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Edition: 5TH 03

Copyright: 2003

Publisher: McGraw-Hill Publishing Company

Published: 2003

International: No

Edition: 5TH 03

Copyright: 2003

Publisher: McGraw-Hill Publishing Company

Published: 2003

International: No

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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.

**New to This Edition :**

- Logic: Coverage of logic has been expanded to provide a more accessible approach to beginning students. There are now two sections devoted to quantifiers, specifically addressing the use of quantifiers in expressing statements of various types. Additional background has been supplied to help students translate English and mathematical statements into statements involving predicates and quantifiers. Material has also been added on using logic to express and work with system specifications, directly relating the topic to the computer science field.
- Mathematical Reasoning: Coverage of the rules of inference and basic types of proof has been expanded and moved into Chapter 1. This allows students a better understanding of proofs of facts about sets and functions in Chapter 1, and in number theory in Chapter 2. Sequences and summations has been moved to Chapter 3 where it immediately precedes the material on mathematical induction. The presentation of mathematical induction has been enhanced with further explanations and interesting examples.
- Probability: Coverage of discrete probability and aspects of probability theory have been expanded and placed in their own chapter (Chapter 5). This allows for more thorough coverage of probability while increasing the accessibility to counting techniques.
- Web site: This text continues its efforts to support and expand upon the textual material with enhanced coverage of topics on its web site. Features include a detailed Web Resources Guide, Additional Exercises and Answers, a Guide to Writing Proofs, and many other elements. The site will continue its expansion throughout the life of the edition, providing tools that students and instructors can use to assess student's understanding of key topics--including implications, quantifiers, proofs, mathematical introduction, and counting problems. Extensive additional examples in key areas and more detailed explanations of key examples in the text will also be provided.

**Features :**

- Accessibility: This text has proven to be easily read and understood by beginning students with no mathematical background beyond college algebra. Once basic mathematical concepts have been carefully developed, more difficult material and applications to other areas of study are presented. There are a few places in the text where calculus is referred to and these places are carefully noted.
- Flexibility: This text has been carefully designed for flexible use. Each chapter is divided into sections of approximately the same length, and each section is divided into subsections that form natural blocks of material teaching. The dependence of chapters on previous material has been minimized.
- Worked Examples: Thorough attention has been paid to the 650+ examples throughout the text. Many new examples have been added, as well as a focused expansion of key examples. These examples are used to illustrate concepts, relate different topics, and introduce applications. In the examples, a question is first posed, then its solution is presented with the appropriate amount of detail.
- Applications: The applications included in this text demonstrate the utility of discrete mathematics in the solution of real-world problems. This text includes applications to a wide variety of areas, including computer science, data networking, psychology, chemistry, engineering, linguistics, biology, business, and the Internet.
- Exercises: There are over 3000 exercises in the text. There is an ample supply of straightforward exercises that develop basic skills, a large number of intermediate exercises, and many challenging exercises. Exercises are stated clearly and unambiguously, and all are carefully graded for level of difficulty. Exercises sets contain special discussions, with exercises, that develop new concepts not covered in the text, permitting students to discover new ideas through their own work. Supplementary exercises follow each chapter and provide a rich and varied set of additional exercises. These exercises are generally more difficult than those in the section exercise sets and integrate different topics more effectively.
- Computer Projects: Each chapter is followed by a set of computer projects. These computer projects tie together what students may have learned in computing and in discrete mathematics. Computer projects that are more difficult than average, from both a mathematical and a programming point of view, are marked with a star, and those that are extremely challenging are marked with two stars.
- Computations and Explorations: A set of computations and explorations is included at the conclusion of each chapter. These exercises are designed to be completed using existing software tools, such as programs that students or instructors have written or mathematical computation packages such as MAPLE or Mathematica.
- Writing Projects: Each chapter is followed by a set of writing projects. To do these projects students need to consult the mathematical literature. Some of these projects are historical in nature and may involve looking up original sources. Others are designed to serve as gateways to new topics and ideas. All are designed to expose students to ideas not covered in depth in the text.
- Historical Information: The background of many topics is succinctly described in the text. Brief biographies of more than 55 mathematicians and computer scientists are included as footnotes. These biographies include information about the lives, careers, and accomplishments of these important contributors to discrete mathematics. In addition, numerous historical footnotes are included that supplement the historical information in the main body of the text.

**Rosen, Kenneth H. : AT&T Laboratories**

Discrete Mathematics and Its Applications, Fifth Edition

Chapter 1: The Foundations: Logic, 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

Chapter 2: The Fundamentals: Algorithms, the Integers, and Matrices

2.1, Algorithms

2.2, The Growth of Functions

2.3, Complexity of Algorithms

2.4, Integers and Algorithms

2.5, Applications of Number Theory

2.6, Matrices

Chapter 3: Mathematical Reasoning, Induction, and Recursion

3.1, Art and Strategy of Proof

3.2, Sequences and Sums

3.3, Mathematical Induction

3.4, Recursive Definitions and Structural Induction

3.5, Recursive Algorithms

3.6, Program Correctness

Chapter 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

Chapter 5: Discrete Probability

5.1, An Introduction to Discrete Probability

5.2, Probability Theory

5.3, Expected Value and Variance

Chapter 6: Advanced Counting Techniques

6.1, Recurrence Relations

6.2, Solving Recurrence Relations

6.3, Divide-and-Conquer Relations

6.4, Generating Functions

6.5, Inclusion-Exclusion

6.6, Applications of Inclusion-Exclusion

Chapter 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

Chapter 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

Chapter 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

Chapter 10: Boolean Algebra

10.1, Boolean Functions

10.2, Representing Boolean Functions

10.3, Logic Gates

10.4, Minimization of Circuits

Chapter 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

Appendix.1: Exponential and Logarithmic Functions

Appendix.2: Pseudocode

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Summary

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.

**New to This Edition :**

- Logic: Coverage of logic has been expanded to provide a more accessible approach to beginning students. There are now two sections devoted to quantifiers, specifically addressing the use of quantifiers in expressing statements of various types. Additional background has been supplied to help students translate English and mathematical statements into statements involving predicates and quantifiers. Material has also been added on using logic to express and work with system specifications, directly relating the topic to the computer science field.
- Mathematical Reasoning: Coverage of the rules of inference and basic types of proof has been expanded and moved into Chapter 1. This allows students a better understanding of proofs of facts about sets and functions in Chapter 1, and in number theory in Chapter 2. Sequences and summations has been moved to Chapter 3 where it immediately precedes the material on mathematical induction. The presentation of mathematical induction has been enhanced with further explanations and interesting examples.
- Probability: Coverage of discrete probability and aspects of probability theory have been expanded and placed in their own chapter (Chapter 5). This allows for more thorough coverage of probability while increasing the accessibility to counting techniques.
- Web site: This text continues its efforts to support and expand upon the textual material with enhanced coverage of topics on its web site. Features include a detailed Web Resources Guide, Additional Exercises and Answers, a Guide to Writing Proofs, and many other elements. The site will continue its expansion throughout the life of the edition, providing tools that students and instructors can use to assess student's understanding of key topics--including implications, quantifiers, proofs, mathematical introduction, and counting problems. Extensive additional examples in key areas and more detailed explanations of key examples in the text will also be provided.

**Features :**

- Accessibility: This text has proven to be easily read and understood by beginning students with no mathematical background beyond college algebra. Once basic mathematical concepts have been carefully developed, more difficult material and applications to other areas of study are presented. There are a few places in the text where calculus is referred to and these places are carefully noted.
- Flexibility: This text has been carefully designed for flexible use. Each chapter is divided into sections of approximately the same length, and each section is divided into subsections that form natural blocks of material teaching. The dependence of chapters on previous material has been minimized.
- Worked Examples: Thorough attention has been paid to the 650+ examples throughout the text. Many new examples have been added, as well as a focused expansion of key examples. These examples are used to illustrate concepts, relate different topics, and introduce applications. In the examples, a question is first posed, then its solution is presented with the appropriate amount of detail.
- Applications: The applications included in this text demonstrate the utility of discrete mathematics in the solution of real-world problems. This text includes applications to a wide variety of areas, including computer science, data networking, psychology, chemistry, engineering, linguistics, biology, business, and the Internet.
- Exercises: There are over 3000 exercises in the text. There is an ample supply of straightforward exercises that develop basic skills, a large number of intermediate exercises, and many challenging exercises. Exercises are stated clearly and unambiguously, and all are carefully graded for level of difficulty. Exercises sets contain special discussions, with exercises, that develop new concepts not covered in the text, permitting students to discover new ideas through their own work. Supplementary exercises follow each chapter and provide a rich and varied set of additional exercises. These exercises are generally more difficult than those in the section exercise sets and integrate different topics more effectively.
- Computer Projects: Each chapter is followed by a set of computer projects. These computer projects tie together what students may have learned in computing and in discrete mathematics. Computer projects that are more difficult than average, from both a mathematical and a programming point of view, are marked with a star, and those that are extremely challenging are marked with two stars.
- Computations and Explorations: A set of computations and explorations is included at the conclusion of each chapter. These exercises are designed to be completed using existing software tools, such as programs that students or instructors have written or mathematical computation packages such as MAPLE or Mathematica.
- Writing Projects: Each chapter is followed by a set of writing projects. To do these projects students need to consult the mathematical literature. Some of these projects are historical in nature and may involve looking up original sources. Others are designed to serve as gateways to new topics and ideas. All are designed to expose students to ideas not covered in depth in the text.
- Historical Information: The background of many topics is succinctly described in the text. Brief biographies of more than 55 mathematicians and computer scientists are included as footnotes. These biographies include information about the lives, careers, and accomplishments of these important contributors to discrete mathematics. In addition, numerous historical footnotes are included that supplement the historical information in the main body of the text.

Author Bio

**Rosen, Kenneth H. : AT&T Laboratories**

Table of Contents

Discrete Mathematics and Its Applications, Fifth Edition

Chapter 1: The Foundations: Logic, 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

Chapter 2: The Fundamentals: Algorithms, the Integers, and Matrices

2.1, Algorithms

2.2, The Growth of Functions

2.3, Complexity of Algorithms

2.4, Integers and Algorithms

2.5, Applications of Number Theory

2.6, Matrices

Chapter 3: Mathematical Reasoning, Induction, and Recursion

3.1, Art and Strategy of Proof

3.2, Sequences and Sums

3.3, Mathematical Induction

3.4, Recursive Definitions and Structural Induction

3.5, Recursive Algorithms

3.6, Program Correctness

Chapter 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

Chapter 5: Discrete Probability

5.1, An Introduction to Discrete Probability

5.2, Probability Theory

5.3, Expected Value and Variance

Chapter 6: Advanced Counting Techniques

6.1, Recurrence Relations

6.2, Solving Recurrence Relations

6.3, Divide-and-Conquer Relations

6.4, Generating Functions

6.5, Inclusion-Exclusion

6.6, Applications of Inclusion-Exclusion

Chapter 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

Chapter 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

Chapter 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

Chapter 10: Boolean Algebra

10.1, Boolean Functions

10.2, Representing Boolean Functions

10.3, Logic Gates

10.4, Minimization of Circuits

Chapter 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

Appendix.1: Exponential and Logarithmic Functions

Appendix.2: Pseudocode

Publisher Info

Publisher: McGraw-Hill Publishing Company

Published: 2003

International: No

Published: 2003

International: No

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.

**New to This Edition :**

- Logic: Coverage of logic has been expanded to provide a more accessible approach to beginning students. There are now two sections devoted to quantifiers, specifically addressing the use of quantifiers in expressing statements of various types. Additional background has been supplied to help students translate English and mathematical statements into statements involving predicates and quantifiers. Material has also been added on using logic to express and work with system specifications, directly relating the topic to the computer science field.
- Mathematical Reasoning: Coverage of the rules of inference and basic types of proof has been expanded and moved into Chapter 1. This allows students a better understanding of proofs of facts about sets and functions in Chapter 1, and in number theory in Chapter 2. Sequences and summations has been moved to Chapter 3 where it immediately precedes the material on mathematical induction. The presentation of mathematical induction has been enhanced with further explanations and interesting examples.
- Probability: Coverage of discrete probability and aspects of probability theory have been expanded and placed in their own chapter (Chapter 5). This allows for more thorough coverage of probability while increasing the accessibility to counting techniques.
- Web site: This text continues its efforts to support and expand upon the textual material with enhanced coverage of topics on its web site. Features include a detailed Web Resources Guide, Additional Exercises and Answers, a Guide to Writing Proofs, and many other elements. The site will continue its expansion throughout the life of the edition, providing tools that students and instructors can use to assess student's understanding of key topics--including implications, quantifiers, proofs, mathematical introduction, and counting problems. Extensive additional examples in key areas and more detailed explanations of key examples in the text will also be provided.

**Features :**

- Accessibility: This text has proven to be easily read and understood by beginning students with no mathematical background beyond college algebra. Once basic mathematical concepts have been carefully developed, more difficult material and applications to other areas of study are presented. There are a few places in the text where calculus is referred to and these places are carefully noted.
- Flexibility: This text has been carefully designed for flexible use. Each chapter is divided into sections of approximately the same length, and each section is divided into subsections that form natural blocks of material teaching. The dependence of chapters on previous material has been minimized.
- Worked Examples: Thorough attention has been paid to the 650+ examples throughout the text. Many new examples have been added, as well as a focused expansion of key examples. These examples are used to illustrate concepts, relate different topics, and introduce applications. In the examples, a question is first posed, then its solution is presented with the appropriate amount of detail.
- Applications: The applications included in this text demonstrate the utility of discrete mathematics in the solution of real-world problems. This text includes applications to a wide variety of areas, including computer science, data networking, psychology, chemistry, engineering, linguistics, biology, business, and the Internet.
- Exercises: There are over 3000 exercises in the text. There is an ample supply of straightforward exercises that develop basic skills, a large number of intermediate exercises, and many challenging exercises. Exercises are stated clearly and unambiguously, and all are carefully graded for level of difficulty. Exercises sets contain special discussions, with exercises, that develop new concepts not covered in the text, permitting students to discover new ideas through their own work. Supplementary exercises follow each chapter and provide a rich and varied set of additional exercises. These exercises are generally more difficult than those in the section exercise sets and integrate different topics more effectively.
- Computer Projects: Each chapter is followed by a set of computer projects. These computer projects tie together what students may have learned in computing and in discrete mathematics. Computer projects that are more difficult than average, from both a mathematical and a programming point of view, are marked with a star, and those that are extremely challenging are marked with two stars.
- Computations and Explorations: A set of computations and explorations is included at the conclusion of each chapter. These exercises are designed to be completed using existing software tools, such as programs that students or instructors have written or mathematical computation packages such as MAPLE or Mathematica.
- Writing Projects: Each chapter is followed by a set of writing projects. To do these projects students need to consult the mathematical literature. Some of these projects are historical in nature and may involve looking up original sources. Others are designed to serve as gateways to new topics and ideas. All are designed to expose students to ideas not covered in depth in the text.
- Historical Information: The background of many topics is succinctly described in the text. Brief biographies of more than 55 mathematicians and computer scientists are included as footnotes. These biographies include information about the lives, careers, and accomplishments of these important contributors to discrete mathematics. In addition, numerous historical footnotes are included that supplement the historical information in the main body of the text.

**Rosen, Kenneth H. : AT&T Laboratories**

Chapter 1: The Foundations: Logic, 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

Chapter 2: The Fundamentals: Algorithms, the Integers, and Matrices

2.1, Algorithms

2.2, The Growth of Functions

2.3, Complexity of Algorithms

2.4, Integers and Algorithms

2.5, Applications of Number Theory

2.6, Matrices

Chapter 3: Mathematical Reasoning, Induction, and Recursion

3.1, Art and Strategy of Proof

3.2, Sequences and Sums

3.3, Mathematical Induction

3.4, Recursive Definitions and Structural Induction

3.5, Recursive Algorithms

3.6, Program Correctness

Chapter 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

Chapter 5: Discrete Probability

5.1, An Introduction to Discrete Probability

5.2, Probability Theory

5.3, Expected Value and Variance

Chapter 6: Advanced Counting Techniques

6.1, Recurrence Relations

6.2, Solving Recurrence Relations

6.3, Divide-and-Conquer Relations

6.4, Generating Functions

6.5, Inclusion-Exclusion

6.6, Applications of Inclusion-Exclusion

Chapter 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

Chapter 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

Chapter 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

Chapter 10: Boolean Algebra

10.1, Boolean Functions

10.2, Representing Boolean Functions

10.3, Logic Gates

10.4, Minimization of Circuits

Chapter 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

Appendix.1: Exponential and Logarithmic Functions

Appendix.2: Pseudocode