Instructors Resource Guide for Discrete Mathematics and Its Applications

【实例简介】Instructors Resource Guide for Discrete Mathematics and Its Applications

Contents

Preface iii
Solutions for Even-numbered Exercises
CHAPTER 1 The Foundations: Logic and Proofs 1
1.1 Propositional Logic 1
1.2 Applications of Propositional Logic 6
1.3 Propositional Equivalences 9
1.4 Predicates and Quantifiers 14
1.5 Nested Quantifiers 20
1.6 Rules of Inference 25
1.7 Introduction to Proofs 28
1.8 Proof Methods and Strategy 31
Supplementary Exercises for Chapter 1 34
CHAPTER 2 Basic Structures: Sets, Functions,
Sequences, Sums, and Matrices 38
2.1 Sets 38
2.2 Set Operations 41
2.3 Functions 45
2.4 Sequences and Summations 53
2.5 Cardinality of Sets 58
2.6 Matrices 61
Supplementary Exercises for Chapter 2 63
CHAPTER 3 Algorithms 66
3.1 Algorithms 66
3.2 The Growth of Functions 73
3.3 Complexity of Algorithms 78
Supplementary Exercises for Chapter 3 82
CHAPTER 4 Number Theory and Cryptography 87
4.1 Divisibility and Modular Arithmetic 87
4.2 Integer Representations and Algorithms 90
4.3 Primes and Greatest Common Divisors 94
4.4 Solving Congruences 101
4.5 Applications of Congruences 107
4.6 Cryptography 109
Supplementary Exercises for Chapter 4 110
v
CHAPTER 5 Induction and Recursion 115
5.1 Mathematical Induction 115
5.2 Strong Induction and Well-Ordering 124
5.3 Recursive Definitions and Structural Induction 130
5.4 Recursive Algorithms 136
5.5 Program Correctness 140
Supplementary Exercises for Chapter 5 141
CHAPTER 6 Counting 152
6.1 The Basics of Counting 152
6.2 The Pigeonhole Principle 158
6.3 Permutations and Combinations 161
6.4 Binomial Coefficients and Identities 164
6.5 Generalized Permutations and Combinations 167
6.6 Generating Permutations and Combinations 172
Supplementary Exercises for Chapter 6 173
CHAPTER 7 Discrete Probability 178
7.1 An Introduction to Discrete Probability 178
7.2 Probability Theory 180
7.3 Bayes’ Theorem 185
7.4 Expected Value and Variance 187
Supplementary Exercises for Chapter 7 192
CHAPTER 8 Advanced Counting Techniques 196
8.1 Applications of Recurrence Relations 196
8.2 Solving Linear Recurrence Relations 203
8.3 Divide-and-Conquer Algorithms
and Recurrence Relations 209
8.4 Generating Functions 213
8.5 Inclusion–Exclusion 224
8.6 Applications of Inclusion–Exclusion 226
Supplementary Exercises for Chapter 8 227
CHAPTER 9 Relations 231
9.1 Relations and Their Properties 231
9.2 n-ary Relations and Their Applications 236
9.3 Representing Relations 238
9.4 Closures of Relations 241
9.5 Equivalence Relations 244
9.6 Partial Orderings 249
Supplementary Exercises for Chapter 9 254
vi
CHAPTER 10 Graphs 258
10.1 Graphs and Graph Models 258
10.2 Graph Terminology and Special Types of Graphs 260
10.3 Representing Graphs and Graph Isomorphism 265
10.4 Connectivity 270
10.5 Euler and Hamilton Paths 276
10.6 Shortest-Path Problems 280
10.7 Planar Graphs 282
10.8 Graph Coloring 284
Supplementary Exercises for Chapter 10 288
CHAPTER 11 Trees 294
11.1 Introduction to Trees 294
11.2 Applications of Trees 297
11.3 Tree Traversal 303
11.4 Spanning Trees 307
11.5 Minimum Spanning Trees 312
Supplementary Exercises for Chapter 11 314
CHAPTER 12 Boolean Algebra 318
12.1 Boolean Functions 318
12.2 Representing Boolean Functions 321
12.3 Logic Gates 323
12.4 Minimization of Circuits 325
Supplementary Exercises for Chapter 12 331
CHAPTER 13 Modeling Computation 334
13.1 Languages and Grammars 334
13.2 Finite-State Machines with Output 338
13.3 Finite-State Machines with No Output 341
13.4 Language Recognition 346
13.5 Turing Machines 349
Supplementary Exercises for Chapter 13 352
APPENDIXES 355
Appendix 1 Axioms for the Real Numbers
and the Positive Integers 355
Appendix 2 Exponential and Logarithmic Functions 356
Appendix 3 Pseudocode 357
Suggested Syllabi NOT YET AVAILABLE
Teaching Suggestions NOT YET AVAILABLE

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