Exam (elaborations)
Solutions Manual for Discrete Mathematics (Classic Version) 5th Edition by John A. Dossey
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Discrete Mathematics (Classic Version) 5th Edition John A. Dossey SOLUTION
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INSTRUCTOR’SSOLUTIONS MANUAL
R
U
DISCRETE MATHEMATICS
SE
FIFTH EDITION
John A. Dossey IS
O
Illinois State University
N
Albert D. Otto
N
Illinois State University
O
Lawrence E. Spence
C
Illinois State University
ED
Charles Vanden Eynden
Illinois State University
M
, Contents
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U
SE
1 An Introduction to Combinatorial Problems and Techniques 1
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1.1 The Time to Complete a Project . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 A Matching Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 A Knapsack Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
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1.4 Algorithms and Their Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 4
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
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2 Sets, Relations, and Functions 6
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2.1 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
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2.3 Partial Ordering Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
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2.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
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3 Coding Theory 14
3.1 Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
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3.2 The Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 The RSA Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Error-Detecting and Error-Correcting Codes . . . . . . . . . . . . . . . . . . 17
3.5 Matrix Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.6 Matrix Codes That Correct All Single-Digit Errors . . . . . . . . . . . . . . . 19
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
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,Table of Contents
4 Graphs 22
4.1 Graphs and Their Representations . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Paths and Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 Shortest Paths and Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 Coloring a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
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4.5 Directed Graphs and Multigraphs . . . . . . . . . . . . . . . . . . . . . . . . 30
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
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5 Trees 36
SE
5.1 Properties of Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.2 Spanning Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Depth-First Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.4 Rooted Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.5
5.6
Binary Trees and Traversals . . . . . . . . . . .
Optimal Binary Trees and Binary Search Trees
Supplementary Exercises . . . . . . . . . . . . IS .
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6 Matching 63
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6.1 Systems of Distinct Representatives . . . . . . . . . . . . . . . . . . . . . . . 63
6.2 Matchings in Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
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6.3 A Matching Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.4 Applications of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 66
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6.5 The Hungarian Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
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7 Network Flows 68
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7.1 Flows and Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7.2 A Flow Augmentation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.3 The Max-Flow Min-Cut Theorem . . . . . . . . . . . . . . . . . . . . . . . . 73
7.4 Flows and Matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
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Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
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, Table of Contents
8 Counting Techniques 78
8.1 Pascal’s Triangle and the Binomial Theorem . . . . . . . . . . . . . . . . . . 78
8.2 Three Fundamental Principles . . . . . . . . . . . . . . . . . . . . . . . . . . 78
8.3 Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . 79
8.4 Arrangements and Selections with Repetitions . . . . . . . . . . . . . . . . . 79
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8.5 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
8.6 The Principle of Inclusion-Exclusion . . . . . . . . . . . . . . . . . . . . . . . 81
Generating Permutations and r-Combinations
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8.7 . . . . . . . . . . . . . . . . . 82
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
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9 Recurrence Relations and Generating Functions IS 84
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9.1 Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
9.2 The Method of Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
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9.3 Linear Difference Equations with Constant Coefficients . . . . . . . . . . . . 91
9.4∗ Analyzing the Efficiency of Algorithms with Recurrence Relations . . . . . . 94
N
9.5 Counting with Generating Functions . . . . . . . . . . . . . . . . . . . . . . . 99
9.6 The Algebra of Generating Functions . . . . . . . . . . . . . . . . . . . . . . 100
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
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10 Combinatorial Circuits and Finite State Machines 105
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10.1 Logical Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
10.2 Creating Combinatorial Circuits . . . . . . . . . . . . . . . . . . . . . . . . . 107
10.3 Karnaugh Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
10.4 Finite State Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
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