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Mathematics for Computer Science PDF

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“mcs-ftl” — 2010/9/8 — 0:40 — page i — #1 Mathematics for Computer Science revisedWednesday8th September,2010,00:40 Eric Lehman GoogleInc. F Thomson Leighton DepartmentofMathematicsandCSAIL,MIT AkamaiTechnologies Albert R Meyer MassachusetsInstituteofTechnology Copyright©2010, EricLehman,FTomLeighton,AlbertRMeyer.Allrightsreserved. “mcs-ftl” — 2010/9/8 — 0:40 — page ii — #2 “mcs-ftl” — 2010/9/8 — 0:40 — page iii — #3 Contents I Proofs 1 Propositions 5 1.1 CompoundPropositions 6 1.2 PropositionalLogicinComputerPrograms 10 1.3 PredicatesandQuantifiers 11 1.4 Validity 19 1.5 Satisfiability 21 2 PatternsofProof 23 2.1 TheAxiomaticMethod 23 2.2 ProofbyCases 26 2.3 ProvinganImplication 27 2.4 Provingan“IfandOnlyIf” 30 2.5 ProofbyContradiction 32 2.6 ProofsaboutSets 33 2.7 Good ProofsinPractice 40 3 Induction 43 3.1 TheWellOrderingPrinciple 43 3.2 OrdinaryInduction 46 3.3 Invariants 56 3.4 StrongInduction 64 3.5 StructuralInduction 69 4 NumberTheory 81 4.1 Divisibility 81 4.2 TheGreatestCommonDivisor 87 4.3 TheFundamentalTheoremofArithmetic 94 4.4 AlanTuring 96 4.5 ModularArithmetic 100 4.6 ArithmeticwithaPrimeModulus 103 4.7 ArithmeticwithanArbitraryModulus 108 4.8 TheRSAAlgorithm 113 “mcs-ftl” — 2010/9/8 — 0:40 — page iv — #4 iv Contents II Structures 5 GraphTheory 121 5.1 Definitions 121 5.2 MatchingProblems 128 5.3 Coloring 143 5.4 GettingfromAtoB inaGraph 147 5.5 Connectivity 151 5.6 AroundandAroundWeGo 156 5.7 Trees 162 5.8 PlanarGraphs 170 6 DirectedGraphs 189 6.1 Definitions 189 6.2 TournamentGraphs 192 6.3 CommunicationNetworks 196 7 RelationsandPartialOrders 213 7.1 BinaryRelations 213 7.2 RelationsandCardinality 217 7.3 RelationsonOneSet 220 7.4 EquivalenceRelations 222 7.5 PartialOrders 225 7.6 PosetsandDAGs 226 7.7 TopologicalSort 229 7.8 ParallelTaskScheduling 232 7.9 Dilworth’sLemma 235 8 StateMachines 237 III Counting 9 SumsandAsymptotics 243 9.1 TheValueofanAnnuity 244 9.2 PowerSums 250 9.3 ApproximatingSums 252 9.4 HangingOutOvertheEdge 257 9.5 DoubleTrouble 269 9.6 Products 272 “mcs-ftl” — 2010/9/8 — 0:40 — page v — #5 v Contents 9.7 AsymptoticNotation 275 10 Recurrences 283 10.1 TheTowersofHanoi 284 10.2 MergeSort 291 10.3 LinearRecurrences 294 10.4 Divide-and-ConquerRecurrences 302 10.5 AFeelforRecurrences 309 11 CardinalityRules 313 11.1 CountingOneThingbyCountingAnother 313 11.2 CountingSequences 314 11.3 TheGeneralizedProductRule 317 11.4 TheDivisionRule 321 11.5 CountingSubsets 324 11.6 SequenceswithRepetitions 326 11.7 CountingPractice: PokerHands 329 11.8 Inclusion-Exclusion 334 11.9 CombinatorialProofs 339 11.10ThePigeonholePrinciple 342 11.11AMagicTrick 346 12 GeneratingFunctions 355 12.1 DefinitionsandExamples 355 12.2 OperationsonGeneratingFunctions 356 12.3 EvaluatingSums 361 12.4 ExtractingCoefficients 363 12.5 SolvingLinearRecurrences 370 12.6 CountingwithGeneratingFunctions 374 13 InfiniteSets 379 13.1 Injections,Surjections,andBijections 379 13.2 CountableSets 381 13.3 PowerSetsAreStrictlyBigger 384 13.4 InfinitiesinComputerScience 386 IV Probability 14 EventsandProbabilitySpaces 391 14.1 Let’sMakeaDeal 391 14.2 TheFourStepMethod 392 “mcs-ftl” — 2010/9/8 — 0:40 — page vi — #6 vi Contents 14.3 StrangeDice 402 14.4 SetTheoryandProbability 411 14.5 InfiniteProbabilitySpaces 413 15 ConditionalProbability 417 15.1 Definition 417 15.2 UsingtheFour-StepMethodtoDetermineConditionalProbability 418 15.3 APosterioriProbabilities 424 15.4 ConditionalIdentities 427 16 Independence 431 16.1 Definitions 431 16.2 IndependenceIsanAssumption 432 16.3 MutualIndependence 433 16.4 PairwiseIndependence 435 16.5 TheBirthdayParadox 438 17 RandomVariablesandDistributions 445 17.1 DefinitionsandExamples 445 17.2 DistributionFunctions 450 17.3 BernoulliDistributions 452 17.4 UniformDistributions 453 17.5 BinomialDistributions 456 18 Expectation 467 18.1 DefinitionsandExamples 467 18.2 ExpectedReturnsinGamblingGames 477 18.3 ExpectationsofSums 483 18.4 ExpectationsofProducts 490 18.5 ExpectationsofQuotients 492 19 Deviations 497 19.1 Variance 497 19.2 Markov’sTheorem 507 19.3 Chebyshev’sTheorem 513 19.4 BoundsforSumsofRandomVariables 516 19.5 MutuallyIndependentEvents 523 20 RandomWalks 533 20.1 UnbiasedRandomWalks 533 20.2 Gambler’sRuin 542 20.3 WalkinginCircles 549 “mcs-ftl” — 2010/9/8 — 0:40 — page 1 — #7 I Proofs “mcs-ftl” — 2010/9/8 — 0:40 — page 2 — #8 “mcs-ftl” — 2010/9/8 — 0:40 — page 3 — #9 Introduction This text explains how to use mathematical models and methods to analyze prob- lems that arise in computer science. The notion of a proof plays a central role in thiswork. Simplyput,aproofisamethodofestablishingtruth. Likebeauty,“truth”some- timesdependsontheeyeofthebeholder,anditshouldnotbesurprisingthatwhat constitutesaproofdiffersamongfields. Forexample, inthejudicialsystem, legal truthisdecidedbyajurybasedontheallowableevidencepresentedattrial. Inthe businessworld,authoritativetruthisspecifiedbyatrustedpersonororganization, or maybe just your boss. In fields such as physics and biology, scientific truth1 is confirmed by experiment. In statistics, probable truth is established by statistical analysisofsampledata. Philosophical proof involves careful exposition and persuasion typically based on a series of small, plausible arguments. The best example begins with “Cogito ergo sum,” a Latin sentence that translates as “I think, therefore I am.” It comes fromthebeginningofa17thcenturyessaybythemathematician/philosopher,Rene´ Descartes, and it is one of the most famous quotes in the world: do a web search onthephraseandyouwillbefloodedwithhits. Deducingyourexistencefromthefactthatyou’rethinkingaboutyourexistence isaprettycoolandpersuasive-soundingidea. However,withjustafewmorelines of argument in this vein, Descartes goes on to conclude that there is an infinitely beneficent God. Whether or not you believe in a beneficent God, you’ll probably agree that any very short proof of God’s existence is bound to be far-fetched. So 1Actually,onlyscientificfalsehoodcanbedemonstratedbyanexperiment—whentheexperiment failstobehaveaspredicted.Butnoamountofexperimentcanconfirmthatthenextexperimentwon’t fail.Forthisreason,scientistsrarelyspeakoftruth,butratheroftheoriesthataccuratelypredictpast, andanticipatedfuture,experiments. “mcs-ftl” — 2010/9/8 — 0:40 — page 4 — #10 4 PartI Proofs eveninmasterfulhands,thisapproachisnotreliable. Mathematicshasitsownspecificnotionof“proof.” Definition. Amathematicalproof ofapropositionisachainoflogicaldeductions leadingtothepropositionfromabasesetofaxioms. The three key ideas in this definition are highlighted: proposition, logical de- duction, and axiom. These three ideas are explained in the following chapters, beginningwithpropositionsinChapter1. Wewillthenprovidelotsofexamplesof proofsandevensomeexamplesof“falseproofs”(thatis,argumentsthatlooklike a proof but that contain missteps, or deductions that aren’t so logical when exam- inedclosely). Falseproofsareoftenevenmoreimportantasexamplesthancorrect proofs, because they are uniquely helpful with honing your skills at making sure eachstepofaprooffollowslogicallyfrompriorsteps. Creating a good proof is a lot like creating a beautiful work of art. In fact, mathematiciansoftenrefertoreallygoodproofsasbeing“elegant”or“beautiful.” As with any endeavor, it will probably take a little practice before your fellow students use such praise when referring to your proofs, but to get you started in therightdirection, wewillprovidetemplatesforthemostusefulprooftechniques in Chapters 2 and 3. We then apply these techniques in Chapter 4 to establish someimportantfactsaboutnumbers;factsthatformtheunderpinningofoneofthe world’smostwidely-usedcryptosystems.

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