Mathematical Reasoning: Writing and Proof. Version 2.0

608 Pages·6.04 MB·english

by  Sundstrom Ted.

Mathematical Reasoning
Writing and Proof
Version2.1
December 3, 2016
Ted Sundstrom
Grand Valley State University
TedSundstrom
DepartmentofMathematics
GrandValleyStateUniversity
Allendale,MI49401
[email protected]
MathematicalReasoning:WritingandProof
PreviousversionsofthisbookwerepublishedbyPearsonEducation,Inc.
Changes Made in Version 2.1
There are no changes in content between Version 2.0 of this book and Version
2.1. A few minor errors in Version 2.0 have been corrected in Version 2.1. In
addition, there are no changes in content between Version 1.1 of this book and
Version2.0. TheonlychangeisthatAppendixC,AnswersandHintsforSelected
Exercises, now contains solutions and hints for more exercises. Those exercises
withananswerorahintintheappendixareprecededbyastar.?/.
License
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ShareAlike3.0UnportedLicense. Thegraphic
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Contents
NotetoStudents vi
Preface viii
SupplementaryMaterialsfortheInstructor . . . . . . . . . . . . . . . . xiv
1 IntroductiontoWritingProofsinMathematics 1
1.1 StatementsandConditionalStatements. . . . . . . . . . . . . . . 1
1.2 ConstructingDirectProofs . . . . . . . . . . . . . . . . . . . . . 15
1.3 Chapter1Summary . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 LogicalReasoning 33
2.1 StatementsandLogicalOperators . . . . . . . . . . . . . . . . . 33
2.2 LogicallyEquivalentStatements . . . . . . . . . . . . . . . . . . 43
2.3 OpenSentencesandSets . . . . . . . . . . . . . . . . . . . . . . 52
2.4 QuantifiersandNegations. . . . . . . . . . . . . . . . . . . . . . 63
2.5 Chapter2Summary . . . . . . . . . . . . . . . . . . . . . . . . . 80
3 ConstructingandWritingProofsinMathematics 82
3.1 DirectProofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2 MoreMethodsofProof . . . . . . . . . . . . . . . . . . . . . . . 102
3.3 ProofbyContradiction . . . . . . . . . . . . . . . . . . . . . . . 116
3.4 UsingCasesinProofs . . . . . . . . . . . . . . . . . . . . . . . . 131
iii
iv Contents
3.5 TheDivisionAlgorithmandCongruence . . . . . . . . . . . . . . 141
3.6 ReviewofProofMethods . . . . . . . . . . . . . . . . . . . . . . 158
3.7 Chapter3Summary . . . . . . . . . . . . . . . . . . . . . . . . . 166
4 MathematicalInduction 169
4.1 ThePrincipleofMathematicalInduction . . . . . . . . . . . . . . 169
4.2 OtherFormsofMathematicalInduction . . . . . . . . . . . . . . 188
4.3 InductionandRecursion . . . . . . . . . . . . . . . . . . . . . . 200
4.4 Chapter4Summary . . . . . . . . . . . . . . . . . . . . . . . . . 213
5 SetTheory 215
5.1 SetsandOperationsonSets. . . . . . . . . . . . . . . . . . . . . 215
5.2 ProvingSetRelationships. . . . . . . . . . . . . . . . . . . . . . 230
5.3 PropertiesofSetOperations . . . . . . . . . . . . . . . . . . . . 244
5.4 CartesianProducts . . . . . . . . . . . . . . . . . . . . . . . . . 254
5.5 IndexedFamiliesofSets . . . . . . . . . . . . . . . . . . . . . . 264
5.6 Chapter5Summary . . . . . . . . . . . . . . . . . . . . . . . . . 277
6 Functions 281
6.1 IntroductiontoFunctions . . . . . . . . . . . . . . . . . . . . . . 281
6.2 MoreaboutFunctions . . . . . . . . . . . . . . . . . . . . . . . . 294
6.3 Injections,Surjections,andBijections . . . . . . . . . . . . . . . 307
6.4 CompositionofFunctions. . . . . . . . . . . . . . . . . . . . . . 323
6.5 InverseFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 334
6.6 FunctionsActingonSets . . . . . . . . . . . . . . . . . . . . . . 349
6.7 Chapter6Summary . . . . . . . . . . . . . . . . . . . . . . . . . 359
7 EquivalenceRelations 362
7.1 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
7.2 EquivalenceRelations. . . . . . . . . . . . . . . . . . . . . . . . 375
7.3 EquivalenceClasses . . . . . . . . . . . . . . . . . . . . . . . . . 387
Contents v
7.4 ModularArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 400
7.5 Chapter7Summary . . . . . . . . . . . . . . . . . . . . . . . . . 411
8 TopicsinNumberTheory 414
8.1 TheGreatestCommonDivisor . . . . . . . . . . . . . . . . . . . 414
8.2 PrimeNumbersandPrimeFactorizations . . . . . . . . . . . . . 426
8.3 LinearDiophantineEquations . . . . . . . . . . . . . . . . . . . 439
8.4 Chapter8Summary . . . . . . . . . . . . . . . . . . . . . . . . . 449
9 FiniteandInfiniteSets 452
9.1 FiniteSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
9.2 CountableSets . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
9.3 UncountableSets . . . . . . . . . . . . . . . . . . . . . . . . . . 476
9.4 Chapter9Summary . . . . . . . . . . . . . . . . . . . . . . . . . 490
A GuidelinesforWritingMathematicalProofs 492
B AnswersfortheProgressChecks 497
C AnswersandHintsforSelectedExercises 536
D ListofSymbols 582
Index 585
Note to Students
Thisbookmaybedifferentthanothermathematicstextbooksyouhaveusedsince
oneofthemaingoalsofthisbookistohelpyoutodeveloptheabilitytoconstruct
andwritemathematicalproofs. Sothisbookisnotjustaboutmathematicalcontent
but isalso aboutthe processof doingmathematics. Alongthe way, youwillalso
learn some importantmathematical topicsthat will help you in your future study
ofmathematics.
Thisbookisdesignednottobe justcasuallyread butrather tobe engaged. It
may seem likea cliche´ (because itis in almostevery mathematics booknow)but
thereistruthinthestatementthatmathematicsisnotaspectatorsport.Tolearnand
understandmathematics,youmustengageintheprocessofdoingmathematics. So
youmustactivelyreadandstudythebook,whichmeanstohaveapencilandpaper
with you and be willing to follow along and fill in missing details. This type of
engagement is noteasy and is often frustrating,but if you do so, you willlearn a
greatdealaboutmathematicsandmoreimportantly,aboutdoingmathematics.
Recognizingthatactivelystudyinga mathematics bookis oftennoteasy, sev-
eralfeaturesofthetextbookhavebeendesignedtohelpyoubecomemoreengaged
asyoustudythematerial. Someofthefeaturesare:
PreviewActivities. WiththeexceptionofSections1.1and3.6,eachsection
(cid:15)
hasexactlytwopreviewactivities.SomePreviewActivitieswillreviewprior
mathematical work that is necessary for the new section. This prior work
maycontainmaterialfrompreviousmathematicalcoursesoritmay contain
material covered earlier in thistext. Other previewactivitieswillintroduce
newconceptsanddefinitionsthatwillbeusedwhenthatsectionisdiscussed
inclass. Itisveryimportantthatyouworkonthesepreviewactivitiesbefore
starting the rest of the section. Please note that answers to these preview
activitiesare notincluded in the text. This bookis designed to be used for
a courseand itisleftup tothediscretionof each individualinstructoras to
howtodistributetheanswerstothepreviewactivities.
vi
NotetoStudents vii
Progress Checks. Several Progress Checks are included in each section.
(cid:15)
These are either shortexercises or short activitiesdesignedto help you de-
termineifyouareunderstandingthematerialasyouarestudyingthematerial
inthesection.Assuch,itisimportanttoworkthroughtheseprogresschecks
totestyourunderstanding,andifnecessary, studythematerialagainbefore
proceedingfurther. Soitisimportanttoattempttheseprogresschecksbefore
checkingtheanswers,whichareareprovidedinAppendixB.
Chapter Summaries. To assistyou with studyingthe material in the text,
(cid:15)
there is a summary at the end of each chapter. The summaries usually list
theimportantdefinitionsintroducedinthechapterandtheimportantresults
proveninthechapter. Ifappropriate,thesummaryalsodescribestheimpor-
tantprooftechniquesdiscussedinthechapter.
AnswersforSelected Exercises. Answersorhintsforseveralexercisesare
(cid:15)
includedinan AppendixC. Thoseexercises withananswerora hintinthe
appendixareprecededbyastar.?/. ThemainchangeinVersion2.0ofthis
textbook from the previous versions is the addition of more exercises with
answersorhintsintheappendix.
Althoughnot part of the textbook, there are now 107 onlineideoswith about
14 hours of content that span the first seven chapters of this book. These videos
arefreelyavailableonlineatGrandValley’sDepartmentofMathematicsYouTube
channelonthisplaylist:
http://gvsu.edu/s/0l1
These online videos were created and developed by Dr. Robert Talbert of Grand
ValleyStateUniversity.
Thereisalsoawebsiteforthetextbookat
https://sites.google.com/site/mathematicalreasoning3ed/
Youmayfindsomethingstherethatcouldbeofhelp. Forexample,therecurrently
is a link to study guides for the sections of this textbook. Good luck with your
studyof mathematics and please make use of the onlinevideosandthe resources
availableinthetextbookandatthewebsiteforthetextbook.Iftherearethingsthat
youthinkwouldbegoodadditionstothebookor theweb site, pleasefeel free to
[email protected].
Preface
MathematicalReasoning: WritingandProof is designedto be a text for the first
course in the college mathematics curriculum that introducesstudentsto the pro-
cesses of constructingand writingproofs and focuses on the formal development
ofmathematics. Theprimarygoalsofthetextaretohelpstudents:
Developlogical thinkingskillsand to develop the abilityto thinkmore ab-
(cid:15)
stractlyinaprooforientedsetting.
Develop the ability to construct and write mathematical proofs using stan-
(cid:15)
dard methods of mathematical proof includingdirect proofs, proof by con-
tradiction,mathematicalinduction,caseanalysis,andcounterexamples.
Developtheabilitytoreadandunderstandwrittenmathematicalproofs.
(cid:15)
Developtalentsforcreativethinkingandproblemsolving.
(cid:15)
Improve theirqualityof communicationin mathematics. Thisincludesim-
(cid:15)
proving writing techniques, reading comprehension, and oral communica-
tioninmathematics.
Betterunderstandthenatureofmathematicsanditslanguage.
(cid:15)
Anotherimportantgoalofthistextistoprovidestudentswithmaterialthatwillbe
neededfortheirfurtherstudyofmathematics.
Thistypeofcoursehasnowbecomeastandardpartofthemathematicsmajorat
manycollegesanduniversities. Itisoftenreferredtoasa“transitioncourse”from
thecalculussequencetotheupper-levelcoursesinthemajor. Thetransitionisfrom
the problem-solving orientation of calculus to the more abstract and theoretical
upper-level courses. This is needed today because many students complete their
study of calculus withoutseeing a formal proof or having constructed a proof of
their own. This is in contrast to many upper-level mathematics courses, where
viii
Preface ix
the emphasis is on the formal development of abstract mathematical ideas, and
theexpectationsarethatstudentswillbeabletoreadandunderstandproofsandbe
abletoconstructandwritecoherent,understandablemathematicalproofs. Students
shouldbeabletousethistextwithabackgroundofonesemesterofcalculus.
Important Features of the Book
Following are some of the important features of this text that will help with the
transitionfromcalculustoupper-levelmathematicscourses.
1. EmphasisonWritinginMathematics
Issuesdealingwithwritingmathematical expositionare addressedthrough-
out the book. Guidelines for writing mathematical proofs are incorporated
intothe book. Theseguidelinesare introducedas neededand begininSec-
tion 1.2. AppendixA contains a summary of all the guidelinesfor writing
mathematicalproofsthatareintroducedthroughoutthetext. Inaddition,ev-
eryattempthasbeenmadetoensurethateverycompletedproofpresentedin
thistextiswrittenaccordingtotheseguidelines.Thisprovidesstudentswith
examplesofwell-writtenproofs.
Oneofthemotivatingfactorsforwritingthisbookwastodevelopatextbook
forthecourse“CommunicatinginMathematics”atGrandValleyStateUni-
versity. This course is part of the university’sSupplemental WritingSkills
Program,andtherewasnotextthatdealtwithwritingissuesinmathematics
thatwassuitableforthiscourse. Thisiswhysomeofthewritingguidelines
in the text deal withthe use of LATEXor a word processor that is capable of
producing the appropriate mathematical symbols and equations. However,
thewritingguidelinescaneasilybeimplementedforcourseswherestudents
donothaveaccesstothistypeofwordprocessing.
2. InstructionintheProcessofConstructingProofs
One of the primary goals of this book is to develop students’ abilities to
constructmathematical proofs. Another goal is to develop their abilitiesto
write the proof in a coherent manner that conveys an understanding of the
prooftothereader. Thesearetwodistinctskills.
InstructiononhowtowriteproofsbeginsinSection1.2andisdevelopedfur-
ther inChapter3. In addition,Chapter 4 isdevotedto developingstudents’
abilitiestoconstructproofsusingmathematicalinduction.
x Preface
Studentsareintroducedtoamethodtoorganizetheirthoughtprocesseswhen
attemptingtoconstructa proofthatusesa so-calledknow-showtable. (See
Section1.2andSection3.1.) Studentsusethistabletoworkbackwardfrom
what it is they are trying to prove while at the same time workingforward
fromtheassumptionsoftheproblem. Theknow-showtablesare usedquite
extensivelyinChapters1and3. However,theexplicituseofknow-showta-
blesisgraduallyreducedandthesetablesarerarelyusedinthelaterchapters.
One reason for this is that these tables may work well when there appears
tobe onlyone wayof provinga certain result. As theproofs become more
complicatedorothermethodsofproof(suchasproofsusingcases)areused,
theseknow-showtablesbecomelessuseful.
So the know-showtables are not to be considered an absolute necessity in
usingthetext. However,theyare usefulforstudentsbeginningtolearnhow
toconstructandwriteproofs. Theyprovideaconvenientwayforstudentsto
organizetheir work. More importantly,they introducestudentsto a way of
thinkingaboutaproblem. Insteadofimmediatelytryingtowriteacomplete
proof,theknow-showtableforcesstudentstostop,think,andaskquestions
suchas
JustexactlywhatisitthatIamtryingtoprove?
(cid:15)
HowcanIprovethis?
(cid:15)
WhatmethodsdoIhavethatmayallowmetoprovethis?
(cid:15)
Whataretheassumptions?
(cid:15)
HowcanIusetheseassumptionstoprovetheresult?
(cid:15)
Being able toask these questionsis a bigstep in constructinga proof. The
nexttask istoanswer the questionsand touse thoseanswersto constructa
proof.
3. EmphasisonActiveLearning
Oneoftheunderlyingpremisesofthistextisthatthebestwaytolearn and
understand mathematics is to be actively involved in the learning process.
However,itisunlikelythatstudentswilllearnallthemathematicsinagiven
course on their own. Students actively involved in learning mathematics
need appropriate materials that will provide guidance and support in their
learningof mathematics. There are several waysthistext promotesactivite
learning.