目录 1 The Foundations: Logic and Proofs 1.1 Propositional Logic 1.2 Applications of Propositional Logic 1.3 Propositional Equivalences 1.4 Predicates and Quantifiers 1.5 Nested Quantifiers 1.6 Rules of Inference 1.7 Introduction to Proofs 1.8 Proof Methods and Strategy End-of-Chapter Material 2 Basic Structures: Sets, Functions, Sequences, Sums and Matrices 2.1 Sets 2.2 Set Operations 2.3 Functions 2.4 Sequences and Summations 2.5 Cardinality of Sets 2.6 Matrices End-of-Chapter Material 3 Algorithms 3.1 Algorithms 3.2 The Growth of Functions 3.3 Complexity of Algorithms End-of-Chapter Material 4 Number Theory and Cryptography 4.1 Divisibility and Modular Arithmetic 4.2 Integer Representations and Algorithms 4.3 Primes and Greatest Common Divisors 4.4 Solving Congruences 4.5 Applications of Congruences 4.6 Cryptography End-of-Chapter Material 5 Induction and Recursion 5.1 Mathematical Induction 5.2 Strong Induction and Well-Ordering 5.3 Recursive Definitions and Structural Induction 5.4 Recursive Algorithms 5.5 Program Correctness End-of-Chapter Material 6 Counting 6.1 The Basics of Counting 6.2 The Pigeonhole Principle 6.3 Permutations and Combinations 6.4 Binomial Coefficients and Identifies 6.5 Generalized Permutations and Combinations 6.6 Generating Permutations and Combinations End-of-Chapter Material 7 Discrete Probability 7.1 An Introduction to Discrete Probability 7.2 Probability Theory 7.3 Bayes' Theorem 7.4 Expected Value and Variance End-of-Chapter Material 8 Advanced Counting Techniques 8.1 Applications of Recurrence Relations 8.2 Solving Linear Recurrence Relations 8.3 Divide-and-Conquer Algorithms and Recurrence Relations 8.4 Generating Functions 8.5 Inclusion-Exclusion 8.6 Applications of Inclusion-Exclusion End-of-Chapter Material 9 Relations 9.1 Relations and Their Properties 9.2 n-ary Relations and Their Applications 9.3 Representing Relations 9.4 Closures of Relations 9.5 Equivalence Relations 9.6 Partial Orderings End-of-Chapter Material 10 Graphs 10.1 Graphs and Graph Models 10.2 Graph Terminology and Special Types of Graphs 10.3 Representing Graphs and Graph Isomorphism 10.4 Connectivity 10.5 Euler and Hamilton Paths 10.6 Shortest-Path Problems 10.7 Planar Graphs 10.8 Graph Coloring End-of-Chapter Material 11 Trees 11.1 Introduction to Trees 11.2 Applications of Trees 11.3 Tree Traversal 11.4 Spanning Trees 11.5 Minimum Spanning Trees End-of-Chapter Material 12 Boolean Algebra 12.1 Boolean Functions 12.2 Representing Boolean Functions 12.3 Logic Gates 12.4 Minimization of Circuits End-of-Chapter Material 13 Modeling Computation 13.1 Languages and Grammars 13.2 Finite-State Machines with Output 13.3 Finite-State Machines with No Output 13.4 Language Recognition 13.5 Turing Machines End-of-Chapter Material Appendices 1 Axioms for the Real Numbers and the Positive Integers 2 Exponential and Logarithmic Functions 3 Pseudocode Suggested Readings B-1 Answers to Odd-Numbered Exercises S-1 Index of Biographies I-1 Index I-2
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