Discrete Mathematics and its Applications is a focused introduction to the primary themes in a discrete mathematics course, as introduced through extensive applications, expansive discussion, and detailed exercise sets. These themes include mathematical reasoning, combinatorial analysis, discrete structures, algorithmic thinking, and enhanced problem-solving skills through modeling. Its intent is to demonstrate the relevance and practicality of discrete mathematics to all students. The Fifth Edition includes a more thorough and linear presentation of logic, proof types and proof writing, and mathematical reasoning. This enhanced coverage will provide students with a solid understanding of the material as it relates to their immediate field of study and other relevant subjects. The inclusion of applications and examples to key topics has been significantly addressed to add clarity to every subject.True to the Fourth Edition, the text-specific web site supplements the subject matter in meaningful ways, offering additional material for students and instructors. Discrete math is an active subject with new discoveries made every year. The continual growth and updates to the web site reflect the active nature of the topics being discussed.The book is appropriate for a one- or two-term introductory discrete mathematics course to be taken by students in a wide variety of majors, including computer science, mathematics, and engineering. College Algebra is the only explicit prerequisite.

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

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Discrete Mathematics and Its Applications

Discrete Mathematics and Its Applications

9780072930337

0072930330

Summary

Table of Contents

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