Discrete Mathematics Applications

1071 Pages·2017·3.47 MB·English

KennethH.Rosen
Discrete
Mathematics
andIts
Applications
SEVENTHEDITION
Discrete
Mathematics
and Its
Applications
Seventh Edition
Kenneth H. Rosen
MonmouthUniversity
(andformerlyAT&TLaboratories)
DISCRETEMATHEMATICSANDITSAPPLICATIONS,SEVENTHEDITION
PublishedbyMcGrawHill,abusinessunitofTheMcGrawHillCompanies,Inc.,1221Avenueofthe
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ISBN 9780073383095
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LibraryofCongressCataloginginPublicationData
Rosen,KennethH.
Discretemathematicsanditsapplications/KennethH.Rosen.—7thed.
p.cm.
Includesindex.
ISBN0–07–338309–0
1.Mathematics.2.Computerscience—Mathematics.I.Title.
QA39.3.R672012
511–dc22
2011011060
www.mhhe.com
AbouttheAuthorvi
Prefacevii
TheCompanionWebsitexvi
TotheStudentxvii
1 TheFoundations:LogicandProofs..................................1
1.1 PropositionalLogic............................................................1
1.2 ApplicationsofPropositionalLogic.............................................16
1.3 PropositionalEquivalences .................................................... 25
1.4 PredicatesandQuantifiers.....................................................36
1.5 NestedQuantifiers............................................................57
1.6 RulesofInference ............................................................69
1.7 IntroductiontoProofs.........................................................80
1.8 ProofMethodsandStrategy....................................................92
EndofChapterMaterial.....................................................109
2 BasicStructures:Sets,Functions,Sequences,Sums,andMatrices.115
2.1 Sets........................................................................115
2.2 SetOperations...............................................................127
2.3 Functions...................................................................138
2.4 SequencesandSummations...................................................156
2.5 CardinalityofSets...........................................................170
2.6 Matrices....................................................................177
EndofChapterMaterial.....................................................185
3 Algorithms........................................................191
3.1 Algorithms..................................................................191
3.2 TheGrowthofFunctions.....................................................204
3.3 ComplexityofAlgorithms....................................................218
EndofChapterMaterial.....................................................232
4 NumberTheoryandCryptography................................237
4.1 DivisibilityandModularArithmetic...........................................237
4.2 IntegerRepresentationsandAlgorithms........................................245
4.3 PrimesandGreatestCommonDivisors ........................................ 257
4.4 SolvingCongruences.........................................................274
4.5 ApplicationsofCongruences..................................................287
4.6 Cryptography............................................................... 294
EndofChapterMaterial.....................................................306
iii
iv Contents
5 InductionandRecursion .......................................... 311
5.1 MathematicalInduction......................................................311
5.2 StrongInductionandWellOrdering...........................................333
5.3 RecursiveDefinitionsandStructuralInduction..................................344
5.4 RecursiveAlgorithms........................................................360
5.5 ProgramCorrectness.........................................................372
EndofChapterMaterial.....................................................377
6 Counting..........................................................385
6.1 TheBasicsofCounting.......................................................385
6.2 ThePigeonholePrinciple.....................................................399
6.3 PermutationsandCombinations...............................................407
6.4 BinomialCoefficientsandIdentities...........................................415
6.5 GeneralizedPermutationsandCombinations...................................423
6.6 GeneratingPermutationsandCombinations....................................434
EndofChapterMaterial.....................................................439
7 DiscreteProbability...............................................445
7.1 AnIntroductiontoDiscreteProbability ........................................ 445
7.2 ProbabilityTheory...........................................................452
7.3 Bayes’Theorem.............................................................468
7.4 ExpectedValueandVariance..................................................477
EndofChapterMaterial.....................................................494
8 AdvancedCountingTechniques...................................501
8.1 ApplicationsofRecurrenceRelations..........................................501
8.2 SolvingLinearRecurrenceRelations..........................................514
8.3 DivideandConquerAlgorithmsandRecurrenceRelations.......................527
8.4 GeneratingFunctions........................................................537
8.5 Inclusion–Exclusion.........................................................552
8.6 ApplicationsofInclusion–Exclusion...........................................558
EndofChapterMaterial.....................................................565
9 Relations..........................................................573
9.1 RelationsandTheirProperties................................................573
9.2 naryRelationsandTheirApplications.........................................583
9.3 RepresentingRelations.......................................................591
9.4 ClosuresofRelations ........................................................ 597
9.5 EquivalenceRelations........................................................607
9.6 PartialOrderings ............................................................ 618
EndofChapterMaterial.....................................................633
Contents v
10 Graphs ............................................................ 641
10.1 GraphsandGraphModels....................................................641
10.2 GraphTerminologyandSpecialTypesofGraphs................................651
10.3 RepresentingGraphsandGraphIsomorphism..................................668
10.4 Connectivity................................................................678
10.5 EulerandHamiltonPaths.....................................................693
10.6 ShortestPathProblems.......................................................707
10.7 PlanarGraphs...............................................................718
10.8 GraphColoring..............................................................727
EndofChapterMaterial.....................................................735
11 Trees...............................................................745
11.1 IntroductiontoTrees.........................................................745
11.2 ApplicationsofTrees.........................................................757
11.3 TreeTraversal...............................................................772
11.4 SpanningTrees..............................................................785
11.5 MinimumSpanningTrees....................................................797
EndofChapterMaterial.....................................................803
12 BooleanAlgebra...................................................811
12.1 BooleanFunctions...........................................................811
12.2 RepresentingBooleanFunctions..............................................819
12.3 LogicGates.................................................................822
12.4 MinimizationofCircuits ..................................................... 828
EndofChapterMaterial.....................................................843
13 ModelingComputation ........................................... 847
13.1 LanguagesandGrammars....................................................847
13.2 FiniteStateMachineswithOutput.............................................858
13.3 FiniteStateMachineswithNoOutput.........................................865
13.4 LanguageRecognition .......................................................878
13.5 TuringMachines.............................................................888
EndofChapterMaterial.....................................................899
Appendixes ....................................................... A1
1 AxiomsfortheRealNumbersandthePositiveIntegers............................1
2 ExponentialandLogarithmicFunctions..........................................7
3 Pseudocode..................................................................11
SuggestedReadingsB1
AnswerstoOddNumberedExercisesS1
PhotoCreditsC1
IndexofBiographiesI1
IndexI2
K
ennethH.RosenhashadalongcareerasaDistinguishedMemberoftheTechnicalStaff
atAT&TLaboratoriesinMonmouthCounty,NewJersey.Hecurrentlyholdstheposition
ofVisitingResearchProfessoratMonmouthUniversity,whereheteachesgraduatecoursesin
computerscience.
Dr.RosenreceivedhisB.S. inMathematicsfromtheUniversityofMichigan,AnnArbor
(1972),andhisPh.D.inMathematicsfromM.I.T.(1976),wherehewrotehisthesisinthearea
ofnumbertheoryunderthedirectionofHaroldStark.BeforejoiningBellLaboratoriesin1982,
heheldpositionsattheUniversityofColorado,Boulder;TheOhioStateUniversity,Columbus;
and the University of Maine, Orono, where he was an associate professor of mathematics.
WhileworkingatAT&TLabs,hetaughtatMonmouthUniversity,teachingcoursesindiscrete
mathematics,codingtheory,anddatasecurity.Hecurrentlyteachescoursesinalgorithmdesign
andincomputersecurityandcryptography.
Dr.Rosenhaspublishednumerousarticlesinprofessional journalsinnumbertheoryand
inmathematicalmodeling.HeistheauthorofthewidelyusedElementaryNumberTheoryand
ItsApplications,publishedbyPearson,currentlyinitssixthedition,whichhasbeentranslated
intoChinese.HeisalsotheauthorofDiscreteMathematicsandItsApplications,publishedby
McGrawHill,currently in itsseventhedition.Discrete Mathematics and ItsApplicationshas
soldmorethan350,000copiesinNorthAmericaduringitslifetime,andhundredsofthousands
of copies throughout the rest of the world. This book has also been translated into Spanish,
French,Greek,Chinese,Vietnamese,andKorean.HeisalsocoauthorofUNIX:TheComplete
Reference;UNIXSystemVRelease4:AnIntroduction;andBestUNIXTipsEver,allpublishedby
OsborneMcGrawHill.Thesebookshavesoldmorethan150,000copies,withtranslationsinto
Chinese,German,Spanish,andItalian.Dr.RosenisalsotheeditoroftheHandbookofDiscrete
andCombinatorialMathematics,publishedbyCRCPress,andheistheadvisoryeditorofthe
CRCseriesofbooksindiscretemathematics,consistingofmorethan55volumesondifferent
aspectsofdiscretemathematics,mostofwhichareintroducedinthisbook.Dr.Rosenservesasan
AssociateEditorforthejournalDiscreteMathematics,whereheworkswithsubmittedpapersin
severalareasofdiscretemathematics,includinggraphtheory,enumeration,andnumbertheory.
Heisalsointerestedinintegratingmathematicalsoftwareintotheeducationalandprofessional
environments, and worked on several projects withWaterloo Maple Inc.’s MapleTM software
in both these areas. Dr. Rosen has also worked with several publishing companies on their
homeworkdeliveryplatforms.
AtBellLaboratoriesandAT&TLaboratories,Dr.Rosenworkedonawiderangeofprojects,
includingoperationsresearchstudies,productlineplanningforcomputersanddatacommuni
cationsequipment,andtechnologyassessment.HehelpedplanAT&T’sproductsandservicesin
theareaofmultimedia,includingvideocommunications,speechrecognition,speechsynthesis,
andimagenetworking.HeevaluatednewtechnologyforusebyAT&Tanddidstandardswork
intheareaofimagenetworking.Healsoinventedmanynewservices,andholdsmorethan55
patents.OneofhismoreinterestingprojectsinvolvedhelpingevaluatetechnologyfortheAT&T
attractionthatwaspartofEPCOTCenter.
vi
I
nwritingthisbook,Iwasguidedbymylongstanding experienceandinterestinteaching
discrete mathematics. For the student, my purpose was to present material in a precise,
readable manner, with the concepts and techniquesof discrete mathematicsclearly presented
anddemonstrated.Mygoalwastoshowtherelevanceandpracticalityofdiscretemathematics
tostudents,whoareoftenskeptical.Iwantedtogivestudentsstudyingcomputerscienceallof
themathematicalfoundationstheyneedfortheirfuturestudies.Iwantedtogivemathematics
students an understanding of important mathematical concepts together with a sense of why
these concepts are important for applications.And most importantly, I wanted to accomplish
thesegoalswithoutwateringdownthematerial.
Fortheinstructor,mypurposewastodesignaflexible,comprehensiveteachingtoolusing
provenpedagogicaltechniquesinmathematics.Iwantedtoprovideinstructorswithapackage
ofmaterialsthattheycouldusetoteachdiscretemathematicseffectivelyandefficientlyinthe
most appropriatemannerfortheirparticularsetofstudents.Ihope that Ihave achievedthese
goals.
Ihavebeenextremelygratifiedbythetremendoussuccessofthistext.Themanyimprove
mentsintheseventheditionhavebeenmadepossiblebythefeedbackandsuggestionsofalarge
numberofinstructorsandstudentsatmanyofthemorethan600NorthAmericanschools,and
atanymanyuniversitiesinpartsoftheworld,wherethisbookhasbeensuccessfullyused.
Thistextisdesignedforaoneortwotermintroductorydiscretemathematicscoursetaken
bystudentsinawidevarietyofmajors,includingmathematics,computerscience,andengineer
ing.Collegealgebraistheonlyexplicitprerequisite,althoughacertaindegreeofmathematical
maturityisneededtostudydiscretemathematicsinameaningfulway.Thisbookhasbeende
signedtomeettheneedsofalmostalltypesofintroductorydiscretemathematicscourses.Itis
highlyflexibleandextremelycomprehensive.Thebookisdesignednotonlytobeasuccessful
textbook, but also to serve as valuable resource students can consult throughout their studies
andprofessionallife.
Goals of a Discrete Mathematics Course
A discrete mathematicscourse has more than one purpose. Students should learn a particular
setofmathematicalfactsandhowtoapplythem;moreimportantly,suchacourseshouldteach
students how to think logically and mathematically. To achieve these goals, this text stresses
mathematical reasoning and the different ways problems are solved. Five important themes
areinterwoveninthistext:mathematicalreasoning,combinatorialanalysis,discretestructures,
algorithmicthinking,andapplicationsandmodeling.Asuccessfuldiscretemathematicscourse
shouldcarefullyblendandbalanceallfivethemes.
1. Mathematical Reasoning: Students must understand mathematical reasoning in order to
read,comprehend,andconstructmathematicalarguments.Thistextstartswithadiscussion
of mathematical logic, which serves as the foundation for the subsequent discussions of
methods of proof. Both the science and the art of constructing proofs are addressed. The
technique of mathematical induction is stressed through many different typesof examples
of such proofs and a careful explanation of why mathematical induction is a valid proof
technique.
vii
viii Preface
2. CombinatorialAnalysis:Animportantproblemsolvingskillistheabilitytocountorenu
merateobjects.Thediscussionofenumerationinthisbookbeginswiththebasictechniques
ofcounting.Thestressisonperformingcombinatorialanalysistosolvecountingproblems
andanalyzealgorithms,notonapplyingformulae.
3. Discrete Structures:A course in discrete mathematics should teach students how to work
with discrete structures, which are the abstract mathematical structures used to represent
discrete objects and relationships between these objects. These discrete structures include
sets,permutations,relations,graphs,trees,andfinitestatemachines.
4. Algorithmic Thinking: Certain classes of problems are solved by the specification of an
algorithm.After an algorithmhas been described, a computer program can be constructed
implementingit.Themathematicalportionsofthisactivity,whichincludethespecification
of the algorithm, the verification that it works properly, and the analysis of the computer
memoryandtimerequiredtoperformit,areallcoveredinthistext.Algorithmsaredescribed
usingbothEnglishandaneasilyunderstoodformofpseudocode.
5. ApplicationsandModeling:Discretemathematicshasapplicationstoalmosteveryconceiv
able area of study. There are many applications to computer science and data networking
in this text, as well as applications to such diverse areas as chemistry, biology, linguistics,
geography,business,andtheInternet.Theseapplicationsare naturalandimportantusesof
discrete mathematics and are not contrived. Modeling with discrete mathematics is an ex
tremelyimportantproblemsolvingskill,whichstudentshavetheopportunitytodevelopby
constructingtheirownmodelsinsomeoftheexercises.
Changes in the Seventh Edition
Although the sixth edition has been an extremely effective text, many instructors, including
longtime users, have requested changes designed to make this book more effective. I have
devotedasignificantamountoftimeandenergytosatisfytheirrequestsandIhaveworkedhard
tofindmyownwaystomakethebookmoreeffectiveandmorecompellingtostudents.
The seventh edition is a major revision, with changes based on input from more than 40
formal reviewers, feedback from students and instructors, and author insights.The result is a
new edition that offers an improved organization of topics making the book a more effective
teaching tool. Substantial enhancements to the material devoted to logic, algorithms, number
theory,andgraphtheorymakethisbookmoreflexibleandcomprehensive.Numerouschanges
in the seventh edition have been designed to help students more easily learn the material.
Additionalexplanationsandexampleshavebeenaddedtoclarifymaterialwherestudentsoften
havedifficulty.Newexercises,bothroutineandchallenging,havebeenadded.Highlyrelevant
applications,includingmanyrelatedtotheInternet,tocomputerscience,andtomathematical
biology,have beenadded.The companionwebsite hasbenefitedfromextensive development
activityandnowprovidestoolsstudentscanusetomasterkeyconceptsandexploretheworld
of discrete mathematics, and many new tools under development will be released in the year
followingpublicationofthisbook.
Ihopethatinstructorswillcloselyexaminethisneweditiontodiscoverhowitmightmeet
their needs. Although it is impractical to list all the changes in this edition, a brief list that
highlightssomekeychanges,listedbythebenefitstheyprovide,maybeuseful.
MoreFlexibleOrganization
Applicationsofpropositionallogicarefoundinanewdedicatedsection,whichbriefly
introduceslogiccircuits.
RecurrencerelationsarenowcoveredinChapter2.
ExpandedcoverageofcountabilityisnowfoundinadedicatedsectioninChapter2.
Preface ix
Separatechaptersnowprovideexpandedcoverageofalgorithms(Chapter3)andnumber
theoryandcryptography(Chapter4).
Moresecondandthirdlevelheadshavebeenusedtobreaksectionsintosmallercoherent
parts.
ToolsforEasierLearning
DifficultdiscussionsandproofshavebeenmarkedwiththefamousBourbaki“dangerous
bend”symbolinthemargin.
New marginal notes make connections, add interesting notes, and provide advice to
students.
Moredetailsandaddedexplanations,inbothproofsandexposition,makeiteasierfor
studentstoreadthebook.
Manynewexercises,both routineand challenging,havebeenadded,while manyex
istingexerciseshavebeenimproved.
EnhancedCoverageofLogic,Sets,andProof
Thesatisfiabilityproblemisaddressedingreaterdepth,withSudokumodeledinterms
ofsatisfiability.
Hilbert’sGrandHotelisusedtohelpexplainuncountability.
Proofsthroughoutthebookhavebeenmademoreaccessiblebyaddingstepsandreasons
behindthesesteps.
Atemplateforproofsbymathematicalinductionhasbeenadded.
Thestepthatappliestheinductivehypothesisinmathematicalinductionproofisnow
explicitlynoted.
Algorithms
Thepseudocodeusedinthebookhasbeenupdated.
Explicitcoverageofalgorithmicparadigms,includingbruteforce,greedyalgorithms,
anddynamicprograming,isnowprovided.
UsefulrulesforbigOestimatesoflogarithms,powers,andexponentialfunctionshave
beenadded.
NumberTheoryandCryptography
Expandedcoverageallowsinstructorstoincludejustalittleoralotofnumbertheory
intheircourses.
Therelationshipbetweenthemodfunctionandcongruenceshasbeenexplainedmore
fully.
ThesieveofEratosthenesisnowintroducedearlierinthebook.
Linearcongruencesandmodularinversesarenowcoveredinmoredetail.
Applicationsofnumbertheory,includingcheckdigitsandhashfunctions,arecovered
ingreatdepth.
Anewsectiononcryptographyintegratespreviouscoverage,andthenotionofacryp
tosystemhasbeenintroduced.
Cryptographicprotocols,includingdigitalsignaturesandkeysharing,arenowcovered.