Schaum's Outline of Discrete Mathematics, 3rd Ed.

488 Pages·2007·8.158 MB·English

by  Seymour Lipschutz, Marc Lipson

SCHAUM’S
OUTLINEOF
TheoryandProblemsof
DISCRETE
MATHEMATICS
SCHAUM’S
OUTLINEOF
TheoryandProblemsof
DISCRETE
MATHEMATICS
ThirdEdition
SEYMOUR LIPSCHUTZ, Ph.D.
Temple University
MARC LARS LIPSON, Ph.D.
University of Virginia
Schaum’s Outline Series
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Copyright © 2007, 1997, 1976 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of America.
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DOI: 10.1036/0071470387
PREFACE
Discretemathematics,thestudyoffinitesystems,hasbecomeincreasinglyimportantasthecomputerage
hasadvanced.Thedigitalcomputerisbasicallyafinitestructure,andmanyofitspropertiescanbeunderstood
andinterpretedwithintheframeworkoffinitemathematicalsystems.Thisbook,inpresentingthemoreessential
material,maybeusedasatextbookforaformalcourseindiscretemathematicsorasasupplementtoallcurrent
texts.
The first three chapters cover the standard material on sets, relations, and functions and algorithms. Next
comechaptersonlogic,counting,andprobability.Wethenhavethreechaptersongraphtheory:graphs,directed
graphs,andbinarytrees.Finallythereareindividualchaptersonpropertiesoftheintegers,languages,machines,
orderedsetsandlattices,andBooleanalgebra,andappendicesonvectorsandmatrices,andalgebraicsystems.
Thechapteronfunctionsandalgorithmsincludesadiscussionofcardinalityandcountablesets,andcomplexity.
Thechaptersongraphtheoryincludediscussionsonplanarity,traversability,minimalpaths,andWarshall’sand
Huffman’salgorithms.Weemphasizethatthechaptershavebeenwrittensothattheordercanbechangedwithout
difficultyandwithoutlossofcontinuity.
Eachchapterbeginswithaclearstatementofpertinentdefinitions,principles,andtheoremswithillustrative
and other descriptive material. This is followed by sets of solved and supplementary problems. The solved
problems serve to illustrate and amplify the material, and also include proofs of theorems.The supplementary
problemsfurnishacompletereviewofthematerialinthechapter.Morematerialhasbeenincludedthancanbe
coveredinmostfirstcourses.Thishasbeendonetomakethebookmoreflexible,toprovideamoreusefulbook
ofreference,andtostimulatefurtherinterestinthetopics.
SeymourLipschutz
MarcLarsLipson
v
Copyright © 2007, 1997, 1976 by The McGraw-Hill Companies, Inc. Click here for terms of use.
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CONTENTS
CHAPTER 1 Set Theory 1
1.1 Introduction 1
1.2 SetsandElements,Subsets 1
1.3VennDiagrams 3
1.4 SetOperations 4
1.5AlgebraofSets,Duality 7
1.6 FiniteSets,CountingPrinciple 8
1.7 ClassesofSets,PowerSets,Partitions 10
1.8 MathematicalInduction 12
SolvedProblems 12
SupplementaryProblems 18
CHAPTER 2 Relations 23
2.1 Introduction 23
2.2 ProductSets 23
2.3 Relations 24
2.4 PictorialRepresentativesofRelations 25
2.5 CompositionofRelations 27
2.6TypesofRelations 28
2.7 ClosureProperties 30
2.8 EquivalenceRelations 31
2.9 PartialOrderingRelations 33
SolvedProblems 34
SupplementaryProblems 40
CHAPTER 3 Functions and Algorithms 43
3.1 Introduction 43
3.2 Functions 43
3.3 One-to-One,Onto,andInvertibleFunctions 46
3.4 MathematicalFunctions,ExponentialandLogarithmicFunctions 47
3.5 Sequences,IndexedClassesofSets 50
3.6 RecursivelyDefinedFunctions 52
3.7 Cardinality 55
3.8AlgorithmsandFunctions 56
3.9 ComplexityofAlgorithms 57
SolvedProblems 60
SupplementaryProblems 66
vii
viii CONTENTS
CHAPTER 4 Logic and Propositional Calculus 70
4.1 Introduction 70
4.2 PropositionsandCompoundStatements 70
4.3 BasicLogicalOperations 71
4.4 PropositionsandTruthTables 72
4.5 TautologiesandContradictions 74
4.6 LogicalEquivalence 74
4.7AlgebraofPropositions 75
4.8 ConditionalandBiconditionalStatements 75
4.9Arguments 76
4.10 PropositionalFunctions,Quantifiers 77
4.11 NegationofQuantifiedStatements 79
SolvedProblems 82
SupplementaryProblems 86
CHAPTER 5 Techniques of Counting 88
5.1 Introduction 88
5.2 BasicCountingPrinciples 88
5.3 MathematicalFunctions 89
5.4 Permutations 91
5.5 Combinations 93
5.6 ThePigeonholePrinciple 94
5.7 TheInclusion–ExclusionPrinciple 95
5.8 TreeDiagrams 95
SolvedProblems 96
SupplementaryProblems 103
CHAPTER 6 Advanced Counting Techniques, Recursion 107
6.1 Introduction 107
6.2 CombinationswithRepetitions 107
6.3 OrderedandUnorderedPartitions 108
6.4 Inclusion–ExclusionPrincipleRevisited 108
6.5 PigeonholePrincipleRevisited 110
6.6 RecurrenceRelations 111
6.7 LinearRecurrenceRelationswithConstantCoefficients 113
6.8 SolvingSecond-OrderHomogeneousLinearRecurrence
Relations 114
6.9 SolvingGeneralHomogeneousLinearRecurrenceRelations 116
SolvedProblems 118
SupplementaryProblems 121
CHAPTER 7 Probability 123
7.1 Introduction 123
7.2 SampleSpaceandEvents 123
7.3 FiniteProbabilitySpaces 126
7.4 ConditionalProbability 127
7.5 IndependentEvents 129
7.6 IndependentRepeatedTrials,BinomialDistribution 130
7.7 RandomVariables 132
CONTENTS ix
7.8 Chebyshev’sInequality,LawofLargeNumbers 135
SolvedProblems 136
SupplementaryProblems 149
CHAPTER 8 Graph Theory 154
8.1 Introduction,DataStructures 154
8.2 GraphsandMultigraphs 156
8.3 Subgraphs,IsomorphicandHomeomorphicGraphs 158
8.4 Paths,Connectivity 159
8.5 TraversableandEulerianGraphs,BridgesofKönigsberg 160
8.6 LabeledandWeightedGraphs 162
8.7 Complete,Regular,andBipartiteGraphs 162
8.8 TreeGraphs 164
8.9 PlanarGraphs 166
8.10 GraphColorings 168
8.11 RepresentingGraphsinComputerMemory 171
8.12 GraphAlgorithms 173
8.13 Traveling-SalesmanProblem 176
SolvedProblems 178
SupplementaryProblems 191
CHAPTER 9 Directed Graphs 201
9.1 Introduction 201
9.2 DirectedGraphs 201
9.3 BasicDefinitions 202
9.4 RootedTrees 204
9.5 SequentialRepresentationofDirectedGraphs 206
9.6 Warshall’sAlgorithm,ShortestPaths 209
9.7 LinkedRepresentationofDirectedGraphs 211
9.8 GraphAlgorithms:Depth-FirstandBreadth-FirstSearches 213
9.9 DirectedCycle-FreeGraphs,TopologicalSort 216
9.10 PruningAlgorithmforShortestPath 218
SolvedProblems 221
SupplementaryProblems 228
CHAPTER 10 Binary Trees 235
10.1 Introduction 235
10.2 BinaryTrees 235
10.3 CompleteandExtendedBinaryTrees 237
10.4 RepresentingBinaryTreesinMemory 239
10.5 TraversingBinaryTrees 240
10.6 BinarySearchTrees 242
10.7 PriorityQueues,Heaps 244
10.8 PathLengths,Huffman’sAlgorithm 248
10.9 General(OrderedRooted)TreesRevisited 251
SolvedProblems 252
SupplementaryProblems 259
x CONTENTS
CHAPTER 11 Properties of the Integers 264
11.1 Introduction 264
11.2 OrderandInequalities,AbsoluteValue 265
11.3 MathematicalInduction 266
11.4 DivisionAlgorithm 267
11.5 Divisibility,Primes 269
11.6 GreatestCommonDivisor,EuclideanAlgorithm 270
11.7 FundamentalTheoremofArithmetic 273
11.8 CongruenceRelation 274
11.9 CongruenceEquations 278
SolvedProblems 283
SupplementaryProblems 299
CHAPTER 12 Languages, Automata, Grammars 303
12.1 Introduction 303
12.2Alphabet,Words,FreeSemigroup 303
12.3 Languages 304
12.4 RegularExpressions,RegularLanguages 305
12.5 FiniteStateAutomata 306
12.6 Grammars 310
SolvedProblems 314
SupplementaryProblems 319
CHAPTER 13 Finite State Machines and Turing Machines 323
13.1 Introduction 323
13.2 FiniteStateMachines 323
13.3 GödelNumbers 326
13.4 TuringMachines 326
13.5 ComputableFunctions 330
SolvedProblems 331
SupplementaryProblems 334
CHAPTER 14 Ordered Sets and Lattices 337
14.1 Introduction 337
14.2 OrderedSets 337
14.3 HasseDiagramsofPartiallyOrderedSets 340
14.4 ConsistentEnumeration 342
14.5 SupremumandInfimum 342
14.6 Isomorphic(Similar)OrderedSets 344
14.7 Well-OrderedSets 344
14.8 Lattices 346
14.9 BoundedLattices 348
14.10 DistributiveLattices 349
14.11 Complements,ComplementedLattices 350
SolvedProblems 351
SupplementaryProblems 360