Discrete Mathematics

784 Pages·2007·4.499 MB·English

by  Richard Johnsonbaugh

Discrete
Mathematics
Seventh Edition
Richard Johnsonbaugh
DePaul University, Chicago
Upper Saddle River, New Jersey 07458
LibraryofCongressCataloging-in-PublicationDataonFile.
©2009byPearsonEducation,Inc.
PearsonPrenticeHall
PearsonEducation,Inc.
UpperSaddleRiver,NJ07458
PrintedintheUnitedStatesofAmerica
ISBN 0-13-159318-8
ISBN 978-0-13-159318-3
Contents
Preface XI
1
➜ Sets and Logic 1
1.1 Sets 2
1.2 Propositions 14
1.3 ConditionalPropositionsandLogicalEquivalence 21
1.4 ArgumentsandRulesofInference 31
1.5 Quantifiers 36
1.6 NestedQuantifiers 51
Problem-Solving Corner:Quantifiers 60
Notes 62
ChapterReview 62
ChapterSelf-Test 63
ComputerExercises 64
2
➜ Proofs 66
2.1 MathematicalSystems,DirectProofs,andCounterexamples 67
2.2 MoreMethodsofProof 76
Problem-Solving Corner:ProvingSomeProperties
ofRealNumbers 87
†
2.3 ResolutionProofs 90
2.4 MathematicalInduction 93
Problem-Solving Corner:MathematicalInduction 106
2.5 StrongFormofInductionandtheWell-OrderingProperty 108
Notes 115
ChapterReview 115
ChapterSelf-Test 116
ComputerExercises 116
†Thissectioncanbeomittedwithoutlossofcontinuity. v
vi Contents
3
➜ Functions, Sequences, and Relations 117
3.1 Functions 117
Problem-Solving Corner:Functions 135
3.2 SequencesandStrings 136
3.3 Relations 148
3.4 EquivalenceRelations 159
Problem-Solving Corner:EquivalenceRelations 166
3.5 MatricesofRelations 168
†
3.6 RelationalDatabases 173
Notes 178
ChapterReview 178
ChapterSelf-Test 179
ComputerExercises 180
4
➜ Algorithms 181
4.1 Introduction 181
4.2 ExamplesofAlgorithms 186
4.3 AnalysisofAlgorithms 193
Problem-Solving Corner:DesignandAnalysisofanAlgorithm 211
4.4 RecursiveAlgorithms 213
Notes 220
ChapterReview 221
ChapterSelf-Test 221
ComputerExercises 222
5
➜ Introduction to Number Theory 223
5.1 Divisors 223
5.2 RepresentationsofIntegersandIntegerAlgorithms 234
5.3 TheEuclideanAlgorithm 248
Problem-Solving Corner:MakingPostage 259
5.4 TheRSAPublic-KeyCryptosystem 260
Notes 263
ChapterReview 263
ChapterSelf-Test 263
ComputerExercises 264
6
➜ Counting Methods and the Pigeonhole
Principle 265
6.1 BasicPrinciples 265
Problem-Solving Corner:Counting 277
†Thissectioncanbeomittedwithoutlossofcontinuity.
Contents vii
6.2 PermutationsandCombinations 278
Problem-Solving Corner:Combinations 291
6.3 GeneralizedPermutationsandCombinations 293
6.4 AlgorithmsforGeneratingPermutationsandCombinations 299
†
6.5 IntroductiontoDiscreteProbability 305
†
6.6 DiscreteProbabilityTheory 309
6.7 BinomialCoefficientsandCombinatorialIdentities 320
6.8 ThePigeonholePrinciple 325
Notes 330
ChapterReview 330
ChapterSelf-Test 330
ComputerExercises 332
7
➜ Recurrence Relations 333
7.1 Introduction 333
7.2 SolvingRecurrenceRelations 345
Problem-Solving Corner:RecurrenceRelations 358
7.3 ApplicationstotheAnalysisofAlgorithms 361
Notes 373
ChapterReview 373
ChapterSelf-Test 374
ComputerExercises 374
8
➜ Graph Theory 376
8.1 Introduction 377
8.2 PathsandCycles 388
Problem-Solving Corner:Graphs 399
8.3 HamiltonianCyclesandtheTravelingSalespersonProblem 400
8.4 AShortest-PathAlgorithm 407
8.5 RepresentationsofGraphs 412
8.6 IsomorphismsofGraphs 417
8.7 PlanarGraphs 425
†
8.8 InstantInsanity 431
Notes 435
ChapterReview 436
ChapterSelf-Test 437
ComputerExercises 438
9
➜ Trees 440
9.1 Introduction 440
9.2 TerminologyandCharacterizationsofTrees 448
Problem-Solving Corner:Trees 453
†Thissectioncanbeomittedwithoutlossofcontinuity.
viii Contents
9.3 SpanningTrees 454
9.4 MinimalSpanningTrees 461
9.5 BinaryTrees 467
9.6 TreeTraversals 474
9.7 DecisionTreesandtheMinimumTimeforSorting 480
9.8 IsomorphismsofTrees 486
†
9.9 GameTrees 496
Notes 505
ChapterReview 505
ChapterSelf-Test 506
ComputerExercises 508
10
➜ Network Models 510
10.1 Introduction 510
10.2 AMaximalFlowAlgorithm 516
10.3 TheMaxFlow,MinCutTheorem 524
10.4 Matching 528
Problem-Solving Corner:Matching 533
Notes 534
ChapterReview 535
ChapterSelf-Test 536
ComputerExercises 536
11
➜ Boolean Algebras and Combinatorial
Circuits 537
11.1 CombinatorialCircuits 537
11.2 PropertiesofCombinatorialCircuits 544
11.3 BooleanAlgebras 549
Problem-Solving Corner:BooleanAlgebras 554
11.4 BooleanFunctionsandSynthesisofCircuits 556
11.5 Applications 561
Notes 570
ChapterReview 570
ChapterSelf-Test 571
ComputerExercises 572
12
➜ Automata, Grammars, and Languages 573
12.1 SequentialCircuitsandFinite-StateMachines 573
12.2 Finite-StateAutomata 579
12.3 LanguagesandGrammars 585
†Thissectioncanbeomittedwithoutlossofcontinuity.
Contents ix
12.4 NondeterministicFinite-StateAutomata 595
12.5 RelationshipsBetweenLanguagesandAutomata 602
Notes 608
ChapterReview 609
ChapterSelf-Test 609
ComputerExercises 611
13
➜ Computational Geometry 612
13.1 TheClosest-PairProblem 612
13.2 AnAlgorithmtoComputetheConvexHull 617
Notes 625
ChapterReview 625
ChapterSelf-Test 625
ComputerExercises 626
Appendix 627
A
➜ Matrices 627
B
➜ Algebra Review 631
C
➜ Pseudocode 644
References 650
Hints and Solutions to Selected Exercises 655
Index 754
Symbols
LOGIC
p∨q porq;page15 P ≡ Q P and Q arelogicallyequivalent;page26
p∧q pandq;page15 ∀ forall;page38
¬p not p;page17 ∃ thereexists;page41
p →q if p,thenq;page21 ∴ therefore;page32
p ↔q pifandonlyifq;page25
SETNOTATION
{x ,...,x } setconsistingoftheelementsx ,...,x ;page2
1 n 1 n
{x | p(x)} setconsistingofthoseelementsx satisfyingproperty p(x);page2
Z,Z−,Z+,Znonneg setsofintegers,negativeintegers,positiveintegers,nonnegativeintegers;pages2–3
Q,Q−,Q+,Qnonneg setsofrationalnumbers,negativerationalnumbers,positiverationalnumbers,
nonnegativerationalnumbers;pages2–3
R,R−,R+,Rnonneg setsofrealnumbers,negativerealnumbers,positiverealnumbers,
nonnegativerealnumbers;pages2–3
x ∈ X x isanelementof X;page3
x ∈/ X x isnotanelementof X;page3
X =Y setequality(X andY havethesameelements);page3
|X| cardinalityof X (numberofelementsin X);page3
∅ emptyset;pages3
X ⊆Y X isasubsetofY;page4
X ⊂Y X isapropersubsetofY;page5
P(X) powersetof X (allsubsetsof X);page5
X ∪Y X unionY (allelementsin X orY);page6
(cid:2)n
X unionof X ,..., X (allelementsthatbelongtoatleastoneof X , X ,..., X );page9
i 1 n 1 2 n
i=1
(cid:2)∞
X unionof X , X ,...(allelementsthatbelongtoatleastoneof X , X ,...);page9
i 1 2 1 2
i=1
∪S unionofS (allelementsthatbelongtoatleastonesetinS);page9
X ∩Y X intersectY (allelementsin X andY);page6
(cid:3)n
X intersectionof X ,..., X (allelementsthatbelongtoeveryoneof X , X ,..., X );page9
i 1 n 1 2 n
i=1
(cid:3)∞
X intersectionof X , X ,...(allelementsthatbelongtoeveryoneof X , X ,...);page9
i 1 2 1 2
i=1
∩S intersectionofS (allelementsthatbelongtoeverysetinS);page9
X −Y setdifference(allelementsin X butnotinY);page6
X complementof X (allelementsnotin X);page7
(x, y) orderedpair;page10
(x ,...,x ) n-tuple;page10
1 n
X ×Y Cartesianproductof X andY [pairs(x, y)withx in X and y inY];page10
X ×X ×···×X Cartesianproductof X , X ,..., X (n-tupleswithx ∈ X );page11
1 2 n 1 2 n i i
X(cid:15)Y symmetricdifferenceof X andY;page13
RELATIONS
xRy (x, y)isin R(x isrelatedto y bytherelation R);page148
[x] equivalenceclasscontainingx;page161
R−1 inverserelation[all(y,x)with(x, y)in R];page155
R ◦ R compositionofrelations;page155
2 1
x (cid:17) y xRy;page154
FUNCTIONS
f(x) valueassignedtox;page119
f:X →Y functionfrom X toY;page118
f ◦g compositionof f andg;page129
f−1 inversefunction[all(y,x)with(x, y)in f];pages127–128
f(n) = O(g(n)) |f(n)|≤C|g(n)|fornsufficientlylarge;page195
f(n) =(cid:2)(g(n)) c|g(n)|≤|f(n)|fornsufficientlylarge;page195
f(n) =(cid:3)(g(n)) c|g(n)|≤|f(n)|≤C|g(n)|fornsufficientlylarge;page195
COUNTING
C(n,r) numberofr-combinationsofann-elementset(n!/[(n−r)!r!]);page282
P(n,r) numberofr-permutationsofann-elementset[n(n−1)···(n−r +1)];page280
GRAPHS
G =(V, E) graphG withvertexsetV andedgeset E;page378
(v,w) edge;page378
δ(v) degreeofvertexv;page392
(v ,...,v ) pathfromv tov ;page388
1 n 1 n
(v ,...,v ),v =v cycle;page391
1 n 1 n
K completegraphonnvertices;page382
n
K completebipartitegraphonm andnvertices;page384
m,n
w(i, j) weightofedge(i, j);page407
F flowinedge(i, j);page511
ij
C capacityofedge(i, j);page511
ij
(P, P) cutinanetwork;page524
PROBABILITY
P(x) probabilityofoutcomex;page309
P(E) probabilityofevent E;page310
P(E|F) conditionalprobabilityof E given F [P(E ∩F)/P(F)];page314
BOOLEANALGEBRASANDCIRCUITS
x ∨y x or y (1ifx or y is1,0otherwise);page538
x ∧y x and y (1ifx and y are1,0otherwise);page538
x ⊕y exclusive-ORofx and y (0ifx = y,1otherwise);page556
x notx (0ifx is1,1ifx is0);page538
x ↓ y x NOR y (0ifx or y is1,1otherwise);page569
x ↑ y x NAND y (0ifx and y are1,1otherwise);page563
ORgate;page538
ANDgate;page538
NOTgate(inverter);page538
NORgate;page569
NANDgate;page563
STRINGS,GRAMMARS,ANDLANGUAGES
λ nullstring;page142
|s| lengthofthestrings;page143
st concatenationofstringss andt (s followedbyt);page143
an aa···a(na’s);page142
X∗ setofallstringsover X;page142
X+ setofallnonnullstringsover X;page142
α →β productioninagrammar;page585
α ⇒β β isderivablefromα;page586
α ⇒α ⇒···⇒α derivationofα fromα ;page586
1 2 n n 1
L(G) languagegeneratedbythegrammarG;page586
S ::=T Backusnormalform(BNF);page587
S ::=T |T S ::=T ,S ::=T ;page587
1 2 1 2
Ac(A) setofstringsacceptedby A;page581
MATRICES
(a ) matrixwithentriesa ;page627
ij ij
A= B matrices Aand B areequal(Aand B arethesamesizeandtheircorrespondingentriesareequal);page627
A+B matrixsum;page628
cA scalarproduct;page628
−A (−1)A;page628
A−B A+(−B);page628
AB matrixproduct;page628
An matrixproduct AA···A(n A’s);page629