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Kenneth H. Rosen
Discrete Mathematics Applications and Its Eighth Edition
Preface Inwritingthisbook,Iwasguidedbymylong-standingexperienceandinterestinteaching discretemathematics.Forthestudent,mypurposewastopresentmaterialinaprecise,readablemanner,withtheconceptsandtechniquesofdiscretemathematicsclearlypresentedand demonstrated.Mygoalwastoshowtherelevanceandpracticalityofdiscretemathematicsto students,whoareoftenskeptical.Iwantedtogivestudentsstudyingcomputerscienceallof themathematicalfoundationstheyneedfortheirfuturestudies.Iwantedtogivemathematics studentsanunderstandingofimportantmathematicalconceptstogetherwithasenseofwhy theseconceptsareimportantforapplications.Andmostimportantly,Iwantedtoaccomplish thesegoalswithoutwateringdownthematerial.
Fortheinstructor,mypurposewastodesignaflexible,comprehensiveteachingtoolusing provenpedagogicaltechniquesinmathematics.Iwantedtoprovideinstructorswithapackage ofmaterialsthattheycouldusetoteachdiscretemathematicseffectivelyandefficientlyinthe mostappropriatemannerfortheirparticularsetofstudents.IhopethatIhaveachievedthese goals.
Ihavebeenextremelygratifiedbythetremendoussuccessofthistext,includingitsuse bymorethanonemillionstudentsaroundtheworldoverthelast30yearsanditstranslation intomanydifferentlanguages.Themanyimprovementsintheeightheditionhavebeenmade possiblebythefeedbackandsuggestionsofalargenumberofinstructorsandstudentsatmany ofthemorethan600NorthAmericanschools,andatmanyuniversitiesindifferentpartsofthe world,wherethisbookhasbeensuccessfullyused.Ihavebeenabletosignificantlyimprovethe appealandeffectivenessofthisbookeditiontoeditionbecauseofthefeedbackIhavereceived andthesignificantinvestmentsthathavebeenmadeintheevolutionofthebook.
Thistextisdesignedforaone-ortwo-termintroductorydiscretemathematicscoursetaken bystudentsinawidevarietyofmajors,includingmathematics,computerscience,andengineering.Collegealgebraistheonlyexplicitprerequisite,althoughacertaindegreeofmathematical maturityisneededtostudydiscretemathematicsinameaningfulway.Thisbookhasbeendesignedtomeettheneedsofalmostalltypesofintroductorydiscretemathematicscourses.Itis highlyflexibleandextremelycomprehensive.Thebookisdesignednotonlytobeasuccessful textbook,butalsotoserveasavaluableresourcestudentscanconsultthroughouttheirstudies andprofessionallife.
Adiscretemathematicscoursehasmorethanonepurpose.Studentsshouldlearnaparticular setofmathematicalfactsandhowtoapplythem;moreimportantly,suchacourseshouldteach studentshowtothinklogicallyandmathematically.Toachievethesegoals,thistextstresses mathematicalreasoningandthedifferentwaysproblemsaresolved.Fiveimportantthemesare interwoveninthistext:mathematicalreasoning,combinatorialanalysis,discretestructures,algorithmicthinking,andapplicationsandmodeling.Asuccessfuldiscretemathematicscourse shouldcarefullyblendandbalanceallfivethemes.
1. MathematicalReasoning: Studentsmustunderstandmathematicalreasoninginordertoread, comprehend,andconstructmathematicalarguments.Thistextstartswithadiscussionof mathematicallogic,whichservesasthefoundationforthesubsequentdiscussionsofmethods ofproof.Boththescienceandtheartofconstructingproofsareaddressed.Thetechniqueof vii
mathematicalinductionisstressedthroughmanydifferenttypesofexamplesofsuchproofs andacarefulexplanationofwhymathematicalinductionisavalidprooftechnique.
2. CombinatorialAnalysis: Animportantproblem-solvingskillistheabilitytocountorenumerateobjects.Thediscussionofenumerationinthisbookbeginswiththebasictechniques ofcounting.Thestressisonperformingcombinatorialanalysistosolvecountingproblems andanalyzealgorithms,notonapplyingformulae.
3. DiscreteStructures: Acourseindiscretemathematicsshouldteachstudentshowtowork withdiscretestructures,whicharetheabstractmathematicalstructuresusedtorepresent discreteobjectsandrelationshipsbetweentheseobjects.Thesediscretestructuresinclude sets,permutations,relations,graphs,trees,andfinite-statemachines.
4. AlgorithmicThinking: Certainclassesofproblemsaresolvedbythespecificationofan algorithm.Afteranalgorithmhasbeendescribed,acomputerprogramcanbeconstructed implementingit.Themathematicalportionsofthisactivity,whichincludethespecification ofthealgorithm,theverificationthatitworksproperly,andtheanalysisofthecomputer memoryandtimerequiredtoperformit,areallcoveredinthistext.Algorithmsaredescribed usingbothEnglishandaneasilyunderstoodformofpseudocode.
5. ApplicationsandModeling: Discretemathematicshasapplicationstoalmosteveryconceivableareaofstudy.Therearemanyapplicationstocomputerscienceanddatanetworking inthistext,aswellasapplicationstosuchdiverseareasaschemistry,biology,linguistics, geography,business,andtheInternet.Theseapplicationsarenaturalandimportantusesof discretemathematicsandarenotcontrived.Modelingwithdiscretemathematicsisanextremelyimportantproblem-solvingskill,whichstudentshavetheopportunitytodevelopby constructingtheirownmodelsinsomeoftheexercises.
ChangesintheEighthEdition Althoughtheseventheditionhasbeenanextremelyeffectivetext,manyinstructorshaverequestedchangestomakethebookmoreusefultothem.Ihavedevotedasignificantamountof timeandenergytosatisfytheirrequestsandIhaveworkedhardtofindmyownwaystoimprove thebookandtokeepitup-to-date.
Theeightheditionincludeschangesbasedoninputfrommorethan20formalreviewers, feedbackfromstudentsandinstructors,andmyinsights.TheresultisaneweditionthatIexpectwillbeamoreeffectiveteachingtool.Numerouschangesintheeightheditionhavebeen designedtohelpstudentslearnthematerial.Additionalexplanationsandexampleshavebeen addedtoclarifymaterialwherestudentshavehaddifficulty.Newexercises,bothroutineand challenging,havebeenadded.Highlyrelevantapplications,includingmanyrelatedtotheInternet,tocomputerscience,andtomathematicalbiology,havebeenadded.Thecompanion websitehasbenefitedfromextensivedevelopment;itnowprovidesextensivetoolsstudentscan usetomasterkeyconceptsandtoexploretheworldofdiscretemathematics.Furthermore,additionaleffectiveandcomprehensivelearningandassessmenttoolsareavailable,complementing thetextbook.
Ihopethatinstructorswillcloselyexaminethisneweditiontodiscoverhowitmightmeet theirneeds.Althoughitisimpracticaltolistallthechangesinthisedition,abrieflistthat highlightssomekeychanges,listedbythebenefitstheyprovide,maybeuseful.
ChangesintheEighthEdition Thisneweditionofthebookincludesmanyenhancements,updates,additions,andedits,all designedtomakethebookamoreeffectiveteachingtoolforamoderndiscretemathematics course.Instructorswhohaveusedthebookpreviouslywillnoticeoverallchangesthathavebeen madethroughoutthebook,aswellasspecificchanges.Themostnotablerevisionsaredescribed here.
OverallChanges ▶ Expositionhasbeenimprovedthroughoutthebookwithafocusonprovidingmoreclarity tohelpstudentsreadandcomprehendconcepts.
▶ Manyproofshavebeenenhancedbyaddingmoredetailsandexplanations,andbyremindingthereaderoftheproofmethodsused.
▶ Newexampleshavebeenadded,oftentomeetneedsidentifiedbyreviewersortoillustratenewmaterial.Manyoftheseexamplesarefoundinthetext,butothersareavailable onlyonthecompanionwebsite.
▶ Manynewexercises,bothroutineandchallenging,addressneedsidentifiedbyinstructorsorcovernewmaterial,whileothersstrengthenandbroadenexistingexercise sets.
▶ Moresecondandthirdlevelheadshavebeenusedtobreaksectionsintosmallercoherentparts,andanewnumberingschemehasbeenusedtoidentifysubsectionsofthe book.
▶ Theonlineresourcesforthisbookhavebeengreatlyexpanded,providingextensivesupportforbothinstructorsandstudents.Theseresourcesaredescribedlaterinthefront matter.
TopicCoverage ▶ Logic Severallogicalpuzzleshavebeenintroduced.Anewexampleexplainshowto modelthe n-queensproblemasasatisfiabilityproblemthatisbothconciseandaccessible tostudents.
▶ Settheory Multisetsarenowcoveredinthetext.(Previouslytheywereintroducedin theexercises.)
▶ Algorithms Thestringmatchingproblem,animportantalgorithmformanyapplications,includingspellchecking,key-wordsearching,string-matching,andcomputational biology,isnowdiscussed.Thebrute-forcealgorithmforsolvingstring-matchingexercisesispresented.
▶ Numbertheory Theneweditionincludesthelatestnumericalandtheoreticdiscoveriesrelatingtoprimesandopenconjecturesaboutthem.TheextendedEuclideanalgorithm,aone-passalgorithm,isnowdiscussedinthetext.(Previouslyitwascoveredin theexercises.)
▶ Cryptography Theconceptofhomomorphicencryption,anditsimportancetocloud computing,isnowcovered.
▶ Mathematicalinduction Thetemplateforproofsbymathematicalinductionhas beenexpanded.Itisnowplacedinthetextbeforeexamplesofproofbymathematical induction.
▶ Countingmethods Thecoverageofthedivisionruleforcountinghasbeenexpanded.
▶ Datamining Associationrules—keyconceptsindatamining—arenowdiscussed inthesectionon n-aryrelations.Also,theJaccardmetric,whichisusedtofindthe distancebetweentwosetsandwhichisusedindatamining,isintroducedinthe exercises.
▶ Graphtheoryapplications Anewexampleillustrateshowsemanticnetworks,an importantstructureinartificialintelligence,canbemodeledusinggraphs.
▶ Biographies
NewbiographiesofWiles,Bhaskaracharya,delaVall ´ ee-Poussin, Hadamard,Zhang,andGentryhavebeenadded.Existingbiographieshavebeenexpandedandupdated.ThisaddsdiversitybyincludingmorehistoricallyimportantEastern mathematicians,majornineteenthandtwentiethcenturyresearchers,andcurrentlyactive twenty-firstcenturymathematiciansandcomputerscientists. FeaturesoftheBook
ACCESSIBILITY Thistexthasproventobeeasilyreadandunderstoodbymanybeginningstudents.Therearenomathematicalprerequisitesbeyondcollegealgebraforalmostall thecontentsofthetext.Studentsneedingextrahelpwillfindtoolsonthecompanionwebsite forbringingtheirmathematicalmaturityuptothelevelofthetext.Thefewplacesinthebook wherecalculusisreferredtoareexplicitlynoted.Moststudentsshouldeasilyunderstandthe pseudocodeusedinthetexttoexpressalgorithms,regardlessofwhethertheyhaveformally studiedprogramminglanguages.Thereisnoformalcomputerscienceprerequisite.
Eachchapterbeginsataneasilyunderstoodandaccessiblelevel.Oncebasicmathematical conceptshavebeencarefullydeveloped,moredifficultmaterialandapplicationstootherareas ofstudyarepresented.
FLEXIBILITY Thistexthasbeencarefullydesignedforflexibleuse.Thedependence ofchaptersonpreviousmaterialhasbeenminimized.Eachchapterisdividedintosectionsof approximatelythesamelength,andeachsectionisdividedintosubsectionsthatformnatural blocksofmaterialforteaching.Instructorscaneasilypacetheirlecturesusingtheseblocks.
WRITINGSTYLE Thewritingstyleinthisbookisdirectandpragmatic.Precisemathematicallanguageisusedwithoutexcessiveformalismandabstraction.Carehasbeentakento balancethemixofnotationandwordsinmathematicalstatements.
MATHEMATICALRIGORANDPRECISION Alldefinitionsandtheoremsinthistext arestatedextremelycarefullysothatstudentswillappreciatetheprecisionoflanguageand rigorneededinmathematics.Proofsaremotivatedanddevelopedslowly;theirstepsareall carefullyjustified.Theaxiomsusedinproofsandthebasicpropertiesthatfollowfromthem areexplicitlydescribedinanappendix,givingstudentsaclearideaofwhattheycanassumein aproof.Recursivedefinitionsareexplainedandusedextensively.
WORKEDEXAMPLES Over800examplesareusedtoillustrateconcepts,relatedifferenttopics,andintroduceapplications.Inmostexamples,aquestionisfirstposed,thenits solutionispresentedwiththeappropriateamountofdetail.
APPLICATIONS Theapplicationsincludedinthistextdemonstratetheutilityofdiscrete mathematicsinthesolutionofreal-worldproblems.Thistextincludesapplicationstoawide varietyofareas,includingcomputerscience,datanetworking,psychology,chemistry,engineering,linguistics,biology,business,andtheInternet.
ALGORITHMS Resultsindiscretemathematicsareoftenexpressedintermsofalgorithms;hence,keyalgorithmsareintroducedinmostchaptersofthebook.Thesealgorithms areexpressedinwordsandinaneasilyunderstoodformofstructuredpseudocode,whichis describedandspecifiedinAppendix3.Thecomputationalcomplexityofthealgorithmsinthe textisalsoanalyzedatanelementarylevel.
HISTORICALINFORMATION Thebackgroundofmanytopicsissuccinctlydescribed inthetext.Briefbiographiesof89mathematiciansandcomputerscientistsareincludedas
footnotes.Thesebiographiesincludeinformationaboutthelives,careers,andaccomplishments oftheseimportantcontributorstodiscretemathematics,andimages,whenavailable,aredisplayed.
Inaddition,numeroushistoricalfootnotesareincludedthatsupplementthehistoricalinformationinthemainbodyofthetext.Effortshavebeenmadetokeepthebookup-to-dateby reflectingthelatestdiscoveries.
KEYTERMSANDRESULTS Alistofkeytermsandresultsfollowseachchapter.The keytermsincludeonlythemostimportantthatstudentsshouldlearn,andnoteverytermdefined inthechapter.
EXERCISES Thereareover4200exercisesinthetext,withmanydifferenttypesofquestionsposed.Thereisanamplesupplyofstraightforwardexercisesthatdevelopbasicskills,a largenumberofintermediateexercises,andmanychallengingexercises.Exercisesarestated clearlyandunambiguously,andallarecarefullygradedforlevelofdifficulty.Exercisesetscontainspecialdiscussionsthatdevelopnewconceptsnotcoveredinthetext,enablingstudentsto discovernewideasthroughtheirownwork.
Exercisesthataresomewhatmoredifficultthanaveragearemarkedwithasinglestar, ∗ ; thosethataremuchmorechallengingaremarkedwithtwostars, ∗∗ .Exerciseswhosesolutions requirecalculusareexplicitlynoted.Exercisesthatdevelopresultsusedinthetextareclearly identifiedwiththerightpointinghandsymbol, .Answersoroutlinedsolutionstoalloddnumberedexercisesareprovidedatthebackofthetext.Thesolutionsincludeproofsinwhich mostofthestepsareclearlyspelledout.
REVIEWQUESTIONS Asetofreviewquestionsisprovidedattheendofeachchapter. Thesequestionsaredesignedtohelpstudentsfocustheirstudyonthemostimportantconcepts andtechniquesofthatchapter.Toanswerthesequestionsstudentsneedtowritelonganswers, ratherthanjustperformcalculationsorgiveshortreplies.
SUPPLEMENTARYEXERCISESETS Eachchapterisfollowedbyarichandvaried setofsupplementaryexercises.Theseexercisesaregenerallymoredifficultthanthoseinthe exercisesetsfollowingthesections.Thesupplementaryexercisesreinforcetheconceptsofthe chapterandintegratedifferenttopicsmoreeffectively.
COMPUTERPROJECTS Eachchapterisfollowedbyasetofcomputerprojects.The approximately150computerprojectstietogetherwhatstudentsmayhavelearnedincomputing andindiscretemathematics.Computerprojectsthataremoredifficultthanaverage,fromboth amathematicalandaprogrammingpointofview,aremarkedwithastar,andthosethatare extremelychallengingaremarkedwithtwostars.
COMPUTATIONSANDEXPLORATIONS Asetofcomputationsandexplorationsis includedattheconclusionofeachchapter.Theseexercises(approximately120intotal)aredesignedtobecompletedusingexistingsoftwaretools,suchasprogramsthatstudentsorinstructorshavewrittenormathematicalcomputationpackagessuchasMapleTM orMathematicaTM . Manyoftheseexercisesgivestudentstheopportunitytouncovernewfactsandideasthrough computation.(Someoftheseexercisesarediscussedinthe ExploringDiscreteMathematics companionworkbooksavailableonline.)
WRITINGPROJECTS Eachchapterisfollowedbyasetofwritingprojects.Todothese projectsstudentsneedtoconsultthemathematicalliterature.Someoftheseprojectsarehistoricalinnatureandmayinvolvelookinguporiginalsources.Othersaredesignedtoserveas gatewaystonewtopicsandideas.Allaredesignedtoexposestudentstoideasnotcoveredin depthinthetext.Theseprojectstiemathematicalconceptstogetherwiththewritingprocessand
helpexposestudentstopossibleareasforfuturestudy.(Suggestedreferencesfortheseprojects canbefoundonlineorintheprinted Student’sSolutionsGuide.)
APPENDICES
Therearethreeappendicestothetext.Thefirstintroducesaxiomsforreal numbersandthepositiveintegers,andillustrateshowfactsareproveddirectlyfromtheseaxioms.Thesecondcoversexponentialandlogarithmicfunctions,reviewingsomebasicmaterial usedheavilyinthecourse.Thethirdspecifiesthepseudocodeusedtodescribealgorithmsin thistext.
SUGGESTEDREADINGS
Alistofsuggestedreadingsfortheoverallbookandforeach chapterisprovidedaftertheappendices.Thesesuggestedreadingsincludebooksatorbelow thelevelofthistext,moredifficultbooks,expositoryarticles,andarticlesinwhichdiscoveries indiscretemathematicswereoriginallypublished.Someofthesepublicationsareclassics,publishedmanyyearsago,whileothershavebeenpublishedinthelastfewyears.Thesesuggested readingsarecomplementedbythemanylinkstovaluableresourcesavailableonthewebthat canbefoundonthewebsiteforthisbook.
HowtoUseThisBook Thistexthasbeencarefullywrittenandconstructedtosupportdiscretemathematicscourses atseverallevelsandwithdifferingfoci.Thefollowingtableidentifiesthecoreandoptional sections.Anintroductoryone-termcourseindiscretemathematicsatthesophomorelevelcan bebasedonthecoresectionsofthetext,withothersectionscoveredatthediscretionofthe instructor.Atwo-termintroductorycoursecanincludealltheoptionalmathematicssectionsin additiontothecoresections.Acoursewithastrongcomputerscienceemphasiscanbetaught bycoveringsomeoralloftheoptionalcomputersciencesections.Instructorscanfindsample syllabiforawiderangeofdiscretemathematicscoursesandteachingsuggestionsforusingeach sectionofthetextcanbefoundinthe Instructor’sResourceGuide availableonthewebsitefor thisbook.
Instructorsusingthisbookcanadjustthelevelofdifficultyoftheircoursebychoosing eithertocoverortoomitthemorechallengingexamplesattheendofsections,aswellas themorechallengingexercises.Thechapterdependencychartshownheredisplaysthestrong dependencies.Astarindicatesthatonlyrelevantsectionsofthechapterareneededforstudy ofalaterchapter.Weakdependencieshavebeenignored.Moredetailscanbefoundinthe Instructor’sResourceGuide
Chapter 9* Chapter 10* Chapter 11
7
Ancillaries 1
2*
3*
4*
5*
6*
12
13
8
STUDENT’SSOLUTIONSGUIDE Thisstudentmanual,availableseparately,contains fullsolutionstoallodd-numberedexercisesintheexercisesets.Thesesolutionsexplainwhy aparticularmethodisusedandwhyitworks.Forsomeexercises,oneortwootherpossible approachesaredescribedtoshowthataproblemcanbesolvedinseveraldifferentways.Suggestedreferencesforthewritingprojectsfoundattheendofeachchapterarealsoincludedin thisvolume.Alsoincludedareaguidetowritingproofsandanextensivedescriptionofcommonmistakesstudentsmakeindiscretemathematics,plussampletestsandasamplecribsheet foreachchapterdesignedtohelpstudentsprepareforexams.
INSTRUCTOR’SRESOURCEGUIDE Thismanual,availableonthewebsiteandin printedformbyrequestforinstructors,containsfullsolutionstoeven-numberedexercisesin thetext.Suggestionsonhowtoteachthematerialineachchapterofthebookareprovided, includingthepointstostressineachsectionandhowtoputthematerialintoperspective.It alsoofferssampletestsforeachchapterandatestbankcontainingover1500examquestionsto choosefrom.Answerstoallsampletestsandtestbankquestionsareincluded.Finally,sample syllabiarepresentedforcourseswithdifferingemphasesandstudentabilitylevels.
Acknowledgments Iwouldliketothankthemanyinstructorsandstudentsatavarietyofschoolswhohaveused thisbookandprovidedmewiththeirvaluablefeedbackandhelpfulsuggestions.Theirinput hasmadethisamuchbetterbookthanitwouldhavebeenotherwise.Iespeciallywanttothank JerroldGrossmanandDanJordanfortheirtechnicalreviewsoftheeightheditionandtheir “eagleeyes,”whichhavehelpedensuretheaccuracyandqualityofthisbook.Bothhaveproofreadeverypartofthebookmanytimesasithasgonethroughthedifferentstepsofproduction andhavehelpedeliminateolderrataandpreventedtheinsertionofnewerrata.
ThanksgotoDanJordanforhisworkonthestudentsolutionsmanualandinstructor’s resourceguide.Hehasdoneanadmirablejobupdatingtheseancillaries.JerroldGrossman, theauthoroftheseancillariesforthefirstseveneditionsofthebook,hasprovidedvaluable assistancetoDan.Iwouldalsoliketoexpressmygratitudetothemanypeoplewhohavehelped createandmaintaintheonlineresourcesforthisbook.Inparticular,specialthanksgotoDan JordanandRochusBoernerfortheirextensiveworkimprovingonlinequestionsfortheConnect Site,describedlaterinthispreface.
Ithankthereviewersofthiseighthandallpreviouseditions.Thesereviewershaveprovided muchhelpfulcriticismandencouragementtome.Ihopethiseditionlivesuptotheirhigh expectations.Therehavebeenwellinexcessof200reviewersofthisbooksinceitsfirstedition, withmanyfromcountriesotherthantheUnitedStates.Themostrecentreviewersarelisted here.
RecentReviewers BarbaraAnthony SouthwesternUniversity
PhilipBarry UniversityofMinnesota,Minneapolis
BenkamBobga UniversityofNorthGeorgia
MiklosBona UniversityofFlorida
SteveBrick UniversityofSouthAlabama
KirbyBrown QueensCollege
JohnCarter UniversityofToronto
NarendraChaudhari NanyangTechnologicalUniversity
TimChappell PennValleyCommunityCollege
AllanCochran UniversityofArkansas
DanielCunningham BuffaloStateCollege
H.K.Dai
OklahomaStateUniversity
GeorgeDavis GeorgiaStateUniversity
AndrzejDerdzinski TheOhioStateUniversity
RonaldDotzel UniversityofMissouri-St.Louis
T.J.Duda ColumbusStateCommunityCollege
BruceElenbogen UniversityofMichigan,Dearborn
NormaElias PurdueUniversity, Calumet-Hammond
HerbertEnderton UniversityofCalifornia,LosAngeles
AnthonyEvans WrightStateUniversity
KimFactor MarquetteUniversity
MargaretFleck UniversityofIllinois,Champaign
MelissaGaddini
RobertMorrisUniversity
PeterGillespie FayettevilleStateUniversity
JohannesHattingh GeorgiaStateUniversity
JamesHelmreich MaristCollege
KenHolladay UniversityofNewOrleans
JerryIanni LaGuardiaCommunityCollege
MilagrosIzquierdo LinkopingUniversity
RaviJanardan UniversityofMinnesota,Minneapolis
NorlizaKatuk UniversityofUtaraMalaysia
MonikaKiss SaintLeoUniversity
WilliamKlostermeyer UniversityofNorthFlorida
PrzemoKranz UniversityofMississippi
JaromyKuhl UniversityofWestFlorida
LoredanaLanzani UniversityofArkansas,Fayetteville
FredericLatour CentralConnecticutStateUniversity
StevenLeonhardi WinonaStateUniversity
ChunleiLiu ValdostaStateUniversity
XuLiutong BeijingUniversityofPostsand Telecommunications
VladimirLogvinenko DeAnzaCommunityCollege
TamsenMcGinley SantaClaraUniversity
RamonA.Mata-Toledo JamesMadisonUniversity
TamaraMelnik ComputerSystemsInstitute
OsvaldoMendez UniversityofTexasatElPaso
DarrellMinor ColumbusStateCommunityCollege
KathleenO’Connor QuinsigamondCommunityCollege
KeithOlson UtahValleyUniversity
DimitrisPapamichail TheCollegeofNewJersey
YongyuthPermpoontanalarp KingMongkut’sUniversityof Technology,Thonburi
GalinPiatniskaia UniversityofMissouri,St.Louis
ShawonRahman UniversityofHawaiiatHilo
EricRawdon UniversityofSt.Thomas
StefanRobila MontclairStateUniversity
ChrisRodger AuburnUniversity
SukhitSingh TexasStateUniversity,SanMarcos
DavidSnyder TexasStateUniversity,SanMarcos
WasinSo SanJoseStateUniversity
BogdanSuceava CaliforniaStateUniversity,Fullerton
ChristopherSwanson AshlandUniversity
BonSy QueensCollege
FerejaTahir IllinoisCentralCollege
K.A.Venkatesh PresidencyUniversity
MatthewWalsh Indiana-PurdueUniversity,Fort Wayne
SheriWang UniversityofPhoenix
GideonWeinstein WesternGovernorsUniversity
DavidWilczynski UniversityofSouthernCalifornia
JamesWooland FloridaStateUniversity
Ialsowanttothankthemanystudentswhohaveprovidedsuggestionsandreportederrata. ThestudentsinthediscretemathematicscoursesIhavetaughtatMonmouthUniversity,both undergraduateandgraduatecomputersciencestudents,havehelpedmeimprovethebookin manyways.
TherearemanypeopletothankatMcGraw-HillHigherEducation,thepublisherofthis book,aswellaspeoplewhodidtheproductionworkatAptara.Iwouldalsoliketothankthe originaleditoratRandomHouse,WayneYuhasz,whoseinsightsandskillshelpedensurethe book’ssuccess,aswellasallthemanyotherpreviouseditorsofthisbook.
IwanttoexpressmydeepappreciationtoNoraDevlin,theProductDeveloperwhohas gonefarbeyondherassigneddutiestosupporttheauthor.Shehasdisplayedmanyskillsand virtuesworkingtosolveawidevarietyofproblemsthathaveariseninthedevelopmentofthis newedition.
IamalsogratefultoPeggySelle,theContentProductManager,whomanagedtheproductionprocess.Shehaskepttheproductionontrackandhashelpedresolvemanyissuesthathave ariseduringthisprocess.ThanksgotoSaritaYadav,SeniorProductManagerofAptara,and hercolleaguesatAptara,whoworkeddiligentlytoensuretheproductionqualityofthisedition.
IalsowanttoexpressmyappreciationtothemanyothersintheScience,Engineering,and Mathematics(SEM)DivisionofMcGraw-HillHigherEducationfortheirvaluablesupportfor thisneweditionandtheassociatedmediacontent,including
▶ MikeRyan,VP,Portfolio&LearningContent,HigherEducation
▶ KathleenMcMahon,ManagingDirector,Mathematics&PhysicalSciences
▶ CarolineCelano,Director,Mathematics
▶ AlisonFrederick,MarketingManager
▶ RobinReed,LeadProductDeveloper
▶ SandyLudovissey,Buyer
▶ EgzonShaqiri,Designer
▶ TammyJuran,AssessmentContentProjectManager
▶ CynthiaNorthrup,DirectorofDigitalContent
▶ RuthCzarnecki-Lichstein,BusinessProductManager
▶ MeganPlatt,EditorialCoordinator
▶ LoraNeyensandJolynnKilburg,ProgramManagers
▶ LorraineBuczek,ContentLicensingSpecialist
KennethH.Rosen
0.1 OnlineResources Extensiveefforthasbeendevotedtoproducingvaluablewebresourcesforthisbook.Instructorsshouldmakeaspecialefforttoexploretheseresourcestoidentifythosetheyfeel willhelptheirstudentslearnandexplorediscretemathematics.Theseresourcesareavailablein theOnlineLearningCenter,whichisavailabletoallstudentsandinstructors,andtheConnect Site,designedforinteractiveinstruction,whichinstructorscanchoosetouse.TouseConnect, studentspurchaseonlineaccessforaspecifictimeperiod.
TheOnlineLearningCenter TheOnlineLearningCenter(OLC),accessibleat www.mhhe.com/rosen,includesan InformationCenter,a StudentSite,andan InstructorSite.Keyfeaturesofeachareaaredescribedhere.
0.1.1TheInformationCenter TheInformationCentercontainsbasicinformationaboutthebookincludingtheexpandedtable ofcontents(includingsubsectionheads),thepreface,descriptionsoftheancillaries,andasamplechapter.Italsoprovidesalinkthatcanbeusedtosubmiterratareportsandotherfeedback aboutthebook.
0.1.2StudentSite TheStudentSitecontainsawealthofresourcesavailableforstudentuse,includingthefollowing,tiedintothetextwhereverthespecialiconsdisplayedbelowarefoundinthetext:
▶ ExtraExamples Youcanfindalargenumberofadditionalexamplesonthesite,cov- Extra Examples eringallchaptersofthebook.Theseexamplesareconcentratedinareaswherestudents oftenaskforadditionalmaterial.Althoughmostoftheseexamplesamplifythebasic concepts,more-challengingexamplescanalsobefoundhere.Manynewextraexamples havebeenrecentlyaddedfortheeighthedition.Eachiconinthebookcorrespondsto oneormoreextraexamplesonthewebsite.
▶ InteractiveDemonstrationApplets Theseappletsenableyoutointeractivelyexplore Demo howimportantalgorithmswork,andaretieddirectlytomaterialinthetextwithlinkages toexamplesandexercises.Additionalresourcesareprovidedonhowtouseandapply theseapplets.
▶ SelfAssessments Theseinteractiveguideshelpyouassessyourunderstandingof14 Assessment keyconcepts,providingaquestionbankwhereeachquestionincludesabrieftutorial followedbyamultiple-choicequestion.Ifyouselectanincorrectanswer,adviceisprovidedtohelpyouunderstandyourerror.UsingtheseSelfAssessments,youshouldbe abletodiagnoseyourproblemsandfindappropriatehelp.
▶ WebResourcesGuide Thisguideprovidesannotatedlinkstohundredsofexternal Links websitescontainingrelevantmaterialsuchashistoricalandbiographicalinformation, puzzlesandproblems,discussions,applets,programs,andmore.Theselinksarekeyed tothetextbypagenumber.
AdditionalresourcesintheStudentSiteinclude:
▶ ExploringDiscreteMathematics Thisancillaryprovideshelpforusingacomputeralgebrasystemtodoawiderangeofcomputationsindiscretemathematics.Eachchapterprovides adescriptionofrelevantfunctionsinthecomputeralgebrasystemandhowtheyareused,programstocarryoutcomputationsindiscretemathematics,examples,andexercisesthatcanbe workedusingthiscomputeralgebrasystem.Twoversions, ExploringDiscreteMathematics withMapleTM and ExploringDiscreteMathematicswithMathematicaTM ,areavailable.
▶ ApplicationsofDiscreteMathematics Thisancillarycontains24chapters—eachwith itsownsetofexercises—presentingawidevarietyofinterestingandimportantapplications coveringthreegeneralareasindiscretemathematics:discretestructures,combinatorics,and graphtheory.Theseapplicationsareidealforsupplementingthetextorforindependentstudy.
▶ AGuidetoProof-Writing Thisguideprovidesadditionalhelpforwritingproofs,askill thatmanystudentsfinddifficulttomaster.Byreadingthisguideatthebeginningofthe courseandperiodicallythereafterwhenproofwritingisrequired,youwillberewardedas yourproof-writingabilitygrows.(Alsoavailableinthe Student’sSolutionsGuide.)
▶ CommonMistakesinDiscreteMathematics Thisguideincludesadetailedlistofcommonmisconceptionsthatstudentsofdiscretemathematicsoftenhaveandthekindsoferrors theytendtomake.Youareencouragedtoreviewthislistfromtimetotimetohelpavoid thesecommontraps.(Alsoavailableinthe Student’sSolutionsGuide.)
▶ AdviceonWritingProjects ThisguideoffershelpfulhintsandsuggestionsfortheWriting Projectsinthetext,includinganextensivebibliographyofhelpfulbooksandarticlesfor research,discussionofvariousresourcesavailableinprintandonline,tipsondoinglibrary research,andsuggestionsonhowtowritewell.(Alsoavailableinthe Student’sSolutions Guide.)
0.1.3InstructorSite ThispartofthewebsiteprovidesaccesstoalloftheresourcesontheStudentSite,aswellas theseresourcesforinstructors:
▶ SuggestedSyllabi Detailedcourseoutlinesareshown,offeringsuggestionsfor courseswithdifferentemphasesanddifferentstudentbackgroundsandabilitylevels.
▶ TeachingSuggestions Thisguidecontainsdetailedteachingsuggestionsforinstructors,includingchapteroverviewsfortheentiretext,detailedremarksoneachsection, andcommentsontheexercisesets.
▶ PrintableTests PrintabletestsareofferedinTeXandWordformatforeverychapter, andcanbecustomizedbyinstructors.
▶ PowerPointLectureSlidesandPowerPointFiguresandTables AnextensivecollectionofPowerPointlecturenotesforallchaptersofthetextareprovidedforinstructor use.Inaddition,imagesofallfiguresandtablesfromthetextareprovidedasPowerPoint slides.
0.1.4Connect Acomprehensiveonlinelearningpackagehasbeendevelopedinconjunctionwiththetext.A high-leveldescriptionofthissitewillbeprovidedhere.Interestedinstructorsandstudentscan findoutmoreaboutConnectfromMcGraw-HillHigherEducation.Wheninstructorschooseto usethisoption,studentsintheirclassesmustobtainaccesstoConnectforthistext,eitherby purchasingacopyofthetextbookthatalsoincludesaccessprivilegesorbypurchasingaccess onlywiththeoptionofbuyingaloose-leafversionofthetextbook.
InstructorswhoadoptConnecthaveaccesstoafull-featuredcoursemanagementsystem. Coursemanagementcapabilitiesareprovidedthatallowinstructorstocreateassignments,automaticallyassignandgradehomework,quiz,andtestquestionsfromabankofquestionstied directlytothetext,createandedittheirownquestions,managecourseannouncementsanddue dates,andtrackstudentprogress.
InstructorscancreatetheirownassignmentsusingConnect.Theyselecttheparticularexercisesfromeachsectionofthebookthattheywanttoassign.Theycanalsoassignchapters fromtheSmartBookversionofthetext,whichprovidesanadaptivelearningenvironment.Studentshaveaccesstotheinteractiveversionofthetextbook,theonlinehomeworkexercises,and SmartBook.
InteractiveTextbook Studentshaveaccesstoaneasy-to-useinteractiveversionofthetextbook whentheyuseConnect.Theinteractivesiteprovidesthefullcontentofthetext,aswellasthe manyextraresourcesthatenrichthebook.Theresourcesincludeextraexamples,interactive demonstrations,andself-assessments.
HomeworkandLearningManagementSolution Anextensivelearningmanagementsolution hasbeendevelopedthatinstructorscanusetoconstructhomeworkassignments.Approximately 800onlinequestionsareavailable,includingquestionsfromeverysectionofthetext.These questionsaretiedtothemostcommonlyassignedexercisesinthebook.
Theseonlinequestionshavebeenconstructedtosupportthesameobjectivesasthecorrespondingwrittenhomeworkquestions.ThischallengehasbeenmetbystretchingthecapabilitiesofdifferentquestiontypessupportedbytheConnectplatform.
SmartBook ConnectalsoprovidesanotherextendedonlineversionofthetextintheMcGrawHillSmartBookplatform.TheSmartBookversionofthetextincludesasetofobjectivesforeach chapterofthetext.Acollectionofquestions,calledprobes,isprovidedtoassessstudentunderstandingofeachobjective.Studentsaredirectedtotheappropriatepartofthetexttoreview thematerialneededforeachoftheseobjectives.SmartBookprovidesanadaptivelearningenvironment;itselectsprobesforstudentsbasedontheirperformanceansweringearlierprobes. InstructorscanassignSmartBookasassignmentsorcanhavetheirstudentsuseSmartBookas alearningtool.
TotheStudent Whatisdiscretemathematics? Discretemathematicsisthepartofmathematicsdevoted tothestudyofdiscreteobjects.(Here discrete meansconsistingofdistinctorunconnectedelements.)Thekindsofproblemssolvedusingdiscretemathematicsinclude:
▶ Howmanywaysaretheretochooseavalidpasswordonacomputersystem?
▶ Whatistheprobabilityofwinningalottery?
▶ Istherealinkbetweentwocomputersinanetwork?
▶ HowcanIidentifyspame-mailmessages?
▶ HowcanIencryptamessagesothatnounintendedrecipientcanreadit?
▶ Whatistheshortestpathbetweentwocitiesusingatransportationsystem?
▶ Howcanalistofintegersbesortedsothattheintegersareinincreasingorder?
▶ Howmanystepsarerequiredtodosuchasorting?
▶ Howcanitbeprovedthatasortingalgorithmcorrectlysortsalist?
▶ Howcanacircuitthataddstwointegersbedesigned?
▶ HowmanyvalidInternetaddressesarethere?
Youwilllearnthediscretestructuresandtechniquesneededtosolveproblemssuchasthese.
Moregenerally,discretemathematicsisusedwheneverobjectsarecounted,whenrelationshipsbetweenfinite(orcountable)setsarestudied,andwhenprocessesinvolvingafinitenumberofstepsareanalyzed.Akeyreasonforthegrowthintheimportanceofdiscretemathematics isthatinformationisstoredandmanipulatedbycomputingmachinesinadiscretefashion.
WHYSTUDYDISCRETEMATHEMATICS? Thereareseveralimportantreasonsfor studyingdiscretemathematics.First,throughthiscourseyoucandevelopyourmathematical maturity:thatis,yourabilitytounderstandandcreatemathematicalarguments.Youwillnot getveryfarinyourstudiesinthemathematicalscienceswithouttheseskills.
Second,discretemathematicsisthegatewaytomoreadvancedcoursesinallpartsofthe mathematicalsciences.Discretemathematicsprovidesthemathematicalfoundationsformany computersciencecourses,includingdatastructures,algorithms,databasetheory,automatatheory,formallanguages,compilertheory,computersecurity,andoperatingsystems.Studentsfind thesecoursesmuchmoredifficultwhentheyhavenothadtheappropriatemathematicalfoundationsfromdiscretemathematics.Onestudentsentmeane-mailmessagesayingthatsheused thecontentsofthisbookineverycomputersciencecourseshetook!
Mathcoursesbasedonthematerialstudiedindiscretemathematicsincludelogic,settheory, numbertheory,linearalgebra,abstractalgebra,combinatorics,graphtheory,andprobability theory(thediscretepartofthesubject).
Also,discretemathematicscontainsthenecessarymathematicalbackgroundforsolvingproblemsinoperationsresearch(includingdiscreteoptimization),chemistry,engineering, biology,andsoon.Inthetext,wewillstudyapplicationstosomeoftheseareas.
Manystudentsfindtheirintroductorydiscretemathematicscoursetobesignificantlymore challengingthancoursestheyhavepreviouslytaken.Onereasonforthisisthatoneoftheprimarygoalsofthiscourseistoteachmathematicalreasoningandproblemsolving,ratherthana discretesetofskills.Theexercisesinthisbookaredesignedtoreflectthisgoal.Althoughthere areplentyofexercisesinthistextsimilartothoseaddressedintheexamples,alargepercentage
oftheexercisesrequireoriginalthought.Thisisintentional.Thematerialdiscussedinthetext providesthetoolsneededtosolvetheseexercises,butyourjobistosuccessfullyapplythese toolsusingyourowncreativity.Oneoftheprimarygoalsofthiscourseistolearnhowtoattack problemsthatmaybesomewhatdifferentfromanyyoumayhavepreviouslyseen.Unfortunately,learninghowtosolveonlyparticulartypesofexercisesisnotsufficientforsuccessin developingtheproblem-solvingskillsneededinsubsequentcoursesandprofessionalwork.This textaddressesmanydifferenttopics,butdiscretemathematicsisanextremelydiverseandlarge areaofstudy.Oneofmygoalsasanauthoristohelpyoudeveloptheskillsneededtomaster theadditionalmaterialyouwillneedinyourownfuturepursuits.
Finally,discretemathematicsisanexcellentenvironmentinwhichtolearnhowtoreadand writemathematicalproofs.InadditiontoexplicitmaterialonproofsinChapter1andChapter5, thistextbookcontainsthroughoutmanyproofsoftheoremsandmanyexercisesaskingthestudenttoprovestatements.Thisnotonlydeepensone’sunderstandingofthesubjectmatterbutis alsovaluablepreparationformoreadvancedcoursesinmathematicsandtheoreticalcomputer science.
THEEXERCISES Iwouldliketooffersomeadviceabouthowyoucanbestlearndiscrete mathematics(andothersubjectsinthemathematicalandcomputingsciences).Youwilllearn themostbyactivelyworkingexercises.Isuggestthatyousolveasmanyasyoupossiblycan. Afterworkingtheexercisesyourinstructorhasassigned,Iencourageyoutosolveadditional exercisessuchasthoseintheexercisesetsfollowingeachsectionofthetextandinthesupplementaryexercisesattheendofeachchapter.(Notethekeyexplainingthemarkingspreceding exercises.)
KeytotheExercises nomarkingAroutineexercise ∗ Adifficultexercise
∗∗ Anextremelychallengingexercise
Anexercisecontainingaresultusedinthebook(Table1onthe followingpageshowswheretheseexercisesareused.)
(Requirescalculus)Anexercisewhosesolutionrequirestheuseoflimitsorconcepts fromdifferentialorintegralcalculus
Thebestapproachistotryexercisesyourselfbeforeyouconsulttheanswersectionatthe endofthisbook.Notethattheodd-numberedexerciseanswersprovidedinthetextareanswers onlyandnotfullsolutions;inparticular,thereasoningrequiredtoobtainanswersisomittedin theseanswers.The Student’sSolutionsGuide,availableseparately,providescomplete,worked solutionstoallodd-numberedexercisesinthistext.Whenyouhitanimpassetryingtosolvean odd-numberedexercise,Isuggestyouconsultthe Student’sSolutionsGuide andlookforsome guidanceastohowtosolvetheproblem.Themoreworkyoudoyourselfratherthanpassively readingorcopyingsolutions,themoreyouwilllearn.Theanswersandsolutionstotheevennumberedexercisesareintentionallynotavailablefromthepublisher;askyourinstructorifyou havetroublewiththese.
WEBRESOURCES AllusersofthebookareabletoaccesstheonlineresourcesaccessibleviatheOnlineLearningCenter(OLC)forthebook.YouwillfindmanyExtraExamplesdesignedtoclarifykeyconcepts,SelfAssessmentsforgauginghowwellyouunderstand coretopics,InteractiveDemonstrationsthatexplorekeyalgorithmsandotherconcepts,aWeb ResourcesGuidecontaininganextensiveselectionoflinkstoexternalsitesrelevanttothe worldofdiscretemathematics,extraexplanationsandpracticetohelpyoumastercoreconcepts,addedinstructiononwritingproofsandonavoidingcommonmistakesindiscretemathematics,in-depthdiscussionsofimportantapplications,andguidanceonutilizingMapleTM and
MathematicaTM softwaretoexplorethecomputationalaspectsofdiscretemathematics.Places inthetextwheretheseadditionalonlineresourcesareavailableareidentifiedinthemarginsby specialicons.Formoredetailsontheseandotheronlineresources,seethedescriptionofthe companionwebsiteimmediatelyprecedingthis“TotheStudent”message.
THEVALUEOFTHISBOOK Myintentionistomakeyoursubstantialinvestmentinthis textanexcellentvalue.Thebook,theassociatedancillaries,andcompanionwebsitehavetaken manyyearsofefforttodevelopandrefine.Iamconfidentthatmostofyouwillfindthatthetext andassociatedmaterialswillhelpyoumasterdiscretemathematics,justassomanyprevious studentshave.Eventhoughitislikelythatyouwillnotcoversomechaptersinyourcurrent course,youshouldfindithelpful—asmanyotherstudentshave—toreadtherelevantsections ofthebookasyoutakeadditionalcourses.Mostofyouwillreturntothisbookasausefultool throughoutyourfuturestudies,especiallyforthoseofyouwhocontinueincomputerscience, mathematics,andengineering.Ihavedesignedthisbooktobeagatewayforfuturestudiesand explorations,andtobecomprehensivereference,andIwishyouluckasyoubeginyourjourney.
KennethH.Rosen
1.1 TheFoundations: LogicandProofs Therulesoflogicspecifythemeaningofmathematicalstatements.Forinstance,these ruleshelpusunderstandandreasonwithstatementssuchas“Thereexistsaninteger thatisnotthesumoftwosquares”and“Foreverypositiveinteger n,thesumofthepositiveintegersnotexceeding n is n(n + 1)∕2.”Logicisthebasisofallmathematicalreasoning, andofallautomatedreasoning.Ithaspracticalapplicationstothedesignofcomputingmachines,tothespecificationofsystems,toartificialintelligence,tocomputerprogramming,to programminglanguages,andtootherareasofcomputerscience,aswellastomanyotherfields ofstudy.
Tounderstandmathematics,wemustunderstandwhatmakesupacorrectmathematical argument,thatis,aproof.Onceweproveamathematicalstatementistrue,wecallitatheorem. Acollectionoftheoremsonatopicorganizewhatweknowaboutthistopic.Tolearnamathematicaltopic,apersonneedstoactivelyconstructmathematicalargumentsonthistopic,and notjustreadexposition.Moreover,knowingtheproofofatheoremoftenmakesitpossibleto modifytheresulttofitnewsituations.
Everyoneknowsthatproofsareimportantthroughoutmathematics,butmanypeoplefind itsurprisinghowimportantproofsareincomputerscience.Infact,proofsareusedtoverify thatcomputerprogramsproducethecorrectoutputforallpossibleinputvalues,toshowthat algorithmsalwaysproducethecorrectresult,toestablishthesecurityofasystem,andtocreate artificialintelligence.Furthermore,automatedreasoningsystemshavebeencreatedtoallow computerstoconstructtheirownproofs.
Inthischapter,wewillexplainwhatmakesupacorrectmathematicalargumentandintroducetoolstoconstructthesearguments.Wewilldevelopanarsenalofdifferentproofmethods thatwillenableustoprovemanydifferenttypesofresults.Afterintroducingmanydifferent methodsofproof,wewillintroduceseveralstrategiesforconstructingproofs.Wewillintroducethenotionofaconjectureandexplaintheprocessofdevelopingmathematicsbystudying conjectures.
PropositionalLogic 1.1.1Introduction
Therulesoflogicgiveprecisemeaningtomathematicalstatements.Theserulesareusedtodistinguishbetweenvalidandinvalidmathematicalarguments.Becauseamajorgoalofthisbook istoteachthereaderhowtounderstandandhowtoconstructcorrectmathematicalarguments, webeginourstudyofdiscretemathematicswithanintroductiontologic.
Besidestheimportanceoflogicinunderstandingmathematicalreasoning,logichasnumerousapplicationstocomputerscience.Theserulesareusedinthedesignofcomputercircuits, theconstructionofcomputerprograms,theverificationofthecorrectnessofprograms,andin manyotherways.Furthermore,softwaresystemshavebeendevelopedforconstructingsome, butnotall,typesofproofsautomatically.Wewilldiscusstheseapplicationsoflogicinthisand laterchapters.
1.1.2Propositions Ourdiscussionbeginswithanintroductiontothebasicbuildingblocksoflogic—propositions. A proposition isadeclarativesentence(thatis,asentencethatdeclaresafact)thatiseither trueorfalse,butnotboth.
EXAMPLE1 Allthefollowingdeclarativesentencesarepropositions.
1.Washington,D.C.,isthecapitaloftheUnitedStatesofAmerica. Extra Examples 2.TorontoisthecapitalofCanada.
3.1 + 1 = 2.
4.2 + 2 = 3.
Propositions1and3aretrue,whereas2and4arefalse.
SomesentencesthatarenotpropositionsaregiveninExample2.
EXAMPLE2 Considerthefollowingsentences.
1.Whattimeisit?
2.Readthiscarefully.
3. x + 1 = 2.
4. x + y = z.
Sentences1and2arenotpropositionsbecausetheyarenotdeclarativesentences.Sentences3 and4arenotpropositionsbecausetheyareneithertruenorfalse.Notethateachofsentences3 and4canbeturnedintoapropositionifweassignvaluestothevariables.Wewillalsodiscuss otherwaystoturnsentencessuchastheseintopropositionsinSection1.4. ◂
Weuseletterstodenote propositionalvariables (or sententialvariables),thatis,variables thatrepresentpropositions,justaslettersareusedtodenotenumericalvariables.Theconventionallettersusedforpropositionalvariablesare p,q,r,s, … . The truthvalue ofaproposition
Source:NationalLibraryof Medicine
ARISTOTLE(384 B C E.–322 B C E.) AristotlewasborninStagirus(Stagira)innorthernGreece.Hisfather
Links wasthepersonalphysicianoftheKingofMacedonia.BecausehisfatherdiedwhenAristotlewasyoung, Aristotlecouldnotfollowthecustomoffollowinghisfather’sprofession.Aristotlebecameanorphanata youngagewhenhismotheralsodied.Hisguardianwhoraisedhimtaughthimpoetry,rhetoric,andGreek. Attheageof17,hisguardiansenthimtoAthenstofurtherhiseducation.AristotlejoinedPlato’sAcademy, wherefor20yearsheattendedPlato’slectures,laterpresentinghisownlecturesonrhetoric.WhenPlatodiedin 347 B.C.E.,AristotlewasnotchosentosucceedhimbecausehisviewsdifferedtoomuchfromthoseofPlato. Instead,AristotlejoinedthecourtofKingHermeaswhereheremainedforthreeyears,andmarriedtheniece oftheKing.WhenthePersiansdefeatedHermeas,AristotlemovedtoMytileneand,attheinvitationofKing PhilipofMacedonia,hetutoredAlexander,Philip’sson,wholaterbecameAlexandertheGreat.Aristotletutored AlexanderforfiveyearsandafterthedeathofKingPhilip,hereturnedtoAthensandsetuphisownschool, calledtheLyceum.
Aristotle’sfollowerswerecalledtheperipatetics,whichmeans“towalkabout,”becauseAristotleoftenwalkedaroundashe discussedphilosophicalquestions.AristotletaughtattheLyceumfor13yearswherehelecturedtohisadvancedstudentsinthe morningandgavepopularlecturestoabroadaudienceintheevening.WhenAlexandertheGreatdiedin323 B.C.E.,abacklashagainst anythingrelatedtoAlexanderledtotrumped-upchargesofimpietyagainstAristotle.AristotlefledtoChalcistoavoidprosecution. HeonlylivedoneyearinChalcis,dyingofastomachailmentin322 B.C.E.
Aristotlewrotethreetypesofworks:thosewrittenforapopularaudience,compilationsofscientificfacts,andsystematic treatises.Thesystematictreatisesincludedworksonlogic,philosophy,psychology,physics,andnaturalhistory.Aristotle’swritings werepreservedbyastudentandwerehiddeninavaultwhereawealthybookcollectordiscoveredthemabout200yearslater.They weretakentoRome,wheretheywerestudiedbyscholarsandissuedinneweditions,preservingthemforposterity.
istrue,denotedbyT,ifitisatrueproposition,andthetruthvalueofapropositionisfalse,denotedbyF,ifitisafalseproposition.Propositionsthatcannotbeexpressedintermsofsimpler propositionsarecalled atomicpropositions
Theareaoflogicthatdealswithpropositionsiscalledthe propositionalcalculus or propositionallogic.ItwasfirstdevelopedsystematicallybytheGreekphilosopherAristotlemorethan 2300yearsago.
Wenowturnourattentiontomethodsforproducingnewpropositionsfromthosethatwe Links alreadyhave.ThesemethodswerediscussedbytheEnglishmathematicianGeorgeBoolein 1854inhisbook TheLawsofThought. Manymathematicalstatementsareconstructedbycombiningoneormorepropositions.Newpropositions,called compoundpropositions,areformed fromexistingpropositionsusing logicaloperators.
Definition1 Let p beaproposition.The negationofp,denotedby ¬p (alsodenotedby p),isthestatement
“Itisnotthecasethat p.”
Theproposition ¬p isread“not p.”Thetruthvalueofthenegationof p, ¬p,istheopposite ofthetruthvalueof p.
Remark: Thenotationforthenegationoperatorisnotstandardized.Although ¬p and p arethe mostcommonnotationsusedinmathematicstoexpressthenegationof p,othernotationsyou mightseeare ∼p, p, p′ ,Np,and !p
EXAMPLE3 Findthenegationoftheproposition
“Michael’sPCrunsLinux” andexpressthisinsimpleEnglish. Extra Examples
Solution: Thenegationis
“ItisnotthecasethatMichael’sPCrunsLinux.”
Thisnegationcanbemoresimplyexpressedas
“Michael’sPCdoesnotrunLinux.”
EXAMPLE4 Findthenegationoftheproposition
“Vandana’ssmartphonehasatleast32GBofmemory” andexpressthisinsimpleEnglish.
Solution: Thenegationis
“ItisnotthecasethatVandana’ssmartphonehasatleast32GBofmemory.”
Thisnegationcanalsobeexpressedas
“Vandana’ssmartphonedoesnothaveatleast32GBofmemory” orevenmoresimplyas
“Vandana’ssmartphonehaslessthan32GBofmemory.”
TABLE1 The TruthTablefor theNegationofa Proposition.
Table1displaysthe truthtable forthenegationofaproposition p.Thistablehasarowfor eachofthetwopossibletruthvaluesof p.Eachrowshowsthetruthvalueof ¬p corresponding tothetruthvalueof p forthisrow.
Thenegationofapropositioncanalsobeconsideredtheresultoftheoperationofthe negationoperator onaproposition.Thenegationoperatorconstructsanewpropositionfrom asingleexistingproposition.Wewillnowintroducethelogicaloperatorsthatareusedtoform newpropositionsfromtwoormoreexistingpropositions.Theselogicaloperatorsarealsocalled connectives
Definition2 Let p and q bepropositions.The conjunction of p and q,denotedby p ∧ q,istheproposition “ p and q.”Theconjunction p ∧ q istruewhenboth p and q aretrueandisfalseotherwise.
Table2displaysthetruthtableof p ∧ q.Thistablehasarowforeachofthefourpossible combinationsoftruthvaluesof p and q.Thefourrowscorrespondtothepairsoftruthvalues TT,TF,FT,andFF,wherethefirsttruthvalueinthepairisthetruthvalueof p andthesecond truthvalueisthetruthvalueof q.
Notethatinlogictheword“but”sometimesisusedinsteadof“and”inaconjunction.For example,thestatement“Thesunisshining,butitisraining”isanotherwayofsaying“Thesun isshininganditisraining.”(Innaturallanguage,thereisasubtledifferenceinmeaningbetween “and”and“but”;wewillnotbeconcernedwiththisnuancehere.)
EXAMPLE5
Definition3
Findtheconjunctionofthepropositions p and q where p istheproposition“Rebecca’sPChas morethan16GBfreeharddiskspace”and q istheproposition“TheprocessorinRebecca’sPC runsfasterthan1GHz.”
Solution: Theconjunctionofthesepropositions, p ∧ q,istheproposition“Rebecca’sPChas morethan16GBfreeharddiskspace,andtheprocessorinRebecca’sPCrunsfasterthan 1GHz.”Thisconjunctioncanbeexpressedmoresimplyas“Rebecca’sPChasmorethan 16GBfreeharddiskspace,anditsprocessorrunsfasterthan1GHz.”Forthisconjunction tobetrue,bothconditionsgivenmustbetrue.Itisfalsewhenoneorbothoftheseconditions arefalse. ◂
Let p and q bepropositions.The disjunction of p and q,denotedby p ∨ q,istheproposition “ p or q.”Thedisjunction p ∨ q isfalsewhenboth p and q arefalseandistrueotherwise.
Table3displaysthetruthtablefor p ∨ q.
TABLE2 TheTruthTablefor theConjunctionofTwo Propositions.
TABLE3 TheTruthTablefor theDisjunctionofTwo Propositions.
EXAMPLE6
Theuseoftheconnective or inadisjunctioncorrespondstooneofthetwowaystheword or isusedinEnglish,namely,asan inclusiveor.Adisjunctionistruewhenatleastoneofthe twopropositionsistrue.Thatis, p ∨ q istruewhenboth p and q aretrueorwhenexactlyone of p and q istrue.
Translatethestatement“Studentswhohavetakencalculusorintroductorycomputersciencecan takethisclass”inastatementinpropositionallogicusingthepropositions p:“Astudentwho hastakencalculuscantakethisclass”and q:“Astudentwhohastakenintroductorycomputer sciencecantakethisclass.”
Solution: Weassumethatthisstatementmeansthatstudentswhohavetakenbothcalculusand introductorycomputersciencecantaketheclass,aswellasthestudentswhohavetakenonly oneofthetwosubjects.Hence,thisstatementcanbeexpressedas p ∨ q,theinclusiveor,or disjunction,of p and q. ◂
EXAMPLE7 Whatisthedisjunctionofthepropositions p and q,where p and q arethesamepropositionsas inExample5?
Solution: Thedisjunctionof p and q, p ∨ q,istheproposition
Extra Examples “Rebecca’sPChasatleast16GBfreeharddiskspace,ortheprocessorinRebecca’sPC runsfasterthan1GHz.”
ThispropositionistruewhenRebecca’sPChasatleast16GBfreeharddiskspace,whenthe PC’sprocessorrunsfasterthan1GHz,andwhenbothconditionsaretrue.Itisfalsewhenboth oftheseconditionsarefalse,thatis,whenRebecca’sPChaslessthan16GBfreeharddisk spaceandtheprocessorinherPCrunsat1GHzorslower.
Besidesitsuseindisjunctions,theconnective or isalsousedtoexpressan exclusiveor Unlikethedisjunctionoftwopropositions p and q,theexclusiveorofthesetwopropositionsis truewhenexactlyoneof p and q istrue;itisfalsewhenboth p and q aretrue(andwhenboth arefalse).
Definition4 Let p and q bepropositions.The exclusiveor of p and q,denotedby p ⊕ q (or p XOR q),is thepropositionthatistruewhenexactlyoneof p and q istrueandisfalseotherwise.
GEORGEBOOLE(1815–1864) GeorgeBoole,thesonofacobbler,wasborninLincoln,England,in Links November1815.Becauseofhisfamily’sdifficultfinancialsituation,Boolestruggledtoeducatehimselfwhile supportinghisfamily.Nevertheless,hebecameoneofthemostimportantmathematiciansofthe1800s.Althoughheconsideredacareerasaclergyman,hedecidedinsteadtogointoteaching,andsoonafterward openedaschoolofhisown.Inhispreparationforteachingmathematics,Boole—unsatisfiedwithtextbooks ofhisday—decidedtoreadtheworksofthegreatmathematicians.WhilereadingpapersofthegreatFrench mathematicianLagrange,Boolemadediscoveriesinthecalculusofvariations,thebranchofanalysisdealing withfindingcurvesandsurfacesbyoptimizingcertainparameters.
Source:LibraryofCongress Washington,D.C.20540 USA[LC-USZ62-61664]
In1848Boolepublished TheMathematicalAnalysisofLogic,thefirstofhiscontributionsto symboliclogic.In1849hewasappointedprofessorofmathematicsatQueen’sCollegeinCork, Ireland.In1854hepublished TheLawsofThought,hismostfamouswork.Inthisbook,Boole introducedwhatisnowcalled Booleanalgebra inhishonor.Boolewrotetextbooksondifferentialequationsandondifference equationsthatwereusedinGreatBritainuntiltheendofthenineteenthcentury.Boolemarriedin1855;hiswifewasthenieceof theprofessorofGreekatQueen’sCollege.In1864Boolediedfrompneumonia,whichhecontractedasaresultofkeepingalecture engagementeventhoughhewassoakingwetfromarainstorm.
EXAMPLE8
ThetruthtablefortheexclusiveoroftwopropositionsisdisplayedinTable4.
Let p and q bethepropositionsthatstate“Astudentcanhaveasaladwithdinner”and“Astudent canhavesoupwithdinner,”respectively.Whatis p ⊕ q,theexclusiveorof p and q?
Solution: Theexclusiveorof p and q isthestatementthatistruewhenexactlyoneof p and q istrue.Thatis, p ⊕ q isthestatement“Astudentcanhavesouporsalad,butnotboth,with dinner.”Notethatthisisoftenstatedas“Astudentcanhavesouporasaladwithdinner,”without explicitlystatingthattakingbothisnotpermitted. ◂
EXAMPLE9 Expressthestatement“IwilluseallmysavingstotraveltoEuropeortobuyanelectriccar”in propositionallogicusingthestatement p:“IwilluseallmysavingstotraveltoEurope”andthe statement q:“Iwilluseallmysavingstobuyanelectriccar.”
Solution: Totranslatethisstatement,wefirstnotethattheorinthisstatementmustbeanexclusiveorbecausethisstudentcaneitheruseallhisorhersavingstotraveltoEuropeoruseall thesesavingstobuyanelectriccar,butcannotbothgotoEuropeandbuyanelectriccar.(This isclearbecauseeitheroptionrequiresallhissavings.)Hence,thisstatementcanbeexpressed as p ⊕ q
1.1.3ConditionalStatements Wewilldiscussseveralotherimportantwaysinwhichpropositionscanbecombined.
Definition5 Let p and q bepropositions.The conditionalstatementp → q istheproposition“if p,then q.”Theconditionalstatement p → q isfalsewhen p istrueand q isfalse,andtrueotherwise. Intheconditionalstatement p → q, p iscalledthe hypothesis (or antecedent or premise)and q iscalledthe conclusion (or consequence).
Thestatement p → q iscalledaconditionalstatementbecause p → q assertsthat q istrue Assessment ontheconditionthat p holds.Aconditionalstatementisalsocalledan implication. Thetruthtablefortheconditionalstatement p → q isshowninTable5.Notethatthestatement p → q istruewhenboth p and q aretrueandwhen p isfalse(nomatterwhattruthvalue q has).
TABLE4 TheTruthTablefor theExclusiveOrofTwo Propositions.
TABLE5 TheTruthTablefor theConditionalStatement p → q.
Youmighthavetrouble understandinghow “unless”isusedin conditionalstatements unlessyoureadthis paragraphcarefully.
Becauseconditionalstatementsplaysuchanessentialroleinmathematicalreasoning,avarietyofterminologyisusedtoexpress p → q.Youwillencountermostifnotallofthefollowing waystoexpressthisconditionalstatement:
“if p,then q ”“ p implies q ” “if p, q ”“ p onlyif q ” “ p issufficientfor q ” “asufficientconditionfor q is p ” “ q if p ”“ q whenever p ” “ q when p ”“ q isnecessaryfor p ” “anecessaryconditionfor p is q ”“ q followsfrom p ” “ q unless ¬p ”“ q providedthat p ”
Ausefulwaytounderstandthetruthvalueofaconditionalstatementistothinkofanobligationoracontract.Forexample,thepledgemanypoliticiansmakewhenrunningforofficeis
“IfIamelected,thenIwilllowertaxes.” Ifthepoliticianiselected,voterswouldexpectthispoliticiantolowertaxes.Furthermore,if thepoliticianisnotelected,thenvoterswillnothaveanyexpectationthatthispersonwilllower taxes,althoughthepersonmayhavesufficientinfluencetocausethoseinpowertolowertaxes. Itisonlywhenthepoliticianiselectedbutdoesnotlowertaxesthatvoterscansaythatthe politicianhasbrokenthecampaignpledge.Thislastscenariocorrespondstothecasewhen p is truebut q isfalsein p → q.
Similarly,considerastatementthataprofessormightmake:
“Ifyouget100%onthefinal,thenyouwillgetanA.” Ifyoumanagetoget100%onthefinal,thenyouwouldexpecttoreceiveanA.Ifyoudonot get100%,youmayormaynotreceiveanAdependingonotherfactors.However,ifyoudoget 100%,buttheprofessordoesnotgiveyouanA,youwillfeelcheated.
Remark: Becausesomeofthedifferentwaystoexpresstheimplication p implies q canbe confusing,wewillprovidesomeextraguidance.Torememberthat“p onlyif q”expressesthe samethingas“if p,then q,”notethat“p onlyif q”saysthat p cannotbetruewhen q isnottrue. Thatis,thestatementisfalseif p istrue,but q isfalse.When p isfalse, q maybeeithertrueor false,becausethestatementsaysnothingaboutthetruthvalueof q.
Forexample,supposeyourprofessortellsyou
“YoucanreceiveanAinthecourseonlyifyourscoreonthefinalisatleast90%.”
Then,ifyoureceiveanAinthecourse,thenyouknowthatyourscoreonthefinalisat least90%.IfyoudonotreceiveanA,youmayormaynothavescoredatleast90%onthe final.Becarefulnottouse“q onlyif p”toexpress p → q becausethisisincorrect.Theword “only”playsanessentialrolehere.Toseethis,notethatthetruthvaluesof“q onlyif p”and p → q aredifferentwhen p and q havedifferenttruthvalues.Toseewhy“q isnecessaryfor p ” isequivalentto“if p,then q,”observethat“q isnecessaryfor p”meansthat p cannotbetrue unless q istrue,orthatif q isfalse,then p isfalse.Thisisthesameassayingthatif p istrue, then q istrue.Toseewhy“p issufficientfor q”isequivalentto“if p,then q,”notethat“p is sufficientfor q”meansif p istrue,itmustbethecasethat q isalsotrue.Thisisthesameas sayingthatif p istrue,then q isalsotrue.
Torememberthat“q unless ¬p”expressesthesameconditionalstatementas“if p,then q,”notethat“q unless ¬p”meansthatif ¬p isfalse,then q mustbetrue.Thatis,thestatement“q unless ¬p”isfalsewhen p istruebut q isfalse,butitistrueotherwise.Consequently, “ q unless ¬p”and p → q alwayshavethesametruthvalue.
EXAMPLE10
WeillustratethetranslationbetweenconditionalstatementsandEnglishstatementsin Example10.
Let p bethestatement“Marialearnsdiscretemathematics”and q thestatement“Mariawillfind agoodjob.”Expressthestatement p → q asastatementinEnglish.
Solution: Fromthedefinitionofconditionalstatements,weseethatwhen p isthestatement
Extra Examples “Marialearnsdiscretemathematics”and q isthestatement“Mariawillfindagoodjob,” p → q representsthestatement
“IfMarialearnsdiscretemathematics,thenshewillfindagoodjob.”
TherearemanyotherwaystoexpressthisconditionalstatementinEnglish.Amongthemost naturaloftheseare
“Mariawillfindagoodjobwhenshelearnsdiscretemathematics.”
“ForMariatogetagoodjob,itissufficientforhertolearndiscretemathematics.” and
“Mariawillfindagoodjobunlessshedoesnotlearndiscretemathematics.” ◂
Notethatthewaywehavedefinedconditionalstatementsismoregeneralthanthemeaning attachedtosuchstatementsintheEnglishlanguage.Forinstance,theconditionalstatementin Example10andthestatement
“Ifitissunny,thenwewillgotothebeach” arestatementsusedinnormallanguagewherethereisarelationshipbetweenthehypothesis andtheconclusion.Further,thefirstofthesestatementsistrueunlessMarialearnsdiscrete mathematics,butshedoesnotgetagoodjob,andthesecondistrueunlessitisindeedsunny, butwedonotgotothebeach.Ontheotherhand,thestatement
“IfJuanhasasmartphone,then2 + 3 = 5” istruefromthedefinitionofaconditionalstatement,becauseitsconclusionistrue.(Thetruth valueofthehypothesisdoesnotmatterthen.)Theconditionalstatement
“IfJuanhasasmartphone,then2 + 3 = 6”
istrueifJuandoesnothaveasmartphone,eventhough2 + 3 = 6isfalse.Wewouldnotuse theselasttwoconditionalstatementsinnaturallanguage(exceptperhapsinsarcasm),because thereisnorelationshipbetweenthehypothesisandtheconclusionineitherstatement.Inmathematicalreasoning,weconsiderconditionalstatementsofamoregeneralsortthanweusein English.Themathematicalconceptofaconditionalstatementisindependentofacause-andeffectrelationshipbetweenhypothesisandconclusion.Ourdefinitionofaconditionalstatement specifiesitstruthvalues;itisnotbasedonEnglishusage.Propositionallanguageisanartificial language;weonlyparallelEnglishusagetomakeiteasytouseandremember.
Theif-thenconstructionusedinmanyprogramminglanguagesisdifferentfromthat usedinlogic.Mostprogramminglanguagescontainstatementssuchas if p then S,where p isapropositionand S isaprogramsegment(oneormorestatementstobeexecuted). (Althoughthislooksasifitmightbeaconditionalstatement, S isnotaproposition,but ratherisasetofexecutableinstructions.)Whenexecutionofaprogramencounterssuch astatement, S isexecutedif p istrue,but S isnotexecutedif p isfalse,asillustratedin Example11.
EXAMPLE11
Whatisthevalueofthevariable x afterthestatement
if 2 + 2 = 4 then x := x + 1
if x = 0beforethisstatementisencountered?(Thesymbol:= standsforassignment.Thestatement x := x + 1meanstheassignmentofthevalueof x + 1to x.)
Solution: Because2 + 2 = 4istrue,theassignmentstatement x := x + 1isexecuted.Hence, x hasthevalue0 + 1 = 1afterthisstatementisencountered.
CONVERSE,CONTRAPOSITIVE,ANDINVERSE Wecanformsomenewconditional statementsstartingwithaconditionalstatement p → q.Inparticular,therearethreerelated conditionalstatementsthatoccursooftenthattheyhavespecialnames.Theproposition q → p iscalledthe converse of p → q.The contrapositive of p → q istheproposition ¬q → ¬p.The proposition ¬p → ¬q iscalledthe inverse of p → q.Wewillseethatofthesethreeconditionalstatementsformedfrom p → q,onlythecontrapositivealwayshasthesametruthvalue as p → q.
Wefirstshowthatthecontrapositive, ¬q → ¬p,ofaconditionalstatement p → q always hasthesametruthvalueas p → q.Toseethis,notethatthecontrapositiveisfalseonlywhen ¬p isfalseand ¬q istrue,thatis,onlywhen p istrueand q isfalse.Wenowshowthatneitherthe converse, q → p,northeinverse, ¬p → ¬q,hasthesametruthvalueas p → q forallpossible truthvaluesof p and q.Notethatwhen p istrueand q isfalse,theoriginalconditionalstatement isfalse,buttheconverseandtheinversearebothtrue. Rememberthatthe contrapositive,but neithertheconverseor inverse,ofaconditional statementisequivalent toit.
Whentwocompoundpropositionsalwayshavethesametruthvalues,regardlessofthetruth valuesofitspropositionalvariables,wecallthem equivalent.Hence,aconditionalstatement anditscontrapositiveareequivalent.Theconverseandtheinverseofaconditionalstatement arealsoequivalent,asthereadercanverify,butneitherisequivalenttotheoriginalconditional statement.(WewillstudyequivalentpropositionsinSection1.3.)Takenotethatoneofthemost commonlogicalerrorsistoassumethattheconverseortheinverseofaconditionalstatement isequivalenttothisconditionalstatement.
WeillustratetheuseofconditionalstatementsinExample12.
EXAMPLE12
Extra Examples
Findthecontrapositive,theconverse,andtheinverseoftheconditionalstatement
“Thehometeamwinswheneveritisraining.”
Solution: Because“q whenever p”isoneofthewaystoexpresstheconditionalstatement p → q, theoriginalstatementcanberewrittenas
“Ifitisraining,thenthehometeamwins.”
Consequently,thecontrapositiveofthisconditionalstatementis
“Ifthehometeamdoesnotwin,thenitisnotraining.”
Theconverseis
“Ifthehometeamwins,thenitisraining.”
Theinverseis
“Ifitisnotraining,thenthehometeamdoesnotwin.”
Onlythecontrapositiveisequivalenttotheoriginalstatement.
