Journey Into Discrete Mathematics

402 Pages·2018·12.023 MB·English

by  Owen Byer, Deirdre L. Smeltzer, Kenneth L. Wantz

AMS/MAA TEXTBOOKS
VOL 41
Journey into
Discrete Mathematics
Owen D. Byer, Deirdre L. Smeltzer,
and Kenneth L. Wantz
CommitteeonBooks
JenniferJ.Quinn,Chair
MAATextbooksEditorialBoard
StanleyE.Seltzer,Editor
WilliamRobertGreen,Co-Editor
BelaBajnok SuzanneLynneLarson JeffreyL.Stuart
MatthiasBeck JohnLorch RonD.Taylor,Jr.
HeatherAnnDye MichaelJ.McAsey ElizabethThoren
CharlesR.Hampton VirginiaNoonburg RuthVanderpool
2010MathematicsSubjectClassification.Primary97K20,97K30,97F60,97N70,97E30.
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LibraryofCongressCataloging-in-PublicationData
Names:Byer,Owen,author.|Smeltzer,DeirdreL.,author.|Wantz,KennethL.,1965-author.
Title:Journeyintodiscretemathematics/OwenD.Byer,DeirdreL.Smeltzer,KennethL.Wantz.
Description: Providence,RhodeIsland: MAAPress,animprintoftheAmericanMathematicalSociety,
[2018]|Series:AMS/MAAtextbooks;volume41|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2018023584|ISBN9781470446963(alk.paper)
Subjects: LCSH:Settheory. |Numbertheory. |Mathematicalanalysis. |Combinatorialanalysis. |AMS:
Mathematicseducation–Combinatorics,graphtheory,probabilitytheory,statistics–Combinatorics.
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ory. msc|Mathematicseducation–Arithmetic,numbertheory–Numbertheory. msc|Mathematics
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Contents
Preface vii
WhatIsDiscreteMathematics? vii
GoalsoftheBook vii
FeaturesoftheBook viii
CourseOutline ix
Acknowledgments xi
1 ConvinceMe! 1
1.1 OpeningProblems 2
1.2 Solutions 4
2 Mini-Theories 9
2.1 Introduction 9
2.2 DivisibilityofIntegers 14
2.3 Matrices 22
3 LogicandSets 31
3.1 Propositions 31
3.2 Sets 33
3.3 LogicalOperatorsandTruthTables 38
3.4 OperationsonSets 44
3.5 TruthValuesofCompoundPropositions 51
3.6 SetIdentities 54
3.7 InfiniteSetsandParadoxes 58
4 LogicandProof 67
4.1 LogicalEquivalences 67
4.2 Predicates 73
4.3 NestedQuantifiers 77
4.4 RulesofInference 83
4.5 MethodsofProof 91
5 RelationsandFunctions 101
5.1 Relations 101
5.2 PropertiesofRelationsonaSet 106
5.3 Functions 113
5.4 Sequences 123
iii
iv Contents
6 Induction 133
6.1 InductiveandDeductiveThinking 133
6.2 Well-OrderingPrinciple 135
6.3 MethodofMathematicalInduction 137
6.4 StrongInduction 147
6.5 ProofoftheDivisionTheorem 151
7 NumberTheory 155
7.1 Primes 155
7.2 TheEuclideanAlgorithm 159
7.3 LinearDiophantineEquations 165
7.4 Congruences 169
7.5 Applications 176
7.6 AdditionalProblems 178
8 Counting 183
8.1 WhatIsCounting? 183
8.2 CountingTechniques 184
8.3 PermutationsandCombinations 193
8.4 TheBinomialTheorem 203
8.5 AdditionalProblems 207
9 GraphTheory 211
9.1 TheLanguageofGraphs 212
9.2 TraversingEdgesandVisitingVertices 221
9.3 VertexColorings 230
9.4 Trees 238
9.5 ProofsofEuler’sandOre’sTheorems 249
10 InvariantsandMonovariants 253
10.1 Invariants 253
10.2 Monovariants 259
11 TopicsinCounting 267
11.1 Inclusion-Exclusion 267
11.2 ThePigeonholePrinciple 274
11.3 MultinomialCoefficients 278
11.4 CombinatorialIdentities 282
11.5 OccupancyProblems 286
12 TopicsinNumberTheory 299
12.1 MoreonPrimes 299
12.2 IntegersinOtherBases 303
12.3 MoreonCongruences 311
12.4 NonlinearDiophantineEquations 318
12.5 Cryptography: Rabin’sMethod 319
Contents v
13 TopicsinGraphTheory 327
13.1 PlanarGraphs 327
13.2 ChromaticPolynomials 331
13.3 SpanningTreeAlgorithms 337
13.4 PathandCircuitAlgorithms 342
Hints 355
ListofNames 381
Bibliography 383
Index 385

Preface
What Is Discrete Mathematics?
Theworddiscreteinmathematicsisincontrasttothewordcontinuous. Forexample,
thesetofintegersisdiscrete,whilethesetofrealnumbersiscontinuous. Thus,dis-
cretemathematicsdescribesacollectionofbranchesofmathematicswiththecommon
characteristicthattheyfocusonthestudyofthingsconsistingofseparate,irreducible,
oftenfiniteparts. Althoughlargelyneglectedintypicalprecollegemathematicscur-
ricula,discretemathematicsisessentialfordevelopinglogicandproblem-solvingabil-
ities. Questionslocatedwithintherealmofdiscretemathematicsnaturallyinvitecre-
ativityandinnovativethinkingthatgobeyondformulas. Furthermore,thecultivation
oflogicalthinkingformsanecessaryfoundationforproof-writing. Forthesereasons,
discretemathematicsiscriticalforundergraduatestudyofbothmathematicsandcom-
puterscience.
Goals of the Book
Simplystated,thegoalofJourneyintoDiscreteMathematicsistonurturethedevelop-
mentofskillsneededtolearnanddomathematics. Theseskillsincludetheabilityto
read,write,andappreciateagoodmathematicalproof,aswellasabasicfluencywith
coremathematicaltopicssuchassets,relationsandfunctions,graphtheory,andnum-
ber theory. The content and the corresponding requisite mathematical thinking are
appropriate for students in computer science and other problem-solving disciplines,
butthecontentpresentationandthenatureoftheproblemsetsreinforcetheprimary
goaloftrainingmathematicians. Throughoutthebook,weemphasizethelanguageof
mathematicsandtheessentialsofproof-writing,andweunderscorethattheprocessis
veryimportantinmathematics.
Entry-leveldiscretemathematicsservesasanexcellentgatewaytoupper-levelmath-
ematics by priming students’ minds for upper-level concepts. Journey into Discrete
Mathematicsisdesignedforuseinthefirstnoncalculuscourseofamathematicsma-
jor, employing a writing style that models a high degree of mathematical accuracy
whilemaintainingaccessibilityforearlycollegestudents. Forexample,thetreatment
ofinclusion-exclusionprovidesbothinformalandtechnicallypreciseexplanations.Ul-
timately,thegoalbehindthisapproachiscommunication:wewanttomodelandteach
studentstocommunicatebothaccuratelyandclearly.
vii
viii Preface
JourneyintoDiscreteMathematicsutilizesproblemsandexamplestolaythefoun-
dationforconceptstobeencounteredinfuturemathematicscourses.Forexample,the
chapteronrelationsandfunctionsintroducesstudentstodefinitionssuchasone-to-
oneandonto;severalproblemsinChapter4guidestudentsthroughdefinitionsofcon-
tinuityusingnestedquantifiers;thetreatmentofgreatestcommondivisorforeshadows
findingtheGCDofpolynomialfunctions;thebinomialandmultinomialtheoremsare
presented as tools for combinatorial counting; and Euler’s totient function and Fer-
mat’sLittleTheoremareimportantnumber-theoreticconceptsthatstudentswillsee
againinanabstractalgebracourse. Thehomeworkquestionsaredividedintosections
accordingtodifficulty,spanningthegamutfromroutinetoquitechallenging.Thefirst
sectiongenerallyincludesexercisesthataremoreroutineorcomputational,meantto
givestudentsachancetopracticegiventechniques,whilethelattersectionsgenerally
consistofproblemsthatrequirecreativity,synthesisofmultipleconcepts,orproofs.
This book takes the time to describe the origins of important discrete math top-
icsaswellastheconnectionsbetweenconcepts. Thetreatmentofmatricesreferences
ArthurCayley’sfirstuseofmatrices;theintroductionofFibonaccinumbersisplaced
withinhistoricalcontext; theworkoninductivethinkingandproofbyinductionex-
hibitscareformakingconnectionswithdeductivethinking,theWell-OrderingPrinci-
ple,andothermathematicalconcepts. Inspirationalquotesthroughoutthebookand
theincorporationofthefirstnamesofmathematiciansinexamplesandexercises(with
acorrespondingsummarylistprovidingabriefbiographyforeachonementioned)con-
tributetofamiliarizingstudentswiththenamesofkeyfigureswithindiscretemathe-
matics.
Features of the Book
ConvinceMeChapter. This opening chapter contains a selection of interesting, non-
standard problems of varying degrees of difficulty. Readers are invited to think cre-
ativelyandarguepersuasivelyastheyworktofindsolutions. Thisprocesscultivates
anunderstandingoftheimportanceofmakingagoodmathematicalargument,while
setting the tone for the problem-solving nature of the book. Moreover, many of the
solutionsinthischapterforeshadowthemathematicaltechniquesandtheoremsthat
willbeencounteredlaterinthebook.
HookProblems. InthemanneroftheConvinceMeproblems,eachchapterbegins
withanintriguingandchallengingproblemintendedtocapturethereader’sinterest.
Eachhookproblemcanbesolvedusingtechniquestobedevelopedinthechapterand
usuallyreappearslaterinthechapter,eitherasanexampleorasahomeworkproblem.
PresentationofLogic.Chapters3and4ofthebookcombinethetopicsofsets,logic,
andproof-writinginadistinctiveway. Thisapproachhelpstohighlightthehighlevel
ofcongruitybetweenconceptssuchasDeMorgan’sLawsforsetsandlogic,member-
shiptablesandtruthtables,logicaloperatorsandsetoperators. Thechapteronlogic
andproof-writingappearsearlyinthebooktohelpstudentsbridgethegapbetween
intuitivethinkingandtheformalpresentationofanargument,bothofwhicharenec-
essaryinmathematics.
FirstThoughtsandFurtherThoughts. Solutionstomanyexamplesinthebookare
preceded by “First Thoughts”, describing the initial thought process that one might
engageinwhenfirstconsideringanewproblem. Thisisintendedtobebothhelpful
CourseOutline ix
andreassuringtostudentswhomightbeintimidatedbyseeingfinalpolishedproofs
andassumingthat“real”mathematicianscanproducetheseimmediately,withoutin-
termediarystrugglesorfailedattempts. FirstThoughtshelptrainstudentsintheways
that mathematicians actually operate. Similarly, “Further Thoughts” often follow a
solutioninordertoprovideadditionalinsightabout,alternativeapproachesto,orex-
tensionsofthegivensolution;onceagain,thegoalistocultivateaspiritofdoingmath-
ematics.
AdvancedTopicChapters.Severalcoretopics(counting,numbertheory,andgraph
theory)areaddressedtwiceinthisbook:firstinanintroductorychaptercoveringstan-
dardcontentandlaterinachapterwithextendedoptional(often,butnotalways,more
advanced)material. Thisprovidesinstructorswithflexibilitytocustomizethecourse,
dependingontheirparticulargoals,orexpandbeyondatypicalfirstcourseindiscrete
mathematics.
Course Outline
Thisbookisdesignedtobeusedasastand-alonetextforathree-creditorfour-credit
discretemathematicscourseforaveragetoaboveaveragemathmajorswhoarelearn-
ing to write proofs; however, since there is more material in this book than can be
coveredinasinglesemester,instructorswillhavetomakesomechoices. Forstudents
whohavealreadyhadanintroduction-to-proofscourse,selectportionsofthefirstsix
chaptersofthebookcanbecoveredratherquickly, andthelasthalfofthebookcan
serveasamaintextforajunior-levelcourseincombinatorics. Ifthisbookissupple-
mented with a few extra topics (such as probability, solving recurrence relations, or
finite-statemachines), thenthereisenoughmaterialforatwo-semestersequencein
discretemathematics.
Withthatgoalinmind,wesuggestonedesignforsuchathree-creditcourse. The
secondcolumninTable0.1onpagexliststhecoresectionswebelieveshouldbecov-
ered.Weestimatethatthecoresectionscanbecoveredinaboutthirty-fourfifty-minute
lectures. Theremainingclassperiodscouldbeusedforreviewdays,testingdays,and
optional sections from the third column. The first column of the table lists sections
andchapterscontainingmaterialthatisessentialforstudentstoknowbeforecover-
ingcorrespondingsectionsofthemiddletwocolumns. Themiddlecolumnscontain
materialthatisusedinsectionsandchapterslistedinthefourthcolumn,thoughthey
maynotbeabsoluteprerequisites. Forexample,althoughmatrices(firstaddressedin
Section2.3)alsoappearinChapter9(GraphTheory),oneneednotstudySection2.3
inordertobeabletounderstandtheessentialcomponentsofChapter9.