Discrete Mathematics

773 Pages·2017·23.04 MB·English

by  Richard Johnsonbaugh

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Discrete
Mathematics
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Discrete
Mathematics
Eighth Edition
Richard Johnsonbaugh
DePaul University, Chicago
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330HudsonStreet,NY,NY10013
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Johnsonbaugh,Richard,1941-
Discretemathematics/RichardJohnsonbaugh,DePaulUniversity,Chicago.–Eighthedition.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-0-321-96468-7–ISBN0-321-96468-3
1. Mathematics.2. Computerscience–Mathematics. I.Title.
QA39.2.J652015
510–dc23
2014006017
1 17
ISBN13:978-0-321-96468-7
ISBN10:0-321-96468-3
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Contents
Preface XIII
1
Sets and Logic 1
1.1 Sets 2
1.2 Propositions 14
1.3 ConditionalPropositionsandLogicalEquivalence 20
(cid:2) 1.4 ArgumentsandRulesofInference 31 (cid:2)
1.5 Quantifiers 36
1.6 NestedQuantifiers 49
Problem-Solving Corner:Quantifiers 57
Chapter1Notes 58
Chapter1Review 58
Chapter1Self-Test 60
Chapter1ComputerExercises 60
2
Proofs 62
2.1 MathematicalSystems,DirectProofs,
andCounterexamples 63
2.2 MoreMethodsofProof 72
Problem-Solving Corner:ProvingSomeProperties
ofRealNumbers 83
†
2.3 ResolutionProofs 85
2.4 MathematicalInduction 88
Problem-Solving Corner:MathematicalInduction 100
2.5 StrongFormofInductionandtheWell-OrderingProperty 102
Chapter2Notes 109
Chapter2Review 109
†Thissectioncanbeomittedwithoutlossofcontinuity. v
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vi Contents
Chapter2Self-Test 109
Chapter2ComputerExercises 110
3
Functions, Sequences, and Relations 111
3.1 Functions 111
Problem-Solving Corner:Functions 128
3.2 SequencesandStrings 129
3.3 Relations 141
3.4 EquivalenceRelations 151
Problem-Solving Corner:EquivalenceRelations 158
3.5 MatricesofRelations 160
†
3.6 RelationalDatabases 165
Chapter3Notes 170
Chapter3Review 170
Chapter3Self-Test 171
Chapter3ComputerExercises 172
4
Algorithms 173
4.1 Introduction 173
(cid:2) 4.2 ExamplesofAlgorithms 177 (cid:2)
4.3 AnalysisofAlgorithms 184
Problem-Solving Corner:DesignandAnalysis
ofanAlgorithm 202
4.4 RecursiveAlgorithms 204
Chapter4Notes 211
Chapter4Review 211
Chapter4Self-Test 212
Chapter4ComputerExercises 212
5
Introduction to Number Theory 214
5.1 Divisors 214
5.2 RepresentationsofIntegersandIntegerAlgorithms 224
5.3 TheEuclideanAlgorithm 238
Problem-Solving Corner:MakingPostage 249
5.4 TheRSAPublic-KeyCryptosystem 250
Chapter5Notes 252
Chapter5Review 253
Chapter5Self-Test 253
Chapter5ComputerExercises 254
†Thissectioncanbeomittedwithoutlossofcontinuity.
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Contents vii
6
Counting Methods and the Pigeonhole
Principle 255
6.1 BasicPrinciples 255
Problem-Solving Corner:Counting 267
6.2 PermutationsandCombinations 269
Problem-Solving Corner:Combinations 281
6.3 GeneralizedPermutationsandCombinations 283
6.4 AlgorithmsforGeneratingPermutationsand
Combinations 289
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6.5 IntroductiontoDiscreteProbability 297
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6.6 DiscreteProbabilityTheory 301
6.7 BinomialCoefficientsandCombinatorialIdentities 313
6.8 ThePigeonholePrinciple 319
Chapter6Notes 324
Chapter6Review 324
Chapter6Self-Test 325
Chapter6ComputerExercises 326
7
Recurrence Relations 327
7.1 Introduction 327
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7.2 SolvingRecurrenceRelations 338
Problem-Solving Corner:RecurrenceRelations 350
7.3 ApplicationstotheAnalysisofAlgorithms 353
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7.4 TheClosest-PairProblem 365
Chapter7Notes 370
Chapter7Review 371
Chapter7Self-Test 371
Chapter7ComputerExercises 372
8
Graph Theory 373
8.1 Introduction 373
8.2 PathsandCycles 384
Problem-Solving Corner:Graphs 395
8.3 HamiltonianCyclesandtheTravelingSalesperson
Problem 396
8.4 AShortest-PathAlgorithm 405
8.5 RepresentationsofGraphs 410
8.6 IsomorphismsofGraphs 415
8.7 PlanarGraphs 422
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8.8 InstantInsanity 429
†Thissectioncanbeomittedwithoutlossofcontinuity.
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viii Contents
Chapter8Notes 433
Chapter8Review 434
Chapter8Self-Test 435
Chapter8ComputerExercises 436
9
Trees 438
9.1 Introduction 438
9.2 TerminologyandCharacterizationsofTrees 445
Problem-Solving Corner:Trees 450
9.3 SpanningTrees 452
9.4 MinimalSpanningTrees 459
9.5 BinaryTrees 465
9.6 TreeTraversals 471
9.7 DecisionTreesandtheMinimumTimeforSorting 477
9.8 IsomorphismsofTrees 483
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9.9 GameTrees 493
Chapter9Notes 502
Chapter9Review 502
Chapter9Self-Test 503
Chapter9ComputerExercises 505
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Network Models 506
10.1 Introduction 506
10.2 AMaximalFlowAlgorithm 511
10.3 TheMaxFlow,MinCutTheorem 519
10.4 Matching 523
Problem-Solving Corner:Matching 528
Chapter10Notes 529
Chapter10Review 530
Chapter10Self-Test 530
Chapter10ComputerExercises 531
11
Boolean Algebras and Combinatorial
Circuits 532
11.1 CombinatorialCircuits 532
11.2 PropertiesofCombinatorialCircuits 539
11.3 BooleanAlgebras 544
Problem-Solving Corner:BooleanAlgebras 549
11.4 BooleanFunctionsandSynthesisofCircuits 551
11.5 Applications 556
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Contents ix
Chapter11Notes 564
Chapter11Review 565
Chapter11Self-Test 565
Chapter11ComputerExercises 567
12
Automata, Grammars, and Languages 568
12.1 SequentialCircuitsandFinite-StateMachines 568
12.2 Finite-StateAutomata 574
12.3 LanguagesandGrammars 579
12.4 NondeterministicFinite-StateAutomata 589
12.5 RelationshipsBetweenLanguagesandAutomata 595
Chapter12Notes 601
Chapter12Review 602
Chapter12Self-Test 602
Chapter12ComputerExercises 603
Appendix 605
A
Matrices 605
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B
Algebra Review 609
C
Pseudocode 620
References 627
Hints and Solutions to Selected Exercises 633
Index 735
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