Discrete Mathematics : Proofs, Structures and Applications, Third Edition

847 Pages·2009·5.2 MB·English

Discrete
Mathematics

Boca Raton London New York
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Contents
Contents v
PrefacetotheThirdEdition ix
PrefacetotheSecondEdition xi
PrefacetotheFirstEdition xiii
ListofSymbols xvii
Chapter1: Logic 1
1.1 PropositionsandTruthValues 1
1.2 LogicalConnectivesandTruthTables 2
1.3 TautologiesandContradictions 13
1.4 LogicalEquivalenceandLogicalImplication 15
1.5 TheAlgebraofPropositions 22
1.6 Arguments 26
vi Contents
1.7 FormalProofoftheValidityofArguments 29
1.8 PredicateLogic 35
1.9 ArgumentsinPredicateLogic 45
Chapter2: MathematicalProof 50
2.1 TheNatureofProof 50
2.2 AxiomsandAxiomSystems 51
2.3 MethodsofProof 55
2.4 MathematicalInduction 69
Chapter3: Sets 79
3.1 SetsandMembership 79
3.2 Subsets 85
3.3 OperationsonSets 91
3.4 CountingTechniques 100
3.5 TheAlgebraofSets 104
3.6 FamiliesofSets 111
3.7 TheCartesianProduct 122
3.8 TypesandTypedSetTheory 134
Chapter4: Relations 154
4.1 RelationsandTheirRepresentations 154
4.2 PropertiesofRelations 164
4.3 IntersectionsandUnionsofRelations 171
Contents vii
4.4 EquivalenceRelationsandPartitions 175
4.5 OrderRelations 188
4.6 HasseDiagrams 198
4.7 Application: RelationalDatabases 205
Chapter5: Functions 220
5.1 DefinitionsandExamples 220
5.2 CompositeFunctions 238
5.3 InjectionsandSurjections 246
5.4 BijectionsandInverseFunctions 260
5.5 MoreonCardinality 270
5.6 Databases: FunctionalDependenceandNormalForms 277
Chapter6: MatrixAlgebra 291
6.1 Introduction 291
6.2 SomeSpecialMatrices 294
6.3 OperationsonMatrices 296
6.4 ElementaryMatrices 308
6.5 TheInverseofaMatrix 318
Chapter7: SystemsofLinearEquations 331
7.1 Introduction 331
7.2 MatrixInverseMethod 337
7.3 Gauss–JordanElimination 342
7.4 GaussianElimination 355
viii Contents
Chapter8: AlgebraicStructures 361
8.1 BinaryOperationsandTheirProperties 361
8.2 AlgebraicStructures 370
8.3 MoreaboutGroups 379
8.4 SomeFamiliesofGroups 384
8.5 Substructures 396
8.6 Morphisms 404
8.7 GroupCodes 418
Chapter9: IntroductiontoNumberTheory 436
9.1 Divisibility 437
9.2 PrimeNumbers 449
9.3 LinearCongruences 460
9.4 GroupsinModularArithmetic 473
9.5 PublicKeyCryptography 479
Chapter10:BooleanAlgebra 492
10.1 Introduction 492
10.2 PropertiesofBooleanAlgebras 496
10.3 BooleanFunctions 503
10.4 SwitchingCircuits 520
10.5 LogicNetworks 529
10.6 MinimizationofBooleanExpressions 536
Contents ix
Chapter11:GraphTheory 548
11.1 DefinitionsandExamples 548
11.2 PathsandCycles 561
11.3 IsomorphismofGraphs 575
11.4 Trees 582
11.5 PlanarGraphs 591
11.6 DirectedGraphs 600
Chapter12:ApplicationsofGraphTheory 611
12.1 Introduction 611
12.2 RootedTrees 612
12.3 Sorting 626
12.4 SearchingStrategies 643
12.5 WeightedGraphs 652
12.6 TheShortestPathandTravellingSalesmanProblems 660
12.7 NetworksandFlows 673
ReferencesandFurtherReading 687
HintsandSolutionsto
SelectedExercises 692
Index 798