ISBN13: 978-0201199123

ISBN10: 0201199122

Cover type:

Edition/Copyright: 4TH 99

Publisher: Addison-Wesley Longman, Inc.

Published: 1999

International: No

ISBN10: 0201199122

Cover type:

Edition/Copyright: 4TH 99

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

ISBN10: 0201199122

Cover type:

Edition/Copyright: 4TH 99

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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