Discrete Mathematics and Its Applications

Preface

To the Student

1 The Foundations: Logic, Sets, and Functions

1.1 Logic

1.2 Propositional Equivalences

1.3 Predicates and Quantifiers

1.4 Sets

1.5 Set Operations

1.6 Functions

1.7 Sequences and Summations

1.8 The Growth Functions

2 The Fundamentals: Algorithms, the Integers, and Matrices

2.1 Algorithms

2.2 Complexity of Algorithms

2.3 The Integers and Division

2.4 Integers and Algorithms

2.5 Applications of Number Theory

2.6 Matrice

3 Mathematical Reasoning

3.1 Methods of Proof

3.2 Mathematical Induction

3.3 Recursive Definitions

3.4 Recursive Algorithms

3.5 Program Correctness

4 Counting

4.1 The Basics of Counting

4.2 The Pigeonhole Principle

4.3 Permutations and Combinations

4.4 Discrete Probability

4.5 Probability Theory

4.6 Generalized Permutations and Combinations

4.7 Generating Permutations and Combinations

5 Advanced Counting Techniques

5.1 Recurrence Relations

5.2 Solving Recurrence Relations

5.3 Divide-and-Conquer Relations

5.4 Generating Functions

5.5 Inclusion-Exclusion

5.6 Applications of Inclusion-Exclusion

6 Relations

6.1 Relations and Their Properties

6.2 n-ary Relations and Their Applications

6.3 Representing Relations

6.4 Closures of Relations

6.5 Equivalence Relations

6.6 Partial Orderings

7 Graphs

7.1 Introduction to Graphs

7.2 Graph Terminology

7.3 Representing Graphs and Graph Isomorphism

7.4 Connectivity

7.5 Euler and Hamilton Paths

7.6 Shortest Path Problems

7.7 Planar Graphs

7.8 Graph Coloring

8 Trees

8.1 Introduction to Trees

8.2 Applications of Trees

8.3 Tree Traversal

8.4 Trees and Sorting

8.5 Spanning Trees

8.6 Minimum Spanning Trees

9 Boolean Algebra

9.1 Boolean Functions

9.2 Representing Boolean Functions

9.3 Logic Gates

9.4 Minimization of Circuits

10 Modeling Computation

10.1 Languages and Grammar

10.2 Finite-State Machines with Output

10.3 Finite-State Machines with no Output

10.4 Language Recognition

10.5 Turing Machines

Appendixes A

A.1 Exponential and Logarithmic Functions

A.2 Pseudocode

Suggested Readings

Solutions to Odd-Numbered Exercises

Index of Biographies

Index

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