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Contents

1 Sets and Logic 1.1 Sets 1.2 Propositions 1.3 Conditional Propositions and Logical Equivalence 1.4 Arguments and Rules of Inference 1.5 Quantifiers 1.6 Nested Quantifiers Problem-Solving Corner: Quantifiers 2 Proofs 2.1 Mathematical Systems, Direct Proofs, and Counterexamples 2.2 More Methods of Proof Problem-Solving Corner: Proving Some Properties of Real Numbers 2.3 Resolution Proofs 2.4 Mathematical Induction Problem-Solving Corner: Mathematical Induction 2.5 Strong Form of Induction and the Well-Ordering Property Notes Chapter Review Chapter Self-Test Computer Exercises 3 Functions, Sequences, and Relations 3.1 Functions Problem-Solving Corner: Functions 3.2 Sequences and Strings 3.3 Relations 3.4 Equivalence Relations Problem-Solving Corner: Equivalence Relations 3.5 Matrices of Relations 3.6 Relational Databases 4 Algorithms 4.1 Introduction 4.2 Examples of Algorithms 4.3 Analysis of Algorithms Problem-Solving Corner: Design and Analysis of an Algorithm 4.4 Recursive Algorithms 5 Introduction to Number Theory 5.1 Divisors 5.2 Representations of Integers and Integer Algorithms 5.3 The Euclidean Algorithm Problem-Solving Corner: Making Postage 5.4 The RSA Public-Key Cryptosystem 6 Counting Methods and the Pigeonhole Principle 6.1 Basic Principles Problem-Solving Corner: Counting 6.2 Permutations and Combinations Problem-Solving Corner: Combinations 6.3 Generalized Permutations and Combinations 6.4 Algorithms for Generating Permutations and Combinations 6.5 Introduction to Discrete Probability 6.6 Discrete Probability Theory 6.7 Binomial Coefficients and Combinatorial Identities 6.8 The Pigeonhole Principle 7 Recurrence Relations 7.1 Introduction 7.2 Solving Recurrence Relations Problem-Solving Corner: Recurrence Relations 7.3 Applications to the Analysis of Algorithms 8 Graph Theory 8.1 Introduction 8.2 Paths and Cycles Problem-Solving Corner: Graphs 8.3 Hamiltonian Cycles and the Traveling Salesperson Problem 8.4 A Shortest-Path Algorithm 8.5 Representations of Graphs 8.6 Isomorphisms of Graphs 8.7 Planar Graphs 8.8 Instant Insanity 9 Trees 9.1 Introduction 9.2 Terminology and Characterizations of Trees Problem-Solving Corner: Trees 9.3 Spanning Trees 9.4 Minimal Spanning Trees 9.5 Binary Trees 9.6 Tree Traversals 9.7 Decision Trees and the Minimum Time for Sorting 9.8 Isomorphisms of Trees 9.9 Game Trees 10 Network Models 10.1 Introduction 10.2 A Maximal Flow Algorithm 10.3 The Max Flow, Min Cut Theorem 10.4 Matching Problem-Solving Corner: Matching 11 Boolean Algebras and Combinatorial Circuits 11.1 Combinatorial Circuits 11.2 Properties of Combinatorial Circuits 11.3 Boolean Algebras Problem-Solving Corner: Boolean Algebras 11.4 Boolean Functions and Synthesis of Circuits 11.5 Applications 12 Automata, Grammars, and Languages 12.1 Sequential Circuits and Finite-State Machines 12.2 Finite-State Automata 12.3 Languages and Grammars 12.4 Nondeterministic Finite-State Automata 12.5 Relationships Between Languages and Automata Appendix A Matrices B Algebra Review C Pseudocode References Hints and Solutions to Selected Exercises Index

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