Mathematics of discrete structures for computer science

300 Pages·2012·2.055 MB·English

by  Gordon J. Pace

Gordon J. Pace
Mathematics of
Discrete Structures
for Computer
Science
GordonJ.Pace
DepartmentofComputerScience,
FacultyofInformationand
CommunicationTechnology
UniversityofMalta
Msida,Malta
ISBN978-3-642-29839-4 ISBN978-3-642-29840-0(eBook)
DOI10.1007/978-3-642-29840-0
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012940954
ACMComputingClassification(1998): F.4,F.2,G.2
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DedicatedtoClaudia
and tomyparents
Foreword
Profoundknowledgeandskillsindiscretemathematicsaremandatoryforawelled-
ucatedcomputerscientist.Thereforeacorrespondingcurriculumtypicallyrequires
atleastabasiccourseonthissubject.Thistextbook,whichisbasedonthelectures
given by the author at the University of Malta, is a perfect companion for every
studenttakingsuchacourse.
Whenteachingaformalcomputersciencetopiconecaneitherfollowaso-called
descriptiveoralgorithmicapproach.Traditionallymosttextbooksondiscretemath-
ematicsfollowthedescriptiveapproachmeaningthatthepropertiesofmathematical
structuresarelistedfirstwhilethealgorithmstocomputecorrespondingresultsare
ofsecondaryinterest.Thistextbookfollowsthealgorithmicapproachwhichseems
tobebettersuitedforcomputersciencestudents:Mathematicalstructuresareintro-
ducedanddefinedusingalgorithmswhicharethencharacterizedbytheirproperties
inasecondstep.
Thebookcoversthematerialthatisessentialknowledgeforacomputerscience
student. Starting with propositional and first order logic it discusses sets, relations
andfurtherdiscretemathematicalstructures.Itcontainsachapterontypetheorybe-
foreintroducingthenaturalnumbers.Inthefinalchapteritdiscusseshowtoreason
aboutprogramsandgivesthebasicsofcomputabilitytheory.
Besidesathoroughformalpresentationthebookalwaysgivesaconvincingmo-
tivationforstudyingthecorrespondingstructuresandexplainsthemainideasusing
illustrations. A well chosen set of exercises rounds off each topic. After studying
thepresentedmaterialacomputersciencestudentwillbewellpreparedforfurther,
morespecializedcourses.
Lübeck,Germany MartinLeucker
vii
Preface
Incomputerscience,asinothersciences,mathematicsplaysanimportantrole.In
fact,computersciencehasmoreincommonwithmathematicsthanwiththetradi-
tionalsciences.Considerphysics,forexample.Clearly,mathematicsisrightatthe
coreofthesubject,usedtodescribemodelswhichexplainandpredictthephysical
world. Without mathematics, most of modern physics would have been impossi-
bletodevelop,sinceapurelyqualitativeanalysisdoesnotallowonetoreasonand
deduce what logically follows from an observation or a hypothesised model. No
manner of qualitative reasoning would have enabled Newton to formulate his law
ofgravitybasedonKepler’sclaimsaboutplanetarymotion.Computersciencealso
exploitsmathematicalreasoninginasimilarmanner.Givenacomputersystem,one
can apply mathematical reasoning to prove that it will always calculate your pay-
checkcorrectly.
However,thelinkiseventighterthanthis.Computerscienceisadirectdescen-
dant of mathematics, and computers are nothing but the physical embodiment of
mathematical systems developed long before the first modern computer was even
conceived,letalonebuilt.
Inthisbook,wewillbeexploringthefoundationalmathematicswhichisneces-
saryfortheunderstandingofmoreadvancedcoursesincomputerscience.Whether
youaredesigningadigitalcircuit,acomputerprogramoranewprogramminglan-
guage, mathematics is needed to reason about the design—its correctness, robust-
nessanddependability.
There are two distinct approaches used to present mathematical concepts and
operators.Inthefirstapproachconceptsandoperatorsaredefinedintermsofprop-
ertieswhichtheysatisfy.Basedonthesedefinitions,waysofcomputingtheresultof
applyingtheseoperatorsarethendevelopedandprovedtobecorrect.Incontrast,in
computersciencefrequentlyonetakestheoppositeapproach—onestartsbydefin-
ingwaysofcalculatingtheresultofanoperator,andthenprovesthattheoperator
oneknowshowtocomputesatisfiesanumberofproperties.Thetwoapproachesare,
intheend,equivalent.However,giventhatthisbookisaimedatcomputerscience
students,thelatterapproachisusuallyadopted.
ix
x Preface
Finally,mostsectionsareaccompaniedbyexerciseswhichareusuallynecessary
as part of the learning process. Learning mathematics is surprisingly like learning
to ride a bicycle. You must try to write formal expressions and proofs before you
really understand the concepts you read about. Some of the concepts covered in
thisbookmayinitiallyappeartobedifficult.However,asyouprogressthroughthe
bookandfamiliariseyourselfwiththeconcepts,youwillhopefullystartrequiring
lessefforttofollowthedefinitionsandproofs,andbegintoenjoythebeautyofhow
various concepts in mathematics can be built and reasoned about, based on just a
smallnumberofbasicconcepts.
BuenosAires,Argentina GordonJ.Pace
Acknowledgements
Theoriginalinspirationwhichledtothisbookdatesbackalmosttwodecades,when
IfollowedacourseonmathematicsforsystemspecificationlecturedbyJoeStoyat
theUniversityofOxford.Hisapproachofbuildingupthenecessarymathematical
machinery from the ground up, akin to building a software system, provided the
spark which years later led to the design of the structure and content of a course
in mathematics of discrete structures, and consequently to this book. Without Joe
Stoy’sspark,thisbookwouldneverhavebeen.
Thisbookhasbeenlonginthemaking.Lifekeptgettingintheway.Manystu-
dents would havebenefittedhad the book beenfinished earlier.Instead,they were
exposedto the intendedcontentin my lectures—questions and blank faces expos-
ingthepartsofthebookwhichIhadtorethink.Thanksgoouttothosehundredsof
studentswithoutwhomthebookwouldhavebeenmuchpoorer.
Thanks also go out to those people who used initial drafts of the book to de-
liver and support taught courses in mathematics of discrete structures: Christian
Colombo, Adrian Francalanza, Ingram Bondin, Mark Bonnici, Andrew Calleja,
KarlstonDemanuele,KevinFalzon,AndrewGauciandChristineVella.Theirfeed-
backwasinvaluable.
Finally,theshortestthanksgoouttothosewhosesupportcanneverbefullyac-
knowledged,nomatterhowlongIwriteaboutit—Claudia,DodieandJosie.Thank
you.
xi
Contents
1 WhyMathematics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 WhatIsMathematics? . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 AHistoricalPerspective . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 OntheSuperiorityofFormalReasoning . . . . . . . . . . . . . . 4
1.4 TheMathematicsofComputing . . . . . . . . . . . . . . . . . . . 4
1.4.1 TheAnalysisofPrograms . . . . . . . . . . . . . . . . . . 5
1.4.2 FormalSpecificationofRequirements . . . . . . . . . . . 6
1.4.3 ReasoningAboutPrograms . . . . . . . . . . . . . . . . . 6
1.4.4 TheLimitsofComputing . . . . . . . . . . . . . . . . . . 7
1.4.5 APhysicalEmbodimentofMathematics . . . . . . . . . . 8
1.5 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 PropositionalLogic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 WhatIsaProposition? . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 TheLanguageofDiscourse . . . . . . . . . . . . . . . . . . . . . 10
2.3 ModelTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 TruthTables:PropositionalOperators . . . . . . . . . . . . 16
2.3.2 PropertiesofPropositionalSentences . . . . . . . . . . . . 22
2.3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 ProofTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.1 WhatIsaProof? . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.2 ExampleAxiomsandRulesofInference . . . . . . . . . . 30
2.4.3 WhyProofs? . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.5 ComparingTruthTablesandProofs . . . . . . . . . . . . . . . . . 47
2.5.1 Completeness . . . . . . . . . . . . . . . . . . . . . . . . 48
2.5.2 Soundness . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 MoreAboutPropositionalLogic . . . . . . . . . . . . . . . . . . 48
2.6.1 BooleanAlgebra . . . . . . . . . . . . . . . . . . . . . . . 48
2.6.2 DoingwithLess . . . . . . . . . . . . . . . . . . . . . . . 50
2.6.3 DoingwithEvenLess . . . . . . . . . . . . . . . . . . . . 51
2.6.4 WhatiftheAxiomsAreWrong? . . . . . . . . . . . . . . 52
xiii
xiv Contents
2.7 SomeNotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.7.1 NotationandSymbols . . . . . . . . . . . . . . . . . . . . 53
2.7.2 ProofElegance . . . . . . . . . . . . . . . . . . . . . . . . 54
2.7.3 HowtoComeupwithaProof . . . . . . . . . . . . . . . . 55
2.7.4 NotAllProofsAreEqual . . . . . . . . . . . . . . . . . . 55
3 PredicateCalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1 TheLimitsofPropositionalLogic . . . . . . . . . . . . . . . . . . 57
3.2 TheLanguageofDiscourse . . . . . . . . . . . . . . . . . . . . . 58
3.3 TheUseofVariables . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.1 FreeVariables . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.2 SubstitutionofVariables . . . . . . . . . . . . . . . . . . . 64
3.4 AxiomatisationofPredicateLogic . . . . . . . . . . . . . . . . . 67
3.4.1 UniversalQuantification . . . . . . . . . . . . . . . . . . . 67
3.4.2 ExistentialQuantification . . . . . . . . . . . . . . . . . . 69
3.4.3 ExistentialQuantificationandEquality . . . . . . . . . . . 76
3.5 BeyondPredicateCalculus . . . . . . . . . . . . . . . . . . . . . 76
4 Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.1 WhatAreSets? . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 QueriesAboutSets . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3 ComparingSets . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3.2 SetEquality . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4 ConstructingSets . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4.1 FiniteSets . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4.2 TheEmptySet . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4.3 SetComprehensions . . . . . . . . . . . . . . . . . . . . . 86
4.5 SetOperators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.5.1 SetUnion . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.5.2 SetIntersection . . . . . . . . . . . . . . . . . . . . . . . 93
4.5.3 SetComplement . . . . . . . . . . . . . . . . . . . . . . . 95
4.5.4 SetDifference . . . . . . . . . . . . . . . . . . . . . . . . 97
4.5.5 OtherPropertiesoftheSetOperators . . . . . . . . . . . . 98
4.5.6 GeneralisedSetOperators . . . . . . . . . . . . . . . . . . 99
4.6 SetsasaBooleanAlgebra . . . . . . . . . . . . . . . . . . . . . . 102
4.7 MoreAboutTypes . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.7.1 TypesandSets . . . . . . . . . . . . . . . . . . . . . . . . 104
4.7.2 OnWhyTypesAreDesirable:Russell’sParadox . . . . . . 108
4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1 AnInformalViewofRelations . . . . . . . . . . . . . . . . . . . 111
5.2 FormalisingRelations . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2.1 TheTypeofaRelation. . . . . . . . . . . . . . . . . . . . 113
5.2.2 SomeBasicRelations . . . . . . . . . . . . . . . . . . . . 114
Contents xv
5.2.3 RelationalEquality . . . . . . . . . . . . . . . . . . . . . 115
5.2.4 CombiningRelationsasSets . . . . . . . . . . . . . . . . 115
5.3 DomainandRange. . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.4 BuildingNewRelationsfromOldOnes . . . . . . . . . . . . . . . 119
5.4.1 CompositionofRelations . . . . . . . . . . . . . . . . . . 119
5.4.2 RelationalInverse . . . . . . . . . . . . . . . . . . . . . . 123
5.4.3 RepeatedRelations . . . . . . . . . . . . . . . . . . . . . 127
5.4.4 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.5 OtherRelationalOperators . . . . . . . . . . . . . . . . . . . . . 134
5.6 BeyondBinaryRelations . . . . . . . . . . . . . . . . . . . . . . 138
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6 ClassifyingRelations . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.1 ClassesofRelations . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.1.1 Totality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.1.2 Surjectivity . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.1.3 Injectivity . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.1.4 Functionality . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.1.5 CombiningtheResults. . . . . . . . . . . . . . . . . . . . 148
6.2 RelatingaTypetoItself . . . . . . . . . . . . . . . . . . . . . . . 149
6.2.1 PropertiesofTheseRelations . . . . . . . . . . . . . . . . 149
6.2.2 Order-InducingRelations . . . . . . . . . . . . . . . . . . 152
6.2.3 EquivalenceRelations . . . . . . . . . . . . . . . . . . . . 153
6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7 MoreDiscreteStructures . . . . . . . . . . . . . . . . . . . . . . . . 157
7.1 Multisets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.3 GraphTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
8 DefiningNewStructuredTypes . . . . . . . . . . . . . . . . . . . . . 175
8.1 NotionsandNotation . . . . . . . . . . . . . . . . . . . . . . . . 176
8.1.1 SimpleEnumeratedTypes . . . . . . . . . . . . . . . . . . 176
8.1.2 MoreElaborateTypes . . . . . . . . . . . . . . . . . . . . 177
8.1.3 Self-ReferentialTypes . . . . . . . . . . . . . . . . . . . . 179
8.1.4 ParametrisedTypes . . . . . . . . . . . . . . . . . . . . . 181
8.2 ReasoningAboutNewTypes . . . . . . . . . . . . . . . . . . . . 184
8.2.1 AxiomatisingNewTypes . . . . . . . . . . . . . . . . . . 184
8.2.2 AGeneralInductivePrinciple . . . . . . . . . . . . . . . . 188
8.2.3 StructuralInduction . . . . . . . . . . . . . . . . . . . . . 193
8.3 UsingStructuredTypes . . . . . . . . . . . . . . . . . . . . . . . 194
8.3.1 Three-ValuedLogic . . . . . . . . . . . . . . . . . . . . . 194
8.3.2 ProcessingData . . . . . . . . . . . . . . . . . . . . . . . 198
8.3.3 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
8.3.4 BinaryTrees . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210