Mathematics for Computer Science

918 Pages·2015·10.31 MB·English

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Mathematics for Computer Science
revisedMonday18th May,2015,01:43
Eric Lehman
GoogleInc.
F Thomson Leighton
DepartmentofMathematics
andtheComputerScienceandAILaboratory,
MassachussettsInstituteofTechnology;
AkamaiTechnologies
Albert R Meyer
DepartmentofElectricalEngineeringandComputerScience
andtheComputerScienceandAILaboratory,
MassachussettsInstituteofTechnology
2015,EricLehman,FTomLeighton,AlbertRMeyer.Thisworkisavailableunderthe
termsoftheCreativeCommonsAttribution-ShareAlike3.0license.
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Contents
I Proofs
Introduction 3
0.1 References 4
1 WhatisaProof? 5
1.1 Propositions 5
1.2 Predicates 8
1.3 TheAxiomaticMethod 8
1.4 OurAxioms 9
1.5 ProvinganImplication 11
1.6 Provingan“IfandOnlyIf” 13
1.7 ProofbyCases 15
1.8 ProofbyContradiction 16
1.9 Good ProofsinPractice 17
1.10 References 19
2 TheWellOrderingPrinciple 27
2.1 WellOrderingProofs 27
2.2 TemplateforWellOrderingProofs 28
2.3 FactoringintoPrimes 30
2.4 WellOrderedSets 31
3 LogicalFormulas 41
3.1 PropositionsfromPropositions 42
3.2 PropositionalLogicinComputerPrograms 45
3.3 EquivalenceandValidity 48
3.4 TheAlgebraofPropositions 50
3.5 TheSATProblem 55
3.6 PredicateFormulas 56
3.7 References 61
4 MathematicalDataTypes 81
4.1 Sets 81
4.2 Sequences 86
4.3 Functions 87
4.4 BinaryRelations 89
4.5 FiniteCardinality 93
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iv Contents
5 Induction 115
5.1 OrdinaryInduction 115
5.2 StrongInduction 124
5.3 StrongInductionvs.Inductionvs.WellOrdering 129
5.4 StateMachines 130
6 RecursiveDataTypes 173
6.1 RecursiveDefinitionsandStructuralInduction 173
6.2 StringsofMatchedBrackets 177
6.3 RecursiveFunctionsonNonnegativeIntegers 180
6.4 ArithmeticExpressions 183
6.5 InductioninComputerScience 188
7 InfiniteSets 205
7.1 InfiniteCardinality 206
7.2 TheHaltingProblem 215
7.3 TheLogicofSets 219
7.4 DoesAllThisReallyWork? 222
II Structures
Introduction 241
8 NumberTheory 243
8.1 Divisibility 243
8.2 TheGreatestCommonDivisor 248
8.3 PrimeMysteries 254
8.4 TheFundamentalTheoremofArithmetic 257
8.5 AlanTuring 259
8.6 ModularArithmetic 263
8.7 RemainderArithmetic 265
8.8 Turing’sCode(Version2.0) 268
8.9 MultiplicativeInversesandCancelling 270
8.10 Euler’sTheorem 274
8.11 RSAPublicKeyEncryption 279
8.12 WhathasSATgottodowithit? 281
8.13 References 282
9 Directedgraphs&PartialOrders 317
9.1 VertexDegrees 319
9.2 WalksandPaths 320
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v Contents
9.3 AdjacencyMatrices 323
9.4 WalkRelations 326
9.5 DirectedAcyclicGraphs&Scheduling 327
9.6 PartialOrders 335
9.7 RepresentingPartialOrdersbySetContainment 339
9.8 LinearOrders 340
9.9 ProductOrders 340
9.10 EquivalenceRelations 341
9.11 SummaryofRelationalProperties 343
10 CommunicationNetworks 373
10.1 CompleteBinaryTree 373
10.2 RoutingProblems 373
10.3 NetworkDiameter 374
10.4 SwitchCount 375
10.5 NetworkLatency 376
10.6 Congestion 376
10.7 2-DArray 377
10.8 Butterfly 379
10.9 Benesˇ Network 381
11 SimpleGraphs 393
11.1 VertexAdjacencyandDegrees 393
11.2 SexualDemographicsinAmerica 395
11.3 SomeCommonGraphs 397
11.4 Isomorphism 399
11.5 BipartiteGraphs&Matchings 401
11.6 TheStableMarriageProblem 406
11.7 Coloring 413
11.8 SimpleWalks 417
11.9 Connectivity 419
11.10Forests&Trees 424
11.11References 433
12 PlanarGraphs 473
12.1 DrawingGraphsinthePlane 473
12.2 DefinitionsofPlanarGraphs 473
12.3 Euler’sFormula 484
12.4 BoundingtheNumberofEdgesinaPlanarGraph 485
12.5 ReturningtoK andK 486
5 3;3
12.6 ColoringPlanarGraphs 487
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vi Contents
12.7 ClassifyingPolyhedra 489
12.8 AnotherCharacterizationforPlanarGraphs 492
III Counting
Introduction 501
12.9 References 502
13 SumsandAsymptotics 503
13.1 TheValueofanAnnuity 504
13.2 SumsofPowers 510
13.3 ApproximatingSums 512
13.4 HangingOutOvertheEdge 516
13.5 Products 522
13.6 DoubleTrouble 525
13.7 AsymptoticNotation 528
14 CardinalityRules 551
14.1 CountingOneThingbyCountingAnother 551
14.2 CountingSequences 552
14.3 TheGeneralizedProductRule 555
14.4 TheDivisionRule 559
14.5 CountingSubsets 562
14.6 SequenceswithRepetitions 564
14.7 CountingPractice: PokerHands 567
14.8 ThePigeonholePrinciple 572
14.9 Inclusion-Exclusion 581
14.10CombinatorialProofs 587
14.11References 591
15 GeneratingFunctions 627
15.1 InfiniteSeries 627
15.2 CountingwithGeneratingFunctions 629
15.3 PartialFractions 635
15.4 SolvingLinearRecurrences 638
15.5 FormalPowerSeries 643
15.6 References 646
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vii Contents
IV Probability
Introduction 665
16 EventsandProbabilitySpaces 667
16.1 Let’sMakeaDeal 667
16.2 TheFourStepMethod 668
16.3 StrangeDice 677
16.4 TheBirthdayPrinciple 684
16.5 SetTheoryandProbability 686
16.6 References 690
17 ConditionalProbability 697
17.1 MontyHallConfusion 697
17.2 DefinitionandNotation 698
17.3 TheFour-StepMethodforConditionalProbability 700
17.4 WhyTreeDiagramsWork 702
17.5 TheLawofTotalProbability 710
17.6 Simpson’sParadox 712
17.7 Independence 714
17.8 MutualIndependence 716
18 RandomVariables 739
18.1 RandomVariableExamples 739
18.2 Independence 741
18.3 DistributionFunctions 742
18.4 GreatExpectations 751
18.5 LinearityofExpectation 762
19 DeviationfromtheMean 789
19.1 Markov’sTheorem 789
19.2 Chebyshev’sTheorem 792
19.3 PropertiesofVariance 796
19.4 EstimationbyRandomSampling 800
19.5 ConfidenceversusProbability 806
19.6 SumsofRandomVariables 807
19.7 ReallyGreatExpectations 816
20 RandomWalks 839
20.1 Gambler’sRuin 839
20.2 RandomWalksonGraphs 849
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viii Contents
V Recurrences
Introduction 865
21 Recurrences 867
21.1 TheTowersofHanoi 867
21.2 MergeSort 870
21.3 LinearRecurrences 874
21.4 Divide-and-ConquerRecurrences 881
21.5 AFeelforRecurrences 888
Bibliography 895
GlossaryofSymbols 899
Index 902
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I Proofs
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