Discrete Mathematics

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