoneselfandtoargueclearly,precisely,andconvincinglyhelps
ineverydayinteractionsaswell.Justasotherscanseeifwelookhealthyphysically,
they can also assess our intellectual fitness when they listen to our explanations
or read our writings. Abstract mathematics, perhaps more than any other field,
facilitatesthelearningoftheseessentialskills.
An important goal of this book, therefore, is to help students become more
comfortablewith abstraction.Paradoxically,the moreone understandsan abstract
topic or idea, the less abstract it will seem! Thus, the author’s hope is that his
InvitationtoAbstractMathematicsisaccepted,butthatbythetimestudentsfinish
thebook,theyagreethatthereisnoneedfortheword“abstract”inthetitle—indeed,
thisbookis(just)aboutmathematics.
The prerequisites to the text are minimal; in particular, no specific knowledge
beyond high school mathematics is assumed. Instead, students taking this course
should be willing to explore unusual and often difficult topics and be ready to
facechallenges.Facingandovercomingthesechallengeswillbestudents’ultimate
rewardattheend.
HowCan OneLearn from ThisBook?
Welcome to abstract mathematics! If you are like 99% of the students who have
takenacoursebasedonthisbook,youwillfindthatthecourseischallengingyouin
ix