Table Of Content“mcs” — 2011/5/9 — 20:49 — page i — #1
Mathematics for Computer Science
revisedMonday9th May,2011,20:49
Eric Lehman
GoogleInc.
F Thomson Leighton
DepartmentofMathematicsandCSAIL,MIT
AkamaiTechnologies
Albert R Meyer
MassachusetsInstituteofTechnology
CreativeCommons 2011, EricLehman,FTomLeighton,AlbertRMeyer.
“mcs” — 2011/5/9 — 20:49 — page ii — #2
“mcs” — 2011/5/9 — 20:49 — page iii — #3
Contents
I Proofs
1 WhatisaProof? 7
1.1 Propositions 7
1.2 Predicates 9
1.3 TheAxiomaticMethod 10
1.4 OurAxioms 11
1.5 ProvinganImplication 13
1.6 Provingan“IfandOnlyIf” 15
1.7 ProofbyCases 17
1.8 ProofbyContradiction 18
1.9 Good ProofsinPractice 19
2 TheWellOrderingPrinciple 25
2.1 WellOrderingProofs 25
2.2 TemplateforWellOrderingProofs 26
2.3 SummingtheIntegers 26
2.4 FactoringintoPrimes 28
3 LogicalFormulas 35
3.1 PropositionsfromPropositions 36
3.2 PropositionalLogicinComputerPrograms 39
3.3 EquivalenceandValidity 42
3.4 TheAlgebraofPropositions 44
3.5 TheSATProblem 49
3.6 PredicateFormulas 50
4 MathematicalDataTypes 67
4.1 Sets 67
4.2 Sequences 70
4.3 Functions 71
4.4 BinaryRelations 73
5 InfiniteSets 87
5.1 FiniteCardinality 88
5.2 InfiniteCardinality 90
5.3 TheHaltingProblem 95
5.4 TheLogicofSets 98
“mcs” — 2011/5/9 — 20:49 — page iv — #4
iv Contents
5.5 DoesAllThisReallyWork? 101
6 Induction 113
6.1 OrdinaryInduction 113
6.2 StateMachines 122
6.3 StrongInduction 134
6.4 StrongInductionvs. Inductionvs. WellOrdering 138
7 RecursiveDataTypes 159
7.1 RecursiveDefinitionsandStructuralInduction 159
7.2 StringsofMatchedBrackets 163
7.3 RecursiveFunctionsonNonnegativeIntegers 166
7.4 ArithmeticExpressions 169
7.5 InductioninComputerScience 174
8 NumberTheory 183
8.1 Divisibility 183
8.2 TheGreatestCommonDivisor 189
8.3 TheFundamentalTheoremofArithmetic 195
8.4 AlanTuring 197
8.5 ModularArithmetic 201
8.6 ArithmeticwithaPrimeModulus 204
8.7 ArithmeticwithanArbitraryModulus 209
8.8 TheRSAAlgorithm 214
8.9 WhathasSATgottodowithit? 216
II Structures
9 Directedgraphs&PartialOrders 233
9.1 Digraphs&VertexDegrees 235
9.2 DigraphWalksandPaths 236
9.3 AdjacencyMatrices 239
9.4 PathRelations 242
9.5 DirectedAcyclicGraphs&PartialOrders 243
9.6 WeakPartialOrders 246
9.7 RepresentingPartialOrdersbySetContainment 247
9.8 Path-TotalOrders 248
9.9 ProductOrders 249
9.10 Scheduling 250
9.11 EquivalenceRelations 256
“mcs” — 2011/5/9 — 20:49 — page v — #5
v Contents
9.12 SummaryofRelationalProperties 257
10 CommunicationNetworks 279
10.1 CompleteBinaryTree 279
10.2 RoutingProblems 279
10.3 NetworkDiameter 280
10.4 SwitchCount 281
10.5 NetworkLatency 282
10.6 Congestion 282
10.7 2-DArray 283
10.8 Butterfly 285
10.9 Benes˘ Network 287
11 SimpleGraphs 299
11.1 VertexAdjacencyandDegrees 299
11.2 SexualDemographicsinAmerica 301
11.3 SomeCommonGraphs 303
11.4 Isomorphism 305
11.5 BipartiteGraphs&Matchings 307
11.6 TheStableMarriageProblem 312
11.7 Coloring 319
11.8 Gettingfromutov inaGraph 324
11.9 Connectivity 325
11.10OddCyclesand2-Colorability 329
11.11Forests&Trees 330
12 PlanarGraphs 361
12.1 DrawingGraphsinthePlane 361
12.2 DefinitionsofPlanarGraphs 361
12.3 Euler’sFormula 371
12.4 BoundingtheNumberofEdgesinaPlanarGraph 372
12.5 ReturningtoK andK 373
5 3;3
12.6 AnotherCharacterizationforPlanarGraphs 374
12.7 ColoringPlanarGraphs 375
12.8 ClassifyingPolyhedra 377
13 StateMachines 387
13.1 TheAlternatingBitProtocol 387
13.2 ReasoningAboutWhilePrograms 390
“mcs” — 2011/5/9 — 20:49 — page vi — #6
vi Contents
III Counting
14 SumsandAsymptotics 401
14.1 TheValueofanAnnuity 402
14.2 SumsofPowers 408
14.3 ApproximatingSums 410
14.4 HangingOutOvertheEdge 414
14.5 Products 426
14.6 DoubleTrouble 428
14.7 AsymptoticNotation 431
15 CardinalityRules 449
15.1 CountingOneThingbyCountingAnother 449
15.2 CountingSequences 450
15.3 TheGeneralizedProductRule 453
15.4 TheDivisionRule 457
15.5 CountingSubsets 460
15.6 SequenceswithRepetitions 461
15.7 TheBinomialTheorem 463
15.8 AWordaboutWords 465
15.9 CountingPractice: PokerHands 465
15.10Inclusion-Exclusion 470
15.11CombinatorialProofs 476
15.12ThePigeonholePrinciple 479
15.13AMagicTrick 484
IV Probability
16 EventsandProbabilitySpaces 515
16.1 Let’sMakeaDeal 515
16.2 TheFourStepMethod 516
16.3 StrangeDice 525
16.4 SetTheoryandProbability 533
16.5 ConditionalProbability 537
16.6 Independence 549
16.7 TheBirthdayPrinciple 555
17 RandomVariables 573
17.1 RandomVariableExamples 573
“mcs” — 2011/5/9 — 20:49 — page vii — #7
vii Contents
17.2 Independence 575
17.3 DistributionFunctions 576
17.4 GreatExpectations 585
17.5 LinearityofExpectation 597
18 DeviationfromtheMean 617
18.1 WhytheMean? 617
18.2 Markov’sTheorem 618
18.3 Chebyshev’sTheorem 620
18.4 PropertiesofVariance 624
18.5 EstimationbyRandomSampling 628
18.6 ConfidenceversusProbability 633
18.7 SumsofRandomVariables 634
18.8 ReallyGreatExpectations 644
19 RandomProcesses 661
19.1 Gamblers’Ruin 661
19.2 RandomWalksonGraphs 667
Index 678
“mcs” — 2011/5/9 — 20:49 — page viii — #8
“mcs” — 2011/5/9 — 20:49 — page 1 — #9
I Proofs
“mcs” — 2011/5/9 — 20:49 — page 2 — #10
