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ISBN13: 978-0201199123

ISBN10: 0201199122

Edition: 4TH 99

Copyright: 1999

Publisher: Addison-Wesley Longman, Inc.

Published: 1999

International: No

ISBN10: 0201199122

Edition: 4TH 99

Copyright: 1999

Publisher: Addison-Wesley Longman, Inc.

Published: 1999

International: No

This fourth edition continues to improve on the features that have made it the market leader. The text offers a flexible organization, enabling instructors to adapt the book to their particular courses: discrete mathematics, graph theory, modern algebra, and/or combinatorics. More elementary problems were added, creating a greater variety of level in problem sets, which allows students to perfect skills as they practice. This new edition continues to feature numerous computer science applications--making this the ideal text for preparing students for advanced study.

*FEATURES:*

- This text has an enhanced mathematical approach, with carefully thought out examples, including many examples with computer sciences applications.
- Historical reviews and biographies bring a human element to their assignments.
- Chapter summaries allow students to review what they have learned.

Author Bio

**Grimaldi, Ralph P. : Rose-Hulman Institute of Technology**

**Chapter 1: Fundamentals of Discrete Mathematics**

Fundamental Principles of Counting

The Rules of Sum and Product

Permutations

Combinations:

The Binomial Theorem

Combinations with Repetition: Distributions

An Application in the Physical Sciences (Optional)

Catalan Numbers

**Chapter 2: Fundamentals of Logic**

Basic Connectives and Truth Tables

Logical Equivalence: The Laws of Logic

Logical Implication: Rules of Inference

The Use of Quantifiers

Quantifiers, Definitions, and the Proofs of Theorems

**Chapter 3: Set Theory**

Sets and Subsets

Set Operations and the Laws of Set Theory

Counting and Venn Diagrams

A Word on Probability

**Chapter 4: Properties of the Integers: Mathematical Induction**

The Well-Ordering Principle: Mathematical Induction

Recursive Definitions

The Division Algorithm: Prime Numbers

The Greatest Common Divisor: The Euclidean Algorithm

The Fundamental Theorem of Arithmetic

**Chapter 5: Relations and Functions**

Cartesian Products and Relations

Functions: Plain and One-to-One

Onto Functions: Stirling Numbers of the Second Kind

Special Functions

The Pigeonhole Principle

Function Composition and Inverse Functions

Computational Complexity

Analysis of Algorithms

**Chapter 6: Languages: Finite State Machines**

Language: The Set Theory of Strings

Finite State Machines: A First Encounter

Finite State Machines: A Second Encounter

Relations: The Second Time Around

Relations Revisited: Properties of Relations

Computer Recognition: ZeroOne Matrices and Directed Graphs

Partial Orders: Hasse Diagrams

Equivalence Relations and Partitions

Finite State Machines: The Minimization Process

**Chapter 7: Further Topics in Enumeration**

The Principle of Inclusion and Exclusion

Generalizations of the Principle (Optional)

Derangements: Nothing Is in Its Right Place

Rook Polynomials

Arrangements with Forbidden Positions

**Chapter 8: Generating Functions**

Introductory Examples

Definition and Examples: Calculational Techniques

Partitions of Integers

Exponential Generating Functions

The Summation Operator

**Chapter 9: Recurrence Relations**

The First-Order Linear Recurrence Relation

The Second-Order Linear Recurrence Relation with Constant Coefficients

The Nonhomogeneous Recurrence Relation

The Method of Generating Functions

A Special Kind of Nonlinear Recurrence Relation (Optional)

Divide and Conquer Algorithms (Optional)

**Chapter 10: Graph Theory and Applications**

An Introduction to Graph Theory

Definitions and Examples

Subgraphs, Complements, and Graph Isomorphism

Vertex Degree: Euler Trails and Circuits

Planar Graphs

Hamilton Paths and Cycles

Graph Coloring and Chromatic Polynomials

**Chapter 11: Trees**

Definitions, Properties, and Examples

Rooted Trees

Trees and Sorting Algorithms

Weighted Trees and Prefix Codes

Biconnected Components and Articulation Points

**Chapter 12: Optimization and Matching**

Dijkstra's Shortest Path Algorithm

Minimal Spanning Trees

Transport Networks: The Max-Flow Min-Cut Theorem

Matching Theory

**Chapter 13: Modern Applied Algebra**

Rings and Modular Arithmetic

The Ring Structure: Definition and Examples

Ring Properties and Substructures

The Integers Modulo n

Ring Homomorphisms and Isomorphisms

**Chapter 14: Boolean Algebra and Switching Functions**

Switching Functions: Disjunctive and Conjunctive Normal Forms

Gating Networks: Minimal Sums of Products: Karnaugh Maps

Further Applications: Don't Care Conditions

The Structure of a Boolean Algebra (Optional)

**Chapter 15: Groups, Coding Theory, and Polya's Method of Enumeration**

Definition, Examples, and Elementary Properties

Homomorphisms, Isomorphisms, and Cyclic Groups

Cosets and Lagrange's Theorem

Elements of Coding Theory

The Hamming Metric

The Parity-Check and Generator trices

Group Codes: Decoding with Coset Leaders

Hamming Matrices Counting and Equivalence: Burnside's Theorem

The Cycle Index

The Pattern Inventory: Polya's Method of Enumeration

**Chapter 16: Finite Fields and Combinatorial Designs**

Polynomial Rings

Irreducible Polynomials: Finite Fields

Latin Squares

Finite Geometries and Affine Planes

Block Designs and Projective Planes

Appendices

Exponential and Logarithmic Functions

Matrices, Matrix Operations, and Determinants

Countable and Uncountable Sets

Solutions

Index

**Other Editions for Discrete and Combinatorial Mathematics : An Applied Introduction**

ISBN10: 0201199122

Edition: 4TH 99

Copyright: 1999

Publisher: Addison-Wesley Longman, Inc.

Published: 1999

International: No

This fourth edition continues to improve on the features that have made it the market leader. The text offers a flexible organization, enabling instructors to adapt the book to their particular courses: discrete mathematics, graph theory, modern algebra, and/or combinatorics. More elementary problems were added, creating a greater variety of level in problem sets, which allows students to perfect skills as they practice. This new edition continues to feature numerous computer science applications--making this the ideal text for preparing students for advanced study.

*FEATURES:*

- This text has an enhanced mathematical approach, with carefully thought out examples, including many examples with computer sciences applications.
- Historical reviews and biographies bring a human element to their assignments.
- Chapter summaries allow students to review what they have learned.

Author Bio

**Grimaldi, Ralph P. : Rose-Hulman Institute of Technology**

Table of Contents

**Chapter 1: Fundamentals of Discrete Mathematics**

Fundamental Principles of Counting

The Rules of Sum and Product

Permutations

Combinations:

The Binomial Theorem

Combinations with Repetition: Distributions

An Application in the Physical Sciences (Optional)

Catalan Numbers

**Chapter 2: Fundamentals of Logic**

Basic Connectives and Truth Tables

Logical Equivalence: The Laws of Logic

Logical Implication: Rules of Inference

The Use of Quantifiers

Quantifiers, Definitions, and the Proofs of Theorems

**Chapter 3: Set Theory**

Sets and Subsets

Set Operations and the Laws of Set Theory

Counting and Venn Diagrams

A Word on Probability

**Chapter 4: Properties of the Integers: Mathematical Induction**

The Well-Ordering Principle: Mathematical Induction

Recursive Definitions

The Division Algorithm: Prime Numbers

The Greatest Common Divisor: The Euclidean Algorithm

The Fundamental Theorem of Arithmetic

**Chapter 5: Relations and Functions**

Cartesian Products and Relations

Functions: Plain and One-to-One

Onto Functions: Stirling Numbers of the Second Kind

Special Functions

The Pigeonhole Principle

Function Composition and Inverse Functions

Computational Complexity

Analysis of Algorithms

**Chapter 6: Languages: Finite State Machines**

Language: The Set Theory of Strings

Finite State Machines: A First Encounter

Finite State Machines: A Second Encounter

Relations: The Second Time Around

Relations Revisited: Properties of Relations

Computer Recognition: ZeroOne Matrices and Directed Graphs

Partial Orders: Hasse Diagrams

Equivalence Relations and Partitions

Finite State Machines: The Minimization Process

**Chapter 7: Further Topics in Enumeration**

The Principle of Inclusion and Exclusion

Generalizations of the Principle (Optional)

Derangements: Nothing Is in Its Right Place

Rook Polynomials

Arrangements with Forbidden Positions

**Chapter 8: Generating Functions**

Introductory Examples

Definition and Examples: Calculational Techniques

Partitions of Integers

Exponential Generating Functions

The Summation Operator

**Chapter 9: Recurrence Relations**

The First-Order Linear Recurrence Relation

The Second-Order Linear Recurrence Relation with Constant Coefficients

The Nonhomogeneous Recurrence Relation

The Method of Generating Functions

A Special Kind of Nonlinear Recurrence Relation (Optional)

Divide and Conquer Algorithms (Optional)

**Chapter 10: Graph Theory and Applications**

An Introduction to Graph Theory

Definitions and Examples

Subgraphs, Complements, and Graph Isomorphism

Vertex Degree: Euler Trails and Circuits

Planar Graphs

Hamilton Paths and Cycles

Graph Coloring and Chromatic Polynomials

**Chapter 11: Trees**

Definitions, Properties, and Examples

Rooted Trees

Trees and Sorting Algorithms

Weighted Trees and Prefix Codes

Biconnected Components and Articulation Points

**Chapter 12: Optimization and Matching**

Dijkstra's Shortest Path Algorithm

Minimal Spanning Trees

Transport Networks: The Max-Flow Min-Cut Theorem

Matching Theory

**Chapter 13: Modern Applied Algebra**

Rings and Modular Arithmetic

The Ring Structure: Definition and Examples

Ring Properties and Substructures

The Integers Modulo n

Ring Homomorphisms and Isomorphisms

**Chapter 14: Boolean Algebra and Switching Functions**

Switching Functions: Disjunctive and Conjunctive Normal Forms

Gating Networks: Minimal Sums of Products: Karnaugh Maps

Further Applications: Don't Care Conditions

The Structure of a Boolean Algebra (Optional)

**Chapter 15: Groups, Coding Theory, and Polya's Method of Enumeration**

Definition, Examples, and Elementary Properties

Homomorphisms, Isomorphisms, and Cyclic Groups

Cosets and Lagrange's Theorem

Elements of Coding Theory

The Hamming Metric

The Parity-Check and Generator trices

Group Codes: Decoding with Coset Leaders

Hamming Matrices Counting and Equivalence: Burnside's Theorem

The Cycle Index

The Pattern Inventory: Polya's Method of Enumeration

**Chapter 16: Finite Fields and Combinatorial Designs**

Polynomial Rings

Irreducible Polynomials: Finite Fields

Latin Squares

Finite Geometries and Affine Planes

Block Designs and Projective Planes

Appendices

Exponential and Logarithmic Functions

Matrices, Matrix Operations, and Determinants

Countable and Uncountable Sets

Solutions

Index

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**Other Editions for Discrete and Combinatorial Mathematics : An Applied Introduction**