An Introduction to the Language of Mathematics

191 Pages·2018·1.603 MB·

by  Frédéric Mynard

Frédéric Mynard
An Introduction
to the
Language of
Mathematics
An Introduction to the Language of Mathematics
Fre´de´ric Mynard
An Introduction to the
Language of Mathematics
123
Fre´de´ricMynard
Mathematics
NewJerseyCityUniversity
JerseyCity,NJ,USA
ISBN978-3-030-00640-2 ISBN978-3-030-00641-9 (eBook)
https://doi.org/10.1007/978-3-030-00641-9
LibraryofCongressControlNumber:2018956569
Mathematics Subject Classification: 97E30, 97E40, 97E50, 97E60, 03B05, 03E20, 03E25, 03F07,
06A06,11A05,11A51
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To Dora, for all the love,
support, and inspiration.
Preface
Thistextismeanttobeusedasatextbookforatransitioncoursefrommathematics
centeredoncalculationtechniquestoproof-basedmathematics.MostUSuniversi-
tiesoffersuchaclassaftertwosemestersofCalculus,moreforthesakeofensuring
a minimum of mathematical maturity than because Calculus itself is of any help.
Such a course often focuses on proof writing and is foundational for upper level
mathematicscourses.Proofs,however,areonlyapartofthefoundationsthatneed
tobelaidouttoprepareforupperlevelcourses.Hence,thistextisaboutfoundations
and discusses not only proofs but the kind of system and conventions you need to
buildyourproofsin.Theradicalchangeofperspectiveisoftendisorientingtostu-
dents,andsuchcourses oftenhave high failureratesandleave students frustrated.
For students who have come to think of mathematics as Calculus and its applica-
tions, it comes as somewhat of a shock that mathematics is about something else
entirely.HereiswhatIwanttotellthembeforeengagingonthatroad:
This is a course unlike any other mathematics course you have taken so far.
Whereyouhavebeenfocusingonbasictechniquesandcalculations,youwillfocus
on arguments from now on. Consider that you have been taking pre-mathematics
courses until now and that you are taking a first mathematics (as what mathe-
maticians do) course, where you will learn the language and way of thinking of
mathematicians—mathematicians rather than mathematics—for mathematics is a
humanactivity,drivenbyhumanimpulsestounderstandandtomodelesthetically
andefficiently.
Thisisaformalcourse,focusedonformalism,thoughthegoalofreachingflu-
ency in reading proofs and proof making goes way beyond formalism. As such,
the material is often disorienting for students at first and appears very dry and ab-
stract.Itisnotunlikelearningaforeignlanguagefromscratch:beforeyoucansee
thebeautyofpoetryinthatlanguage,youhavealongwaytogointhesometimes
tedioustaskoflearningvocabularyandgrammar,beforedevelopingthenecessary
intuition.Learningthismaterialwillprobablybemoredifficultthananyothermath-
ematics class before, but it is worth the effort: you will sweat and puff pushing a
very heavy door, but the land on the other side of the door is the secret garden of
mathematicians,alandofbeautyandharmonythatyouwillsurelyenjoyexploring.
vii
viii Preface
What matters at the end is mathematics in action: presenting beautiful argu-
ments and proving meaningful statements, often surprising or far-reaching. But to
getthere,wehavetostartwithbasicvocabulary,definitions,andconceptsthatneed
to be assimilated like those of your mother tongue. That means you need to know
definitionsprecisely,notapproximately,notjustbyexample.Youshouldkeepex-
amplesdistinguishingvariousconceptsinmind,butthisisnosubstituteforaprecise
definition.Aproof isanairtightargumentwithinacertainlogicalsystem.Airtight
is a pretty high standard when it comes to arguments. That means that you can
only prove statements about things that are very specifically defined. You cannot
prove anything at all about something whose exact meaning is unclear to you. I
overemphasize this for a simple reason: many students will fail precisely because
theyremainedcontentwith“havinganidea”ofsuchandsuchaconceptanddidnot
make the effort to truly assimilate the exact definition. Without learning the basic
vocabulary,youcannotevenbegintolearnalanguageortopickupanything.This
isnoexception,solearningdefinitionscarefullywillbeoneofthekeystosuccess
inthiscourse.
To learn a language well, it is not enough to learn its vocabulary, syntax, and
grammar. You need to read a variety of styles and listen to native speakers in or-
dertorecognizestandardpatternsandgraduallydevelopsomeintuition.Similarly,
studyingstandardproofscarefullywillbeessential.Itisnotenoughtounderstand
eachstepofaproof;thiswouldbeenoughtobeconvincedthattheargumentisvalid
andthatthisisindeedaproperproof,butitisinadequatewithregardtothegoalof
training you to write your own proofs. To this end, you should keep going over a
proof,thinkingabouthowitspartsarearticulated,untilyouareabletoreproduceit
onyourown.
Examplesandexercisesaredrawnfromavarietyofsources,whicharenotspecif-
icallyattributedbutincludedinthereferences.Thebooks[19]and[27]havebeen
particularlyinfluentialsources.Therearetwokindsofexercisesinthisbook.Those
scatteredthroughthetextareanintegralpartofthecourseandshouldbeattempted
as you go. Full solutions for these exercises are provided at the end of the book.
Ontheotherhand,additionalexerciseswithoutsolutionsareincludedattheendof
mostsectionsandcanbeusedtoassignhomework.Theinstructormayrequestthe
solutionmanualforthoseexercisesonthebook’swebpage
The reader will quickly notice that the text contains an unusual number of
footnotes—afactthatmayirritatesomereaderswhowouldratherhaveeverything
incorporatedinthetext,andpleaseotherswhowillbehappytoskipthemaltogether
inafirstreading,onlytoreturntotheseadditionalcommentsinasecond,moresys-
tematic,reading.Theinstructormayrequestthesolutionmanualforthoseexercises
onthebook’swebpage.
There are a few results that are stated or alluded to without proof in the text,
because the arguments involved require more sophistication of the reader that one
would expect of the target audience at this point in the course. These and related
resultsareprovedinAppendixAmakingthebookself-contained.
To a large extent, Chapters 2, 3, and 4 are the core of the book, and Chapter 1
couldbecoveredonlylightly.Itintroducesthebasicsoflogicandofthelanguage
Preface ix
of Set Theory, and as such, it is the natural place to start. Yet, in keeping with its
opening remarks on language, it introduces an early, albeit superficial, discussion
of the axioms of Set Theory—a choice that stems from my experience that semi-
philosophical considerations in class discussions often turn out to be very fruitful,
but a choice that some instructors may prefer not to follow. It should not be hard
forsuchinstructorstorecastSection1.4underamorenaivelighttofittheirneeds.
Section 1.7 is the other part of Chapter 1 that one might find a little ambitious
when compared to the opening of Chapter 2. The choice to introduce the notions
of one-to-one, onto, and bijective maps early on (in that section) in the context of
finite sets to illustrate some counting arguments from the set-theoretic viewpoint
hasalsoprovedusefulinmyexperience,whenitistimetoreturntothesenotionsin
thecontextofinfinitecardinalities(Chapter4).Again,itwouldnotbedifficultfor
an instructor tochoose to treat this material (Section 1.7) later, for instance, when
treating functional relations in Section 3.1.1. I should note that Chapter 3 treats
relationsalittlemoreextensivelythanothersimilarbooksandincludesanumberof
informal comments on their nature, particularly regarding equivalence relations. I
havefoundsuchinformaldiscussionstobeanimportantcomplementtotheformal
developmentofthematerial,buthereagain,theinstructorcaneasilyskipwhatshe
seesastoochatty.
I am grateful to the readers of early drafts who caught a number of typos and
errors, particularly my students at NJCU, among which Fadoua Chigr and John
Stulichstandoutforthemeticulosityoftheirreading.MycolleaguesDeborahBen-
nett (NJCU) and Szymon Dolecki (University of Burgundy) made many helpful
suggestions,andsodidtheanonymousrefereesforSpringer.Iamindebtedtoallof
them. Most likely, many imperfections remain, and they are of course my sole re-
sponsibility.Finally,IamalsogratefultoNewJerseyCityUniversity,forextensive
parts of this manuscript were written while I was benefiting from a course release
undertheSeparatelyBudgetedResearchprogram.
JerseyCity,NJ,USA Fre´de´ricMynard
June29,2018
Contents
1 TheLanguageofLogicandSet-Theory........................... 1
1.1 ALanguageforProofs? ..................................... 1
1.2 PropositionalLogic ........................................ 4
1.2.1 Propositions ........................................ 4
1.2.2 BasicLogicalConnectives ............................ 5
1.2.3 FirstLawsofPropositionalLogic ...................... 6
1.2.4 ConditionalandBiconditionalStatements................ 9
1.3 QuantifiersandQuantifiedStatements ......................... 14
1.3.1 Predicates,UniverseofDiscourse,andTruthSets ......... 14
1.3.2 ExistentialandUniversalQuantifiers.................... 15
1.3.3 AWordonSyntax ................................... 18
1.3.4 NegatingQuantifiedStatements ........................ 19
1.4 SomeBasicAxiomsofSetTheory ............................ 21
1.5 Subsets ................................................... 23
1.6 OperationsonSetsandLogicalConnectives .................... 26
1.7 FunctionsandCounting:FirstLook ........................... 34
1.8 ProductSetandFunctions ................................... 43
SuggestedFurtherReadings ...................................... 48
2 OnProofsandWritingMathematics ............................. 51
2.1 DirectProofs .............................................. 52
2.2 ContrapositiveProofs ....................................... 58
2.3 ProofsbyContradiction ..................................... 61
2.4 SpecialFormsofthePremisesoroftheConclusion.............. 65
2.5 Disproving ................................................ 67
2.5.1 Counterexamples .................................... 68
2.6 ProofbyInduction.......................................... 70
2.6.1 SmallestCounterexample ............................. 80
2.6.2 ADetourontheWell-OrderingPrincipleandArithmetic ... 81
xi
xii Contents
2.6.3 StrongInduction..................................... 83
2.6.4 FibonacciNumbers .................................. 86
2.7 AWordonStyle ........................................... 93
2.8 TypesettingMathematics .................................... 93
SuggestedFurtherReadings ...................................... 94
3 Relations...................................................... 95
3.1 GeneralRelations .......................................... 96
3.1.1 FunctionalRelations .................................102
3.2 OrderRelations ............................................107
3.3 Equivalence ...............................................117
3.4 Equivalence,Order,andSetsofNumbers ......................125
SuggestedFurtherReadings ......................................128
4 Cardinality....................................................131
4.1 InfiniteSets ...............................................131
4.2 CountableSets.............................................134
4.3 CardinalityContinuum ......................................137
4.4 InfinitelyManyInfiniteCardinalities!..........................140
4.5 ContinuumHypothesisandtheSurprisinglyComplexNatureof
TruthinMathematics .......................................141
SuggestedFurtherReadings ......................................143
AppendixAComplements...........................................145
A.1 Inclusion-ExclusionandNumberofontoMaps..................145
A.2 Knaster-TarskiFixedPointTheorem...........................147
A.3 InductionRevisitedandWell-OrderedSets .....................148
A.4 Well-OrderandTrichotomy..................................150
A.5 MoreontheAxiomofChoice ................................153
SolutionstoExercisesintheText.....................................157
References.........................................................179
Index .............................................................183