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Mathematical Structures for Computer Science : A Modern Treatment of Discrete Mathematics

by: Gersting

Mathematical Structures for Computer Science : A Modern Treatment of Discrete Mathematics

by: Gersting

ISBN 13:
9780716743583
ISBN 10:
0716743582
Edition: 5th
Format: Hardcover
Copyright: 07/19/2002
Publisher: W. H. Freeman
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Summary

New edition of the classic discrete mathematics text for computer science majors.

Preface

xvii

Note to the Student

Formal Logic

(82)

Statements, Symbolic Representation, and Tautologies

(18)

Connectives and Truth Values

(6)

Tautologies

(2)

Logical Connectives in the Real World

(1)

An Algorithm

(3)

Exercises 1.1

(6)

Propositional Logic

(13)

Valid Arguments

(2)

Derivation Rules for Propositional Logic

(4)

Deduction Method and Other Rules

(2)

Verbal Arguments

(2)

Exercises 1.2

(3)

Quantifiers, Predicates, and Validity

(12)

Quantifiers and Predicates

(3)

Translation

(3)

Validity

(2)

Exercises 1.3

(4)

Predicate Logic

(15)

Derivation Rules for Predicate Logic

(1)

Universal Instantiation

(1)

Existential Instantiation

(1)

Universal Generalization

(1)

Existential Generalization

(1)

More Work with Rules

(5)

Verbal Arguments

(1)

Conclusion

(2)

Exercises 1.4

(3)

Logic Programming

(10)

Prolog

(2)

Horn Clauses and Resolution

(4)

Recursion

(2)

Expert Systems

(1)

Exercises 1.56

(1)

Proof of Correctness

(13)

Assertions

(1)

Assignment Rule

(3)

Conditional Rule

(3)

Exercises 1.6

(2)

Chapter 1 Review

(2)

On the Computer

(1)

Proofs, Recursion, and Analysis of Algorithms

(79)

Proof Techniques

(10)

Theorems and Informal Proofs

(1)

To Prove or Not to Prove

(1)

Exhaustive Proof

(1)

Direct Proof

(1)

Contraposition

(1)

Contradiction

(2)

Serendipity

(1)

Exercises 2.1

(2)

Induction

(15)

First Principle of Induction

(2)

Proofs by Mathematical Induction

(5)

Second Principle of Induction

101

(4)

Exercises 2.2

105

(4)

More on Proof of Correctness

109

(11)

Loop Rule

110

(3)

Euclidean Algorithm

113

(3)

Exercises 2.3

116

(4)

Recursion and Recurrence Relations

120

(28)

Recursive Definitions

120

(1)

Recursively Defined Sequences

121

(2)

Recursively Defined Sets

123

(2)

Recursively Defined Operations

125

(1)

Recursively Defined Algorithms

126

(5)

Solving Recurrence Relations

131

(8)

Exercises 2.4

139

(9)

Analysis of Algorithms

148

(14)

Analysis Using Recurrence Relations (Binary Search)

149

(4)

Upper Bound (Euclidean Algorithm)

153

(1)

Exercises 2.5

154

(3)

Chapter 2 Review

157

(2)

On the Computer

159

(3)

Sets, Combinatorics, and Probability

162

(84)

Sets

163

(25)

Notation

163

(2)

Relationships Between Sets

165

(3)

Sets of Sets

168

(1)

Binary and Unary Operations

168

(2)

Operations on Sets

170

(3)

Set Identities

173

(2)

Countable and Uncountable Sets

175

(3)

Exercises 3.1

178

(10)

Counting

188

(13)

Multiplication Principle

188

(4)

Addition Principle

192

(1)

Using the Principles Together

193

(2)

Decision Trees

195

(1)

Exercises 3.2

196

(5)

Principle of Inclusion and Exclusion; Pigeonhole Principle

201

(8)

Principle of Inclusion and Exclusion

201

(5)

Pigeonhole Principle

206

(1)

Exercises 3.3

207

(2)

Permutations and Combinations

209

(14)

Permutations

209

(2)

Combinations

211

(3)

Eliminating Duplicates

214

(1)

Permutations and Combinations with Repetitions

215

(2)

Exercises 3.4

217

(6)

Probability

223

(13)

Introduction to Finite Probability

223

(2)

Probability Distributions

225

(2)

Conditional Probability

227

(1)

Expected Value

228

(2)

Average Case Analysis of Algorithms

230

(2)

Exercises 3.5

232

(4)

Binomial Theorem

236

(10)

Pascal's Triangle

236

(2)

Binomial Theorem and Its Proof

238

(1)

Applying the Binomial Theorem

239

(1)

Exercises 3.6

240

(2)

Chapter 3 Review

242

(2)

On the Computer

244

(2)

Relations, Functions, and Matrices

246

(93)

Relations

247

(24)

Binary Relations

247

(3)

Properties of Relations

250

(2)

Closures of Relations

252

(2)

Partial Orderings

254

(2)

Equivalence Relations

256

(6)

Exercises 4.1

262

(9)

Topological Sorting

271

(7)

Exercises 4.2

276

(2)

Relations and Databases

278

(11)

Entity-Relationship Model

278

(1)

Relational Model

279

(3)

Operations on Relations

282

(3)

Null Values and Three-Valued Logic

285

(1)

Database Integrity

286

(1)

Exercises 4.3

287

(2)

Functions

289

(32)

Definition

289

(6)

Properties of Functions

295

(1)

Onto Functions

295

(1)

One-to-One Functions

296

(1)

Bijections

297

(1)

Composition of Functions

297

(2)

Inverse Functions

299

(1)

Permutation Functions

300

(3)

How Many Functions

303

(3)

Equivalent Sets

306

(1)

Order of Magnitude of Functions

307

(5)

Exercises 4.4

312

(9)

Matrices

321

(18)

Terminology

321

(2)

Matrix Operations

323

(4)

Boolean Matrices

327

(2)

Exercises 4.5

329

(5)

Chapter 4 Review

334

(2)

On the Computer

336

(3)

Graphs and Trees

339

(71)

Graphs and Their Representations

340

(31)

Definitions of a Graph

340

(3)

Applications of Graphs

343

(2)

Graph Terminology

345

(4)

Isomorphic Graphs

349

(3)

Planar Graphs

352

(5)

Computer Representation of Graphs

357

(1)

Adjacency Matrix

357

(2)

Adjacency List

359

(2)

Exercises 5.1

361

(10)

Trees and Their Representations

371

(16)

Tree Terminology

371

(1)

Applications of Trees

372

(2)

Binary Tree Representation

374

(1)

Tree Traversal Algorithms

375

(5)

Results About Trees

380

(1)

Exercises 5.2

381

(6)

Decision Trees

387

(10)

Searching

388

(2)

Lower Bounds on Searching

390

(1)

Binary Tree Search

391

(1)

Sorting

392

(2)

Exercises 5.3

394

(3)

Huffman Codes

397

(13)

Problem and Trial Solution

397

(2)

Huffman Encoding Algorithm

399

(2)

Justification

401

(2)

Applications of Huffman Codes

403

(1)

Exercises 5.4

404

(3)

Chapter 5 Review

407

(1)

On the Computer

408

(2)

Graph Algorithms

410

(51)

Directed Graphs and Binary Relations; Warshall's Algorithm

411

(14)

Directed Graphs and Binary Relations

412

(2)

Reachability

414

(4)

Warshall's Algorithm

418

(4)

Exercises 6.1

422

(3)

Euler Path and Hamiltonian Circuit

425

(8)

Euler Path Problem

425

(4)

Hamiltonian Circuit Problem

429

(1)

Exercises 6.2

430

(3)

Shortest Path and Minimal Spanning Tree

433

(14)

Shortest-Path Problem

433

(6)

Minimal Spanning Tree Problem

439

(2)

Exercises 6.3

441

(6)

Traversal Algorithms

447

(14)

Depth-First Search

447

(2)

Breadth-First Search

449

(3)

Analysis

452

(1)

Applications

453

(3)

Exercises 6.4

456

(2)

Chapter 6 Review

458

(1)

On the Computer

459

(2)

Boolean Algebra and Computer Logic

461

(57)

Boolean Algebra Structure

462

(17)

Models or Abstractions

463

(1)

Definition and Properties

464

(4)

Isomorphic Boolean Algebras

468

(1)

What Is Isomorphism?

468

(3)

Isomorphism as Applied to Boolean Algebra

471

(3)

Exercises 7.1

474

(5)

Logic Networks

479

(21)

Combinational Networks

479

(1)

Basic Logic Elements

479

(1)

Boolean Expressions

480

(1)

Truth Functions

481

(1)

Networks and Expressions

481

(2)

Canonical Form

483

(2)

Minimization

485

(2)

Programmable Logic Arrays

487

(2)

A Useful Network

489

(2)

Other Logic Elements

491

(1)

Constructing Truth Functions

492

(1)

Exercises 7.2

493

(7)

Minimization

500

(18)

Minimization Process

500

(1)

Karnaugh Map

501

(1)

Maps for Three and Four Variables

502

(2)

Using the Karnaugh Map

504

(4)

Quine-McCluskey Procedure

508

(4)

Exercises 7.3

512

(3)

Chapter 7 Review

515

(1)

On the Computer

516

(2)

Modeling Arithmetic, Computation, and Languages

518

(94)

Algebraic Structures

519

(25)

Definitions and Examples

519

(8)

Basic Results about Groups

527

(3)

Subgroups

530

(3)

Isomorphic Groups

533

(6)

Exercises 8.1

539

(5)

Finite-State Machines

544

(28)

Definition

544

(1)

Examples of Finite-State Machines

545

(3)

Recognition

548

(2)

Regular Sets and Kleene's Theorem

550

(2)

Machine Minimization

552

(1)

Unreachable States

552

(1)

Minimization Procedure

553

(5)

Sequential Networks and Finite-State Machines

558

(4)

Exercises 8.2

562

(10)

Turing Machines

572

(21)

Definition

573

(4)

Turing Machines as Set Recognizers

577

(3)

Turing Machines as Function Computers

580

(2)

Church-Turing Thesis

582

(2)

Decision Problems and Uncomputability

584

(1)

Examples of Decision Problems

584

(2)

Halting Problem

586

(1)

Computational Complexity

587

(3)

Exercises 8.3

590

(3)

Formal Languages

593

(19)

Classes of Grammars

599

(3)

Formal Languages and Computational Devices

602

(1)

Context-Free Grammars

603

(3)

Exercises 8.4

606

(2)

Chapter 8 Review

608

(2)

On the Computer

610

(2)

Appendix A Summation Notation

612

(3)

Appendix B The Logarithm Function

615

(4)

Answers to Practice Problems

619

(34)

Answers to Selected Exercises

653

(58)

Answers to Self-Tests

711

(8)

Index

719

Amazon no longer offers textbook rentals. We do!

Mathematical Structures for Computer Science : A Modern Treatment of Discrete Mathematics

Mathematical Structures for Computer Science : A Modern Treatment of Discrete Mathematics

9780716743583

0716743582

Summary

Table of Contents

Supplemental Materials

Write a Review

Amazon no longer offers textbook rentals. We do!

Mathematical Structures for Computer Science : A Modern Treatment of Discrete Mathematics

Mathematical Structures for Computer Science : A Modern Treatment of Discrete Mathematics

9780716743583

0716743582

Summary

Table of Contents

Supplemental Materials

Write a Review

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