Preface About the Author To the Student LIST OF SYMBOLS 1 The Foundations： Logic and Proofs 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Predicates and Quantifiers 1.4 Nested Quantifiers 1.5 Rules oflnference 1.6 Introduction to Proofs 1.7 Proof Methods and Strategy End-of-Chapter Material 2 Basic Structures： Sets， Functions， Sequences， and Sums 2.1 Sets 2.2 Set Operations 2.3 Functions 2.4 Sequences and Summations End-of-Chapter Material 3 The Fundamentals： Algorithms， the Integers， and Matrices 3.1 Algorithms 3.2 The Growth of Functions 3.3 Complexity of Algorithms 3.4 The Integers and Division 3.5 Primes and Greatest Common Divisors 3.6 Integers and Algorithms 3.7 Applications of Number Theory 3.8 Matrices End-of-Chapter Material 4 Induction and Recursion 4.1 Mathematical Induction 4.2 Strong Induction and Well-Ordering 4.3 Recursive Definitions and Structural Induction 4.4 Recursive Algorithms 4.5 Program Correctness End-of-Chapter Material 5 Counting 5.1 The Basics of Counting 5.2 The Pigeonhole Principle 5.3 Permutations and Combinations 5.4 Binomial Coefficients 5.5 Generalized Permutations and Combinations 5.6 Generating Permutations and Combinations End-of-Chapter Material 6 Discrete Probability 6.1 An Introduction to Discrete Probability 6.2 Probability Theory 6.3 Bayes Theorem 6.4 Expected Value and Variance End-of-Chapter Material 7 Advanced Counting Techniques 7.1 Recurrence Relations 7.2 Solving Linear Recurrence Relations 7.3 Divide-and-Conquer Algorithms and Recurrence Relations 7.4 Generating Functions 7.5 Inclusion-Exclusion 7.6 Applications of Inclusion-Exclusion End-of-Chapter Material 8 Relations 8.1 Relations and Their Properties 8.2 n-ary Relations and Their Applications 8.3 Representing Relations 8.4 Closures of Relations 8.5 Equivalence Relations 8.6 Partial Orderings End-of-Chapter Material 9 Graphs 9.1 Graphs and Graph Models 9.2 Graph Terminology and Special Types of Graphs 9.3 Representing Graphs and Graph Isomorphism 9.4 Connectivity 9.5 Euler and Hamilton Paths 9.6 Shortest-Path Problems 9.7 Planar Graphs 9.8 Graph Coloring End-of-Chapter Material 10 Trees 11 Boolean Algebra 12 Modeling Computation Appendixes Suggested Readings B-1 Answers to Odd-Numbered Exercises @ S-1 Index of Biographies I-1 Index I-2
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