Discrete Mathematics and Its Applications, Fifth Edition

1 The Foundations: Logic and Proof, Sets, and Functions

1.1 Logic

1.2 Propositional Equivalences

1.3 Predicates and Quantifiers

1.4 Nested Quantifiers

1.5 Methods of Proof

1.6 Sets

1.7 Set Operations

1.8 Functions

2 The Fundamentals: Algorithms, the Integers, and Matrices

2.1 Algorithms

2.2 The Growth of Functions

2.3 Complexity of Algorithms

2.4 The Integers and Division

2.5 Applications of Number Theory

2.6 Matrices

3 Mathematical Reasoning, Induction, and Recursion

3.1 Proof Strategy

3.2 Sequences and Summations

3.3 Mathematical Induction

3.4 Recursive Definitions and Structural Induction

3.5 Recursive Algorithms

3.6 Program Correctness

4 Counting

4.1 The Basics of Counting

4.2 The Pigeonhole Principle

4.3 Permutations and Combinations

4.4 Binomial Coefficients

4.5 Generalized Permutations and Combinations

4.6 Generating Permutations and Combinations

5 Discrete Probability

5.1 An Introduction to Discrete Probability

5.2 Probability Theory

5.3 Expected Value and Variance

6 Advanced Counting Techniques

6.1 Recurrence Relations

6.2 Solving Recurrence Relations

6.3 Divide-and-Conquer Algorithms and Recurrence Relations

6.4 Generating Functions

6.5 Inclusion-Exclusion

6.6 Applications of Inclusion-Exclusion

7 Relations

7.1 Relations and Their Properties

7.2 n-ary Relations and Their Applications

7.3 Representing Relations

7.4 Closures of Relations

7.5 Equivalence Relations

7.6 Partial Orderings

8 Graphs

8.1 Introduction to Graphs

8.2 Graph Terminology

8.3 Representing Graphs and Graph Isomorphism

8.4 Connectivity

8.5 Euler and Hamilton Paths

8.6 Shortest-Path Problems

8.7 Planar Graphs

8.8 Graph Coloring

9 Trees

9.1 Introduction to Trees

9.2 Applications of Trees

9.3 Tree Traversal

9.4 Spanning Trees

9.5 Minimum Spanning Trees

10 Boolean Algebra

10.1 Boolean Functions

10.2 Representing Boolean Functions

10.3 Logic Gates

10.4 Minimization of Circuits

11 Modeling Computation

11.1 Languages and Grammars

11.2 Finite-State Machines with Output

11.3 Finite-State Machines with No Output

11.4 Language Recognition

11.5 Turing Machines

Appendixes

A.1 Exponential and Logarithmic Functions

A.2 Pseudocode