Discrete Mathematics

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ISBN: 9780198507178 | 0198507178
Cover: Paperback
Copyright: 2/20/2003

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The long-awaited second edition of Norman Bigg's best-selling Discrete Mathematics, includes new chapters on statements and proof, logical framework, natural numbers, and the integers, in addition to updated chapters from the previous edition. Carefully structured, coherent and comprehensive, each chapter contains tailored exercises and solutions to selected questions, and miscellaneous exercises are presented throughout. This is an invaluable text for students seeking a clear introduction to discrete mathematics, graph theory, combinatorics, number theory and abstract algebra.

PART I FOUNDATIONS

Statements and proofs

Some mathematical statements

(1)

How to do mathematics

(2)

Compound statements

(1)

Existential statements

(1)

Universal statements

(2)

Proof techniques

(1)

Miscellaneous Exercises

(1)

Set notation

Sets of objects and numbers

(1)

Subsets

(1)

Union and intersection

(1)

Miscellaneous Exercises

(2)

The logical framework

The basic logical operations: not, or, and

(1)

Logical equivalence

(1)

if - then

(1)

The converse statement

(1)

The contrapositive statement

(1)

Universal and existential quantifiers

(2)

Miscellaneous Exercises

(1)

Natural numbers

The `laws of algebra'

(1)

Putting the natural numbers in order

(2)

The principle of induction

(1)

Summation formulae

(2)

Recursive definitions

(1)

Other forms of the principle of induction

(2)

Greatest and least members

(1)

How a conjecture becomes a theorem

(3)

Miscellaneous Exercises

(1)

Functions

The concept of a function

(1)

Surjections, injections, bijections

(2)

Composition of functions

(1)

Bijections and inverse functions

(2)

Miscellaneous Exercises

(1)

How to count

Counting as a bijection

(1)

The size of a set

(2)

A counting problem

(1)

Some applications of the pigeonhole principle

(1)

Infinite sets

(2)

Strange properties of infinite sets

(2)

Miscellaneous Exercises

(1)

Integers

Negative numbers

(1)

Equivalence relations

(2)

Classification

(2)

Construction of the integers

(1)

Properties of the integers

(2)

Bounded subsets of Z

(1)

Miscellaneous Exercises

(1)

Divisibility and prime numbers

Divisibility

(1)

Quotient and remainder

(1)

Representation of integers

(1)

The greatest common divisor

(3)

Prime numbers

(1)

Existence and uniqueness of prime factorization

(2)

Miscellaneous Exercises

(2)

Fractions and real numbers

Construction and properties of rational numbers

(1)

Density of fractions

(2)

Decimal representations of fractions

(2)

Real numbers

(2)

Approximations for real numbers

(2)

The greatest lower bound property

(1)

The real numbers are more plentiful than the rationals

(2)

Miscellaneous Exercises

(4)

PART II TECHNIQUES

Principles of counting

The addition principle

(1)

Counting sets of pairs

(3)

Euler's function

(2)

Functions, words, and selections

(2)

Injections as ordered selections without repetition

(1)

Permutations

100

(3)

Miscellaneous Exercises

103

(2)

Subsets and designs

Binomial numbers

105

(3)

Unordered selections with repetition

108

(2)

The binomial theorem

110

(2)

The sieve principle

112

(2)

Some arithmetical applications

114

(4)

Designs

118

(4)

t-designs

122

(2)

Miscellaneous Exercises

124

(2)

Partition, classification, and distribution

Partitions of a set

126

(2)

Classification and equivalence relations

128

(1)

Distributions and the multinomial numbers

129

(3)

Partitions of a positive integer

132

(2)

Classification of permutations

134

(2)

Even and odd permutations

136

(4)

Miscellaneous Exercises

140

(2)

Modular arithmetic

Congruences

142

(2)

Zm and its arithmetic

144

(3)

Invertible elements of Zm

147

(2)

Cyclic constructions for designs

149

(4)

Latin squares

153

(3)

Miscellaneous Exercises

156

(3)

PART III ALGORITHMS AND GRAPHS

Algorithms and their efficiency

What is an algorithm?

159

(2)

The language of programs

161

(3)

Algorithms and programs

164

(1)

Proving that an algorithm is correct

165

(3)

Efficiency of algorithms

168

(2)

Growth rates: the O notation

170

(1)

Comparison of algorithms

171

(2)

Introduction to sorting algorithms

173

(3)

Miscellaneous Exercises

176

(2)

Graphs

Graphs and their representation

178

(1)

Isomorphism of graphs

179

(2)

Degree

181

(2)

Paths and cycles

183

(2)

Trees

185

(2)

Colouring the vertices of a graph

187

(1)

The greedy algorithm for vertex-colouring

188

(3)

Miscellaneous Exercises

191

(2)

Trees, sorting, and searching

Counting the leaves on a rooted tree

193

(3)

Trees and sorting algorithms

196

(3)

Spanning trees and the MST problem

199

(3)

Depth-first search

202

(3)

Breadth-first search

205

(1)

The shortest path problem

206

(2)

Miscellaneous Exercises

208

(2)

Bipartite graphs and matching problems

Relations and bipartite graphs

210

(2)

Edge colourings of graphs

212

(1)

Application of edge colouring to latin squares

213

(3)

Matchings

216

(3)

Maximum matchings

219

(2)

Transversals for families of finite sets

221

(2)

Miscellaneous Exercises

223

(2)

Digraphs, networks, and flows

Digraphs

225

(2)

Networks and critical paths

227

(2)

Flows and cuts

229

(3)

The max-flow min-cut theorem

232

(3)

The labelling algorithm for network flows

235

(4)

Miscellaneous Exercises

239

(1)

Recursive techniques

Generalities about recursion

240

(1)

Linear recursion

241

(2)

Recursive bisection

243

(2)

Recursive optimization

245

(3)

The framework of dynamic programming

248

(2)

Examples of the dynamic programming method

250

(3)

Miscellaneous Exercises

253

(6)

PART IV ALGEBRAIC METHODS

Groups

The axioms for a group

259

(1)

Examples of groups

260

(3)

Basic algebra in groups

263

(2)

The order of a group element

265

(1)

Isomorphism of groups

266

(2)

Cyclic groups

268

(2)

Subgroups

270

(3)

Cosets and Lagrange's theorem

273

(4)

Characterization of cyclic groups

277

(2)

Miscellaneous Exercises

279

(2)

Groups of permutations

Definitions and examples

281

(2)

Orbits and stabilizers

283

(2)

The size of an orbit

285

(3)

The number of orbits

288

(2)

Representation of groups by permutations

290

(2)

Applications to group theory

292

(3)

Miscellaneous Exercises

295

(1)

Rings, fields, and polynomials

Rings

296

(1)

Invertible elements of a ring

297

(2)

Fields

299

(2)

Polynomials

301

(3)

The division algorithm for polynomials

304

(2)

The Euclidean algorithm for polynomials

306

(3)

Factorization of polynomials in theory

309

(1)

Factorization of polynomials in practice

310

(3)

Miscellaneous Exercises

313

(1)

Finite fields and some applications

A field with nine elements

314

(1)

The order of a finite field

315

(2)

Construction of finite fields

317

(1)

The primitive element theorem

318

(4)

Finite fields and latin squares

322

(2)

Finite geometry and designs

324

(3)

Projective planes

327

(3)

Squares in finite fields

330

(3)

Existence of finite fields

333

(3)

Miscellaneous Exercises

336

(2)

Error-correcting codes

Words, codes, and errors

338

(3)

Linear codes

341

(2)

Construction of linear codes

343

(2)

Correcting errors in linear codes

345

(4)

Cyclic codes

349

(2)

Classification and properties of cyclic codes

351

(4)

Miscellaneous Exercises

355

(2)

Generating functions

Power series and their algebraic properties

357

(3)

Partial fractions

360

(4)

The binomial theorem for negative exponents

364

(3)

Generating functions

367

(2)

The homogeneous linear recursion

369

(3)

Non-homogeneous linear recursions

372

(2)

Miscellaneous Exercises

374

(2)

Partitions of a positive integer

Partitions and diagrams

376

(2)

Conjugate partitions

378

(1)

Partitions and generating functions

379

(3)

Generating functions for restricted partitions

382

(2)

A mysterious identity

384

(3)

The calculation of p(n)

387

(1)

Miscellaneous Exercises

388

(2)

Symmetry and counting

The cycle index of a group of permutations

390

(2)

Cyclic and dihedral symmetry

392

(3)

Symmetry in three dimensions

395

(2)

The number of inequivalent colourings

397

(3)

Sets of colourings and their generating functions

400

(2)

Polya's theorem

402

(3)

Miscellaneous Exercises

405

(2)

Answers to exercises

407

(14)

Index

421

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Discrete Mathematics

Summary

Table of Contents

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