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Discrete and Combinatorial Mathematics : An Applied Introduction

by: Grimaldi, Ralph P.

Discrete and Combinatorial Mathematics : An Applied Introduction

by: Grimaldi, Ralph P.

ISBN 13:
9780201199123
ISBN 10:
0201199122
Edition: 4th
Format: Hardcover
Copyright: 10/01/1998
Publisher: ADDISON WESLEY LONGMAN INC
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Summary

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.

PART 1 Fundamentals of Discrete Mathematics

(358)

1 Fundamental Principles of Counting

(44)

1.1 The Rules of Sum and Product

(3)

1.2 Permutations

(9)

1.3 Combinations: The Binomial Theorem

(12)

1.4 Combinations with Repetition

(9)

1.5 An Application in the Physical Sciences (Optional)

(1)

1.6 The Catalan Numbers (Optional)

(5)

1.7 Summary and Historical Review

(6)

2 Fundamentals of Logic

(80)

2.1 Basic Connectives and Truth Tables

(9)

2.2 Logical Equivalence: The Laws of Logic

(12)

2.3 Logical Implication: Rules of Inference

(21)

2.4 The Use of Quantifiers

(18)

2.5 Quantifiers, Definitions, and the Proofs of Theorems

107

(16)

2.6 Summary and Historical Review

123

(4)

3 Set Theory

127

(36)

3.1 Sets and Subsets

127

(12)

3.2 Set Operations and the Laws of Set Theory

139

(12)

3.3 Counting and Venn Diagrams

151

(3)

3.4 A Word on Probability

154

(3)

3.5 Summary and Historical Review

157

(6)

4 Properties of the Integers: Mathematical Induction

163

(54)

4.1 The Well-Ordering Principle: Mathematical Induction

163

(15)

4.2 Recursive Definitions

178

(11)

4.3 The Division Algorithm: Prime Numbers

189

(11)

4.4 The Greatest Common Divisor: The Euclidean Algorithm

200

(7)

4.5 The Fundamental Theorem of Arithmetic

207

(5)

4.6 Summary and Historical Review

212

(5)

5 Relations and Functions

217

(64)

5.1 Cartesian Products and Relations

218

(4)

5.2 Functions: Plain and One-to-One

222

(8)

5.3 Onto Functions: Stirling Numbers of the Second Kind

230

(8)

5.4 Special Functions

238

(6)

5.5 The Pigeonhole Principle

244

(5)

5.6 Function Composition and Inverse Functions

249

(12)

5.7 Computational Complexity

261

(5)

5.8 Analysis of Algorithms

266

(7)

5.9 Summary and Historical Review

273

(8)

6 Languages: Finite State Machines

281

(30)

6.1 Language: The Set Theory of Strings

282

(10)

6.2 Finite State Machines: A First Encounter

292

(7)

6.3 Finite State Machines: A Second Encounter

299

(7)

6.4 Summary and Historical Review

306

(5)

7 Relations: The Second Time Around

311

(48)

7.1 Relations Revisited: Properties of Relations

311

(8)

7.2 Computer Recognition: Zero-One Matrices and Directed Graphs

319

(12)

7.3 Partial Orders: Hasse Diagrams

331

(10)

7.4 Equivalence Relations and Partitions

341

(5)

7.5 Finite State Machines: The Minimization Process

346

(6)

7.6 Summary and Historical Review

352

(7)

PART 2 Further Topics in Enumeration

359

(116)

8 The Principle of Inclusion and Exclusion

361

(26)

8.1 The Principle of Inclusion and Exclusion

361

(8)

8.2 Generalizations of the Principle

369

(5)

8.3 Derangements: Nothing Is in Its Right Place

374

(2)

8.4 Rook Polynomials

376

(3)

8.5 Arrangements with Forbidden Positions

379

(4)

8.6 Summary and Historical Review

383

(4)

9 Generating Functions

387

(28)

9.1 Introductory Examples

387

(3)

9.2 Definition and Examples: Calculational Techniques

390

(10)

9.3 Partitions of Integers

400

(3)

9.4 The Exponential Generating Function

403

(5)

9.5 The Summation Operator

408

(1)

9.6 Summary and Historical Review

409

(6)

10 Recurrence Relations

415

(60)

10.1 The First-Order Linear Recurrence Relation

415

(9)

10.2 The Second-Order Linear Homogeneous Recurrence Relation with Constant Coefficients

424

(10)

10.3 The Nonhomogeneous Recurrence Relation

434

(11)

10.4 The Method of Generating Functions

445

(5)

10.5 A Special Kind of Nonlinear Recurrence Relation (Optional)

450

(10)

10.6 Divide-and-Conquer Algorithms (Optional)

460

(9)

10.7 Summary and Historical Review

469

(6)

PART 3 Graph Theory and Applications

475

(156)

11 An Introduction to Graph Theory

477

(70)

11.1 Definitions and Examples

477

(8)

11.2 Subgraphs, Complements, and Graph Isomorphism

485

(10)

11.3 Vertex Degree: Euler Trails and Circuits

495

(10)

11.4 Planar Graphs

505

(18)

11.5 Hamilton Paths and Cycles

523

(9)

11.6 Graph Coloring and Chromatic Polynomials

532

(8)

11.7 Summary and Historical Review

540

(7)

12 Trees

547

(44)

12.1 Definitions, Properties, and Examples

547

(6)

12.2 Rooted Trees

553

(17)

12.3 Trees and Sorting

570

(5)

12.4 Weighted Trees and Prefix Codes

575

(5)

12.5 Biconnected Components and Articulation Points

580

(5)

12.6 Summary and Historical Review

585

(6)

13 Optimization and Matching

591

(40)

13.1 Dijkstra's Shortest-Path Algorithm

591

(8)

13.2 Minimal Spanning Trees: The Algorithms of Kruskal and Prim

599

(6)

13.3 Transport Networks: The Max-Flow Min-Cut Theorem

605

(11)

13.4 Matching Theory

616

(9)

13.5 Summary and Historical Review

625

(6)

PART 4 Modern Applied Algebra

631

14 Rings and Modular Arithmetic

633

(30)

14.1 The Ring Structure: Definition and Examples

633

(6)

14.2 Ring Properties and Substructures

639

(7)

14.3 The Integers Modulo n

646

(6)

14.4 Ring Homomorphisms and Isomorphisms

652

(5)

14.5 Summary and Historical Review

657

(6)

15 Boolean Algebra and Switching Functions

663

(38)

15.1 Switching Functions: Disjunctive and Conjunctive Normal Forms

663

(9)

15.2 Gating Networks: Minimal Sums of Products: Karnaugh Maps

672

(10)

15.3 Further Applications: Don't-Care Conditions

682

(5)

15.4 The Structure of a Boolean Algebra (Optional)

687

(9)

15.5 Summary and Historical Review

696

(5)

16 Groups, Coding Theory, and Polya's Method of Enumeration

701

(54)

16.1 Definition, Examples, and Elementary Properties

701

(7)

16.2 Homomorphisms, Isomorphisms, and Cyclic Groups

708

(5)

16.3 Cosets and Lagrange's Theorem

713

(2)

16.4 Elements of Coding Theory

715

(5)

16.5 The Hamming Metric

720

(2)

16.6 The Parity-Check and Generator Matrices

722

(5)

16.7 Group Codes: Decoding with Coset Leaders

727

(4)

16.8 Hamming Matrices

731

(2)

16.9 Counting and Equivalence: Burnside's Theorem

733

(7)

16.10 The Cycle Index

740

(4)

16.11 The Pattern Inventory: Polya's Method of Enumeration

744

(5)

16.12 Summary and Historical Review

749

(6)

17 Finite Fields and Combinatorial Designs

755

17.1 Polynomial Rings

755

(8)

17.2 Irreducible Polynomials: Finite Fields

763

(9)

17.3 Latin Squares

772

(6)

17.4 Finite Geometries and Affine Planes

778

(5)

17.5 Block Designs and Projective Planes

783

(5)

17.6 Summary and Historical Review

788

Appendix 1 Exponential and Logarithmic Functions

A-1

(10)

Appendix 2 Matrices, Matrix Operations, and Determinants

A-11

(14)

Appendix 3 Countable and Uncountable Sets

A-25

Solutions

S-1

Index

I-1

Amazon no longer offers textbook rentals. We do!

Discrete and Combinatorial Mathematics : An Applied Introduction

Discrete and Combinatorial Mathematics : An Applied Introduction

9780201199123

0201199122

Summary

Table of Contents

Supplemental Materials

Write a Review

Amazon no longer offers textbook rentals. We do!

Discrete and Combinatorial Mathematics : An Applied Introduction

Discrete and Combinatorial Mathematics : An Applied Introduction

9780201199123

0201199122

Summary

Table of Contents

Supplemental Materials

Write a Review

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