Table Of Content

Kenneth H. Rosen Discrete Mathematics and Its Applications Eighth Edition Discrete Mathematics and Its Applications Eighth Edition Kenneth H. Rosen formerlyAT&TLaboratories DISCRETEMATHEMATICSANDITSAPPLICATIONS,EIGHTHEDITION PublishedbyMcGraw-HillEducation,2PennPlaza,NewYork,NY10121.Copyright(cid:2)c 2019byMcGraw-HillEducation.Allrightsreserved.Printed intheUnitedStatesofAmerica.Previouseditions(cid:2)c 2012,2007,and2003.Nopartofthispublicationmaybereproducedordistributedinanyformor byanymeans,orstoredinadatabaseorretrievalsystem,withoutthepriorwrittenconsentofMcGraw-HillEducation,including,butnotlimitedto,inany networkorotherelectronicstorageortransmission,orbroadcastfordistancelearning. Someancillaries,includingelectronicandprintcomponents,maynotbeavailabletocustomersoutsidetheUnitedStates. Thisbookisprintedonacid-freepaper. 123456789LWI21201918 ISBN 978-1-259-67651-2 MHID 1-259-67651-X ProductDeveloper:NoraDevlin MarketingManager:AlisonFrederick ContentProjectManager:PeggySelle Buyer:SandyLudovissy Design:EgzonShaqiri ContentLicensingSpecialist:LorraineBuczek CoverImage:(cid:2)cKarlDehnam/AlamyStockPhoto Compositor:Aptara,Inc. Allcreditsappearingonpageorattheendofthebookareconsideredtobeanextensionofthecopyrightpage. LibraryofCongressCataloging-in-PublicationData Names:Rosen,KennethH.,author. Title:Discretemathematicsanditsapplications/KennethH.Rosen,Monmouth University(andformerlyAT&TLaboratories). Description:Eighthedition.|NewYork,NY:McGraw-Hill,[2019]|Includes bibliographicalreferencesandindex. Identifiers:LCCN2018008740|ISBN9781259676512(alk.paper)| ISBN125967651X(alk.paper) Subjects:LCSH:Mathematics.|Computerscience–Mathematics. Classification:LCCQA39.3.R672019|DDC511–dc23LCrecordavailableat https://lccn.loc.gov/2018008740 TheInternetaddresseslistedinthetextwereaccurateatthetimeofpublication.Theinclusionofawebsitedoesnotindicateanendorsementbythe authorsorMcGraw-HillEducation,andMcGraw-HillEducationdoesnotguaranteetheaccuracyoftheinformationpresentedatthesesites. mheducation.com/highered Contents AbouttheAuthorvi Prefacevii OnlineResourcesxvi TotheStudentxix 1 TheFoundations:LogicandProofs....................................1 1.1 PropositionalLogic............................................................1 1.2 ApplicationsofPropositionalLogic.............................................17 1.3 PropositionalEquivalences .................................................... 26 1.4 PredicatesandQuantifiers.....................................................40 1.5 NestedQuantifiers............................................................60 1.6 RulesofInference.............................................................73 1.7 IntroductiontoProofs.........................................................84 1.8 ProofMethodsandStrategy....................................................96 End-of-ChapterMaterial.....................................................115 2 BasicStructures:Sets,Functions,Sequences,Sums, andMatrices.......................................................121 2.1 Sets........................................................................121 2.2 SetOperations...............................................................133 2.3 Functions...................................................................147 2.4 SequencesandSummations...................................................165 2.5 CardinalityofSets...........................................................179 2.6 Matrices....................................................................188 End-of-ChapterMaterial.....................................................195 3 Algorithms.........................................................201 3.1 Algorithms..................................................................201 3.2 TheGrowthofFunctions.....................................................216 3.3 ComplexityofAlgorithms....................................................231 End-of-ChapterMaterial.....................................................244 4 NumberTheoryandCryptography..................................251 4.1 DivisibilityandModularArithmetic...........................................251 4.2 IntegerRepresentationsandAlgorithms........................................260 4.3 PrimesandGreatestCommonDivisors ........................................ 271 4.4 SolvingCongruences.........................................................290 4.5 ApplicationsofCongruences ................................................. 303 4.6 Cryptography ............................................................... 310 End-of-ChapterMaterial.....................................................324 iii iv Contents 5 InductionandRecursion............................................331 5.1 MathematicalInduction......................................................331 5.2 StrongInductionandWell-Ordering...........................................354 5.3 RecursiveDefinitionsandStructuralInduction..................................365 5.4 RecursiveAlgorithms........................................................381 5.5 ProgramCorrectness.........................................................393 End-of-ChapterMaterial.....................................................398 6 Counting...........................................................405 6.1 TheBasicsofCounting.......................................................405 6.2 ThePigeonholePrinciple.....................................................420 6.3 PermutationsandCombinations...............................................428 6.4 BinomialCoefficientsandIdentities...........................................437 6.5 GeneralizedPermutationsandCombinations...................................445 6.6 GeneratingPermutationsandCombinations....................................457 End-of-ChapterMaterial.....................................................461 7 DiscreteProbability.................................................469 7.1 AnIntroductiontoDiscreteProbability ........................................469 7.2 ProbabilityTheory...........................................................477 7.3 Bayes’Theorem.............................................................494 7.4 ExpectedValueandVariance.................................................503 End-of-ChapterMaterial.....................................................520 8 AdvancedCountingTechniques ..................................... 527 8.1 ApplicationsofRecurrenceRelations..........................................527 8.2 SolvingLinearRecurrenceRelations .......................................... 540 8.3 Divide-and-ConquerAlgorithmsandRecurrenceRelations......................553 8.4 GeneratingFunctions ........................................................ 563 8.5 Inclusion–Exclusion ......................................................... 579 8.6 ApplicationsofInclusion–Exclusion...........................................585 End-of-ChapterMaterial.....................................................592 9 Relations...........................................................599 9.1 RelationsandTheirProperties................................................599 9.2 n-aryRelationsandTheirApplications.........................................611 9.3 RepresentingRelations.......................................................621 9.4 ClosuresofRelations.........................................................628 9.5 EquivalenceRelations........................................................638 9.6 PartialOrderings ............................................................ 650 End-of-ChapterMaterial.....................................................665 Contents v 10 Graphs.............................................................673 10.1 GraphsandGraphModels....................................................673 10.2 GraphTerminologyandSpecialTypesofGraphs...............................685 10.3 RepresentingGraphsandGraphIsomorphism..................................703 10.4 Connectivity................................................................714 10.5 EulerandHamiltonPaths.....................................................728 10.6 Shortest-PathProblems.......................................................743 10.7 PlanarGraphs...............................................................753 10.8 GraphColoring..............................................................762 End-of-ChapterMaterial.....................................................771 11 Trees...............................................................781 11.1 IntroductiontoTrees.........................................................781 11.2 ApplicationsofTrees ........................................................ 793 11.3 TreeTraversal...............................................................808 11.4 SpanningTrees..............................................................821 11.5 MinimumSpanningTrees....................................................835 End-of-ChapterMaterial.....................................................841 12 BooleanAlgebra....................................................847 12.1 BooleanFunctions...........................................................847 12.2 RepresentingBooleanFunctions .............................................. 855 12.3 LogicGates.................................................................858 12.4 MinimizationofCircuits ..................................................... 864 End-of-ChapterMaterial.....................................................879 13 ModelingComputation ............................................. 885 13.1 LanguagesandGrammars....................................................885 13.2 Finite-StateMachineswithOutput.............................................897 13.3 Finite-StateMachineswithNoOutput.........................................904 13.4 LanguageRecognition ....................................................... 917 13.5 TuringMachines.............................................................927 End-of-ChapterMaterial.....................................................938 Appendices.........................................................A-1 1 AxiomsfortheRealNumbersandthePositiveIntegers..........................A-1 2 ExponentialandLogarithmicFunctions........................................A-7 3 Pseudocode................................................................A-11 SuggestedReadingsB-1 AnswerstoOdd-NumberedExercisesS-1 IndexofBiographiesI-1 IndexI-2 About the Author K enneth H. Rosen received his B.S. in Mathematics from the University of Michigan, Ann Arbor (1972), and his Ph.D. in Mathematics from M.I.T. (1976), where he wrote histhesisinnumbertheoryunderthedirectionofHaroldStark.BeforejoiningBellLaboratories in 1982, he held positions at the University of Colorado, Boulder; The Ohio State University, Columbus;andtheUniversityofMaine,Orono,wherehewasanassociateprofessorofmath- ematics.HeenjoyedalongcareerasaDistinguishedMemberoftheTechnicalStaffatAT&T BellLaboratories(andAT&TLaboratories)inMonmouthCounty,NewJersey.Whileworking atBellLabs,hetaughtatMonmouthUniversity,teachingcoursesindiscretemathematics,cod- ingtheory,anddatasecurity.AfterleavingAT&TLabs,hebecameavisitingresearchprofessor ofcomputerscienceatMonmouthUniversity,wherehehastaughtcoursesinalgorithmdesign, computersecurityandcryptography,anddiscretemathematics. Dr.Rosenhaspublishednumerousarticlesinprofessionaljournalsonnumbertheoryand onmathematicalmodeling.HeistheauthorofthewidelyusedElementaryNumberTheoryand ItsApplications,publishedbyPearson,currentlyinitssixthedition,whichhasbeentranslated intoChinese.HeisalsotheauthorofDiscreteMathematicsandItsApplications,publishedby McGraw-Hill,currentlyinitseighthedition.DiscreteMathematicsandItsApplicationshassold morethan450,000copiesinNorthAmericaduringitslifetime,andhundredsofthousandsof copiesthroughouttherestoftheworld.Thisbookhasalsobeentranslatedintomanylanguages, includingSpanish,French,Portuguese,Greek,Chinese,Vietnamese,andKorean.Heisalsoco- authorofUNIX:TheCompleteReference;UNIXSystemVRelease4:AnIntroduction;andBest UNIX Tips Ever, all published by Osborne McGraw-Hill. These books have sold more than 150,000copies,withtranslationsintoChinese,German,Spanish,andItalian.Dr.Rosenisalso the editor of both the first and second editions (published in 1999 and 2018, respectively) of the Handbook of Discrete and Combinatorial Mathematics, published by CRC Press. He has served as the advisory editor of the CRC series of books in discrete mathematics, sponsoring morethan70volumesondiverseaspectsofdiscretemathematics,manyofwhichareintroduced inthisbook.HeisanadvisoryeditorfortheCRCseriesofmathematicstextbooks,wherehehas helpedmorethan30authorswritebettertexts.Dr.RosenservesasanAssociateEditorforthe journalDiscreteMathematics,wherehehandlespapersinmanyareas,includinggraphtheory, enumeration,numbertheory,andcryptography. Dr. Rosen has had a longstanding interest in integrating mathematical software into the educational and professional environments. He has worked on several projects with Waterloo MapleInc.’sMapleTMsoftwareinboththeseareas.Dr.Rosenhasdevotedagreatdealofenergy toensuringthattheonlinehomeworkforDiscreteMathematicsanditsApplicationsisasuperior teachingtool.Dr.Rosenhasalsoworkedwithseveralpublishingcompaniesontheirhomework deliveryplatforms. At Bell Laboratories and AT&T Laboratories, Dr. Rosen worked on a wide range of projects, including operations research studies, product line planning for computers and data communicationsequipment,technologyassessmentandinnovation,andmanyotherefforts.He helped plan AT&T’s products and services in the area of multimedia, including video com- munications, speech recognition, speech synthesis, and image networking. He evaluated new technologyforusebyAT&Tanddidstandardsworkintheareaofimagenetworking.Healsoin- ventedmanynewservices,andholdsmorethan70patents.Oneofhismoreinterestingprojects involved helping evaluate technology for the AT&T attraction that was part of EPCOT Cen- ter.AfterleavingAT&T,Dr.RosenhasworkedasatechnologyconsultantforGoogleandfor AT&T. vi Preface I nwriting this book, I was guided by mylong-standing experience and interest in teaching discretemathematics.Forthestudent,mypurposewastopresentmaterialinaprecise,read- able manner, with the concepts and techniques of discrete mathematics clearly presented and demonstrated. My goal was to show the relevance and practicality of discrete mathematics to students, who are often skeptical. I wanted to give students studying computer science all of 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 appropriate manner for their particular set of students. I hope that I have achieved these goals. I have been extremely gratified by the tremendous success of this text, including its use by more than one million students around the world over the last 30 years and its translation into many different languages. The many improvements in the eighth edition have been made possiblebythefeedbackandsuggestionsofalargenumberofinstructorsandstudentsatmany ofthemorethan600NorthAmericanschools,andatmanyuniversitiesindifferentpartsofthe world,wherethisbookhasbeensuccessfullyused.Ihavebeenabletosignificantlyimprovethe appealandeffectivenessofthisbookeditiontoeditionbecauseofthefeedbackIhavereceived andthesignificantinvestmentsthathavebeenmadeintheevolutionofthebook. Thistextisdesignedforaone-ortwo-termintroductorydiscretemathematicscoursetaken bystudentsinawidevarietyofmajors,includingmathematics,computerscience,andengineer- ing.Collegealgebraistheonlyexplicitprerequisite,althoughacertaindegreeofmathematical maturityisneededtostudydiscretemathematicsinameaningfulway.Thisbookhasbeende- signedtomeettheneedsofalmostalltypesofintroductorydiscretemathematicscourses.Itis highlyflexibleandextremelycomprehensive.Thebookisdesignednotonlytobeasuccessful textbook,butalsotoserveasavaluableresourcestudentscanconsultthroughouttheirstudies andprofessionallife. Goals of a Discrete Mathematics Course A discrete mathematics course 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 mathematicalreasoningandthedifferentwaysproblemsaresolved.Fiveimportantthemesare interwoveninthistext:mathematicalreasoning,combinatorialanalysis,discretestructures,al- gorithmicthinking, andapplicationsandmodeling.Asuccessfuldiscretemathematicscourse shouldcarefullyblendandbalanceallfivethemes. 1. MathematicalReasoning:Studentsmustunderstandmathematicalreasoninginordertoread, comprehend, and construct mathematical arguments. This text starts with a discussion of mathematicallogic,whichservesasthefoundationforthesubsequentdiscussionsofmethods ofproof.Boththescienceandtheartofconstructingproofsareaddressed.Thetechniqueof vii viii Preface mathematicalinductionisstressedthroughmanydifferenttypesofexamplesofsuchproofs andacarefulexplanationofwhymathematicalinductionisavalidprooftechnique. 2. Combinatorial Analysis:Animportantproblem-solving skillistheabilitytocountorenu- 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 algorithm has 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.Theseapplicationsarenaturalandimportantusesof discrete mathematics and are not contrived. Modeling with discrete mathematics is an ex- tremelyimportantproblem-solvingskill,whichstudentshavetheopportunitytodevelopby constructingtheirownmodelsinsomeoftheexercises. Changes in the Eighth Edition Although the seventh edition has been an extremely effective text, many instructors have re- questedchangestomakethebookmoreusefultothem.Ihavedevotedasignificantamountof timeandenergytosatisfytheirrequestsandIhaveworkedhardtofindmyownwaystoimprove thebookandtokeepitup-to-date. The eighth edition includes changes based on input from more than 20 formal reviewers, feedbackfromstudentsandinstructors,andmyinsights.TheresultisaneweditionthatIex- pectwillbeamoreeffectiveteachingtool.Numerouschangesintheeightheditionhavebeen designedtohelpstudentslearnthematerial.Additionalexplanationsandexampleshavebeen added to clarify material where students have had difficulty. New exercises, both routine and challenging, have been added. Highly relevant applications, including many related to the In- ternet, to computer science, and to mathematical biology, have been added. The companion websitehasbenefitedfromextensivedevelopment;itnowprovidesextensivetoolsstudentscan usetomasterkeyconceptsandtoexploretheworldofdiscretemathematics.Furthermore,addi- tionaleffectiveandcomprehensivelearningandassessmenttoolsareavailable,complementing thetextbook. Ihopethatinstructorswillcloselyexaminethisneweditiontodiscoverhowitmightmeet their needs. Although it is impractical to list all the changes in this edition, a brief list that highlightssomekeychanges,listedbythebenefitstheyprovide,maybeuseful. ChangesintheEighthEdition This new edition of the book includes many enhancements, updates, additions, and edits, all designed to make the book a more effective teaching tool for a modern discrete mathematics course.Instructorswhohaveusedthebookpreviouslywillnoticeoverallchangesthathavebeen madethroughoutthebook,aswellasspecificchanges.Themostnotablerevisionsaredescribed here. Preface ix OverallChanges ▶ Expositionhasbeenimprovedthroughoutthebookwithafocusonprovidingmoreclarity tohelpstudentsreadandcomprehendconcepts. ▶ Many proofs have been enhanced by adding more details and explanations, and by re- mindingthereaderoftheproofmethodsused. ▶ Newexampleshavebeenadded,oftentomeetneedsidentifiedbyreviewersortoillus- tratenewmaterial.Manyoftheseexamplesarefoundinthetext,butothersareavailable onlyonthecompanionwebsite. ▶ Many new exercises, both routine and challenging, address needs identified by instructors or cover new material, while others strengthen and broaden existing exercise sets. ▶ More second and third level heads have been used to break sections into smaller co- herentparts,andanewnumberingschemehasbeenusedtoidentifysubsectionsofthe book. ▶ Theonlineresourcesforthisbookhavebeengreatlyexpanded,providingextensivesup- port for both instructors and students. These resources are described later in the front matter. TopicCoverage ▶ Logic Severallogicalpuzzleshavebeenintroduced.Anewexampleexplainshowto modelthen-queensproblemasasatisfiabilityproblemthatisbothconciseandaccessible tostudents. ▶ Settheory Multisetsarenowcoveredinthetext.(Previouslytheywereintroducedin theexercises.) ▶ Algorithms Thestringmatchingproblem,animportantalgorithmformanyapplica- tions,includingspellchecking,key-wordsearching,string-matching,andcomputational biology, is now discussed. The brute-force algorithm for solving string-matching exer- cisesispresented. ▶ Numbertheory Theneweditionincludesthelatestnumericalandtheoreticdiscov- eriesrelatingtoprimesandopenconjecturesaboutthem.TheextendedEuclideanalgo- rithm,aone-passalgorithm,isnowdiscussedinthetext.(Previouslyitwascoveredin theexercises.) ▶ Cryptography Theconceptofhomomorphicencryption,anditsimportancetocloud computing,isnowcovered. ▶ Mathematical induction The template for proofs by mathematical induction has been expanded. It is now placed in the text before examples of proof by mathematical induction. ▶ Countingmethods Thecoverageofthedivisionruleforcountinghasbeenexpanded. ▶ Data mining Association rules—key concepts in data mining—are now discussed in the section on n-ary relations. Also, the Jaccard metric, which is used to find the distance between two sets and which is used in data mining, is introduced in the exercises. ▶ Graph theory applications A new example illustrates how semantic networks, an importantstructureinartificialintelligence,canbemodeledusinggraphs.