Mathematics A Discrete Introduction

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ISBN: 9780534398989 | 0534398987
Cover: Hardcover
Copyright: 6/6/2005

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Master the fundamentals of discrete mathematics and proof-writing with MATHEMATICS: A DISCRETE INTRODUCTION! With a wealth of learning aids and a clear presentation, the mathematics text teaches you not only how to write proofs, but how to think clearly and present cases logically beyond this course. Though it is presented from a mathematician's perspective, you will learn the importance of discrete mathematics in the fields of computer science, engineering, probability, statistics, operations research, and other areas of applied mathematics. Tools such as Mathspeak, hints, and proof templates prepare you to succeed in this course.

To the Student

How to Read a Mathematics Book

xvi

Exercises

xvii

To the Instructor

xix

Audience and Prerequisites

xix

Topics Covered and Navigating the Sections

xix

Sample Course Outlines

xxi

Special Features

xxi

What's New in This Second Edition

xxiii

Acknowledgments

xxv

This New Edition

xxv

From the First Edition

xxv

Fundamentals

(36)

Joy

(1)

Why?

(1)

The Agony and the Ecstasy

(1)

Exercise

(1)

Definition

(6)

Recap

(1)

Exercises

(3)

Theorem

(8)

The Nature of Truth

(1)

If-Then

(2)

If and Only If

(1)

And, Or, and Not

(1)

What Theorems Are Called

(1)

Vacuous Truth

(1)

Recap

(1)

Exercises

(1)

Proof

(9)

A More Involved Proof

(2)

Proving If-and-Only-If Theorems

(2)

Proving Equations and Inequalities

(1)

Recap

(1)

Exercises

(1)

Counterexample

(2)

Recap

(1)

Exercises

(1)

Boolean Algebra

(10)

More Operations

(1)

Recap

(1)

Exercises

(2)

Chapter 1 Self Test

(3)

Collections

(46)

Lists

(8)

Counting Two-Element Lists

(3)

Longer Lists

(3)

Recap

(1)

Exercises

(2)

Factorial

(4)

Much Ado About 0!

(1)

Product Notation

(1)

Recap

(1)

Exercises

(1)

Sets I: Introduction, Subsets

(9)

Equality of Sets

(2)

Subset

(2)

Counting Subsets

(2)

Power Set

(1)

Recap

(1)

Exercises

(1)

Quantifiers

(6)

There Is

(1)

For All

(1)

Negating Quantified Statements

(1)

Combining Quantifiers

(1)

Recap

(1)

Exercises

(1)

Sets II: Operations

(12)

Union and Intersection

(2)

The Size of a Union

(2)

Difference and Symmetric Difference

(5)

Cartesian Product

(1)

Recap

(1)

Exercises

(2)

Combinatorial Proof: Two Examples

(7)

Recap

(1)

Exercises

(1)

Chapter 2 Self Test

(3)

Counting and Relations

(52)

Relations

(6)

Properties of Relations

(1)

Recap

(1)

Exercises

(2)

Equivalence Relations

(9)

Equivalence Classes

(3)

Recap

(1)

Exercises

(2)

Partitions

(6)

Counting Classes/Parts

100

(2)

Recap

102

(1)

Exercises

102

(2)

Binomial Coefficients

104

(13)

Calculating (n/k)

107

(2)

Pascal's Triangle

109

(2)

A Formula for (n/k)

111

(2)

Recap

113

(1)

Exercises

113

(4)

Counting Multisets

117

(6)

Multisets

117

(2)

Formulas for (n/k)

119

(3)

Recap

122

(1)

Exercises

122

(1)

Inclusion-Exclusion

123

(12)

How to Use Inclusion-Exclusion

126

(3)

Derangements

129

(3)

A Ghastly Formula

132

(1)

Recap

132

(1)

Exercises

132

(1)

Chapter 3 Self Test

133

(2)

More Proof

135

(58)

Contradiction

135

(7)

Proof by Contrapositive

135

(2)

Reductio ad Absurdum

137

(4)

A Matter of Style

141

(1)

Recap

141

(1)

Exercises

141

(1)

Smallest Counterexample

142

(13)

Well-Ordering

148

(5)

Recap

153

(1)

Exercises

153

(1)

And Finally

154

(1)

Induction

155

(16)

The Induction Machine

155

(2)

Theoretical Underpinnings

157

(1)

Proof by Induction

157

(3)

Proving Equations and Inequalities

160

(2)

Other Examples

162

(1)

Strong Induction

163

(2)

A More Complicated Example

165

(3)

A Matter of Style

168

(1)

Recap

168

(1)

Exercises

168

(3)

Recurrence Relations

171

(22)

First-Order Recurrence Relations

172

(3)

Second-Order Recurrence Relations

175

(3)

The Case of the Repeated Root

178

(2)

Sequences Generated by Polynomials

180

(7)

Recap

187

(1)

Exercises

188

Chapter 4 Self Test

180

(13)

Functions

193

(52)

Functions

193

(12)

Domain and Image

195

(1)

Pictures of Functions

196

(1)

Counting Functions

197

(1)

Inverse Functions

198

(4)

Counting Functions, Again

202

(1)

Recap

203

(1)

Exercises

203

(2)

The Pigeonhole Principle

205

(6)

Cantor's Theorem

208

(2)

Recap

210

(1)

Exercises

210

(1)

Composition

211

(5)

Identity Function

214

(1)

Recap

215

(1)

Exercises

215

(1)

Permutations

216

(15)

Cycle Notation

217

(3)

Calculations with Permutations

220

(1)

Transpositions

221

(5)

A Graphical Approach

226

(2)

Recap

228

(1)

Exercises

228

(3)

Symmetry

231

(5)

Symmetries of a Square

231

(1)

Symmetries as Permutations

232

(1)

Combining Symmetries

233

(2)

Formal Definition of Symmetry

235

(1)

Recap

236

(1)

Exercises

236

(1)

Assorted Notation

236

(9)

Big oh

236

(3)

Ω and Θ

239

(1)

Little oh

240

(1)

Floor and Ceiling

241

(1)

Recap

242

(1)

Exercises

242

(1)

Chapter 5 Self Test

242

(3)

Probability

245

(48)

Sample Space

245

(4)

Recap

248

(1)

Exercises

248

(1)

Events

249

(8)

Combining Events

252

(1)

The Birthday Problem

253

(1)

Recap

254

(1)

Exercises

255

(2)

Conditional Probability and Independence

257

(9)

Independence

259

(2)

Independent Repeated Trials

261

(1)

The Monty Hall Problem

262

(1)

Recap

263

(1)

Exercises

263

(3)

Random Variables

266

(5)

Random Variables as Events

267

(2)

Independent Random Variables

269

(1)

Recap

270

(1)

Exercises

270

(1)

Expectation

271

(22)

Linearity of Expectation

276

(3)

Product of Random Variables

279

(3)

Expected Value as a Measure of Centrality

282

(1)

Variance

283

(4)

Recap

287

(1)

Exercises

287

(2)

Chapter 6 Self Test

289

(4)

Number Theory

293

(44)

Dividing

293

(5)

Div and Mod

296

(1)

Recap

297

(1)

Exercises

297

(1)

Greatest Common Divisor

298

(11)

Calculating the gcd

299

(2)

Correctness

301

(1)

How Fast?

302

(2)

An Important Theorem

304

(3)

Recap

307

(1)

Exercises

307

(2)

Modular Arithmetic

309

(11)

A New Context for Basic Operations

309

(1)

Modular Addition and Multiplication

310

(1)

Modular Subtraction

311

(2)

Modular Division

313

(5)

A Note on Notation

318

(1)

Recap

318

(1)

Exercises

318

(2)

The Chinese Remainder Theorem

320

(5)

Solving One Equation

320

(2)

Solving Two Equations

322

(2)

Recap

324

(1)

Exercises

324

(1)

Factoring

325

(12)

Infinitely Many Primes

327

(1)

A Formula for Greatest Common Divisor

328

(1)

Irrationality of √2

329

(2)

Recap

331

(1)

Exercises

331

(4)

Chapter 7 Self Test

335

(2)

Algebra

337

(52)

Groups

337

(10)

Operations

337

(1)

Properties of Operations

338

(2)

Groups

340

(2)

Examples

342

(3)

Recap

345

(1)

Exercises

345

(2)

Group Isomorphism

347

(6)

The Same?

347

(2)

Cyclic Groups

349

(3)

Recap

352

(1)

Exercises

352

(1)

Subgroups

353

(9)

Lagrange's Theorem

356

(3)

Recap

359

(1)

Exercises

359

(3)

Fermat's Little Theorem

362

(8)

First Proof

362

(1)

Second Proof

363

(3)

Third Proof

366

(1)

Euler's Theorem

367

(1)

Primality Testing

368

(1)

Recap

369

(1)

Exercises

369

(1)

Public Key Cryptography I: Introduction

370

(3)

The Problem: Private Communication in Public

370

(1)

Factoring

370

(1)

Words to Numbers

371

(2)

Cryptography and the Law

373

(1)

Recap

373

(1)

Exercises

373

(1)

Public Key Cryptography II: Rabin's Method

373

(7)

Square Roots Modulo n

374

(4)

The Encryption and Decryption Procedures

378

(1)

Recap

379

(1)

Exercises

379

(1)

Public Key Cryptography III: RSA

380

(9)

The RSA Encryption and Decryption Functions

381

(2)

Security

383

(1)

Recap

384

(1)

Exercises

384

(1)

Chapter 8 Self Test

385

(4)

Graphs

389

(60)

Fundamentals of Graph Theory

389

(10)

Map Coloring

389

(2)

Three Utilities

391

(1)

Seven Bridges

391

(1)

What Is a Graph?

392

(1)

Adjacency

393

(1)

A Matter of Degree

394

(2)

Further Notation and Vocabulary

396

(1)

Recap

397

(1)

Exercises

397

(2)

Subgraphs

399

(7)

Induced and Spanning Subgraphs

400

(2)

Cliques and Independent Sets

402

(1)

Complements

403

(1)

Recap

404

(1)

Exercises

404

(2)

Connection

406

(7)

Walks

406

(1)

Paths

407

(3)

Disconnection

410

(1)

Recap

411

(1)

Exercises

411

(2)

Trees

413

(8)

Cycles

413

(1)

Forests and Trees

413

(1)

Properties of Trees

414

(2)

Leaves

416

(2)

Spanning Trees

418

(1)

Recap

419

(1)

Exercises

420

(1)

Eulerian Graphs

421

(6)

Necessary Conditions

422

(1)

Main Theorems

423

(2)

Unfinished Business

425

(1)

Recap

426

(1)

Exercises

426

(1)

Coloring

427

(8)

Core Concepts

427

(2)

Bipartite Graphs

429

(4)

The Ease of Two-Coloring and the Difficulty of Three-Coloring

433

(1)

Recap

434

(1)

Exercises

434

(1)

Planar Graphs

435

(14)

Dangerous Curves

435

(1)

Embedding

436

(1)

Euler's Formula

437

(3)

Nonplanar Graphs

440

(2)

Coloring Planar Graphs

442

(2)

Recap

444

(1)

Exercises

444

(2)

Chapter 9 Self Test

446

(3)

Partially Ordered Sets

449

(38)

Fundamentals of Partially Ordered Sets

449

(6)

What Is a Poset?

449

(3)

Notation and Language

452

(2)

Recap

454

(1)

Exercises

454

(1)

Max and Min

455

(3)

Recap

457

(1)

Exercises

457

(1)

Linear Orders

458

(3)

Recap

460

(1)

Exercises

461

(1)

Linear Extensions

461

(8)

Sorting

465

(2)

Linear Extensions of Infinite Posets

467

(1)

Recap

468

(1)

Exercises

468

(1)

Dimension

469

(8)

Realizers

469

(2)

Dimension

471

(2)

Embedding

473

(3)

Recap

476

(1)

Exercises

476

(1)

Lattices

477

(10)

Meet and Join

477

(2)

Lattices

479

(2)

Recap

481

(1)

Exercises

482

(1)

Chapter 10 Self Test

483

(4)

Appendices

487

(68)

A Lots of Hints and Comments; Some Answers

487

(28)

B Solutions to Self Tests

515

(29)

Chapter 1

515

(1)

Chapter 2

516

(2)

Chapter 3

518

(2)

Chapter 4

520

(4)

Chapter 5

524

(2)

Chapter 6

526

(4)

Chapter 7

530

(2)

Chapter 8

532

(3)

Chapter 9

535

(4)

Chapter 10

539

(5)

C Glossary

544

(8)

D Fundamentals

552

(3)

Numbers

552

(1)

Operations

552

(1)

Ordering

553

(1)

Complex Numbers

553

(1)

Substitution

553

(2)

Index

555

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Mathematics A Discrete Introduction

Summary

Table of Contents

Supplemental Materials