Exploring Discrete Mathematics With Maple

by Rosen, Kenneth H.; Devitt, John S.; Vasiga, Troy; McCarron, James; Murray, Eithne; Roskos, Ed

Edition:

4th

ISBN13:

9780070541283

ISBN10:

0070541280

Format:

Paperback

Pub. Date:

8/1/1996

Publisher(s):

McGraw Hill College Div

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Summary

This is the first supplement in discrete mathematics to concentrate on the computational aspects of the computer algebra system Maple. Detailed instructions for the use of Maple are included in an introductory chapter and in each subsequent chapter. Each chapter includes discussion of selected Computational and Exploration exercises in the corresponding chapter of Ken Rosen's text Discrete Math and It's Applications, Third Edition. New exercises and projects are included in each chapter to encourage further exploration of discrete mathematics using Maple. All of the Maple code in this supplement is available online via the Waterloo Maple Web site, in addition to new Maple routines that have been created which extend the current capabilities of Maple.

Preface/Acknowledgments

vii

Introduction

(1)

Structure of This Volume

(1)

Interactive Maple

(3)

An Exmple

(2)

A First Encounter with Maple

(4)

An Interest Rate Problem

(3)

Programming Preliminaries

(2)

Getting Help

(2)

Basic Programming Constructions

(6)

Iteration

(3)

Structuring

(1)

Branching

(2)

Premature Loop Exit

(1)

Saving and Reading

(1)

Executing System Programs within Maple

(1)

Packages

(3)

Maple Versions

(1)

On-Line Material

(1)

Using Anonymous FTP

(1)

Maple Worksheets

(1)

Communicating with the Authors

(1)

Exercises/Problems

(2)

Logic, Sets and Functions

(28)

Logic

(8)

Bit Operations

(1)

Bit Strings

(1)

A Maple Programming Example

(1)

Loops and Truth Tables

(3)

Using Maple to Check Logical Arguments

(1)

Quantifiers and Propositions

(3)

Sets

(5)

Set Operations

(3)

Functions and Maple

(7)

Tables

(5)

Functional Composition

(2)

Growth of Functions

(1)

Computations and Explorations

(2)

Exercises/Projects

(2)

The Fundamentals

(46)

Implementing Algorithms in Maple

(6)

Procedure Execution

(1)

Local and Global Variables

(4)

Measuring Time Complexity

(3)

Number Theory

(9)

Basic Number Theory

(1)

Greatest Common Divisors and Least Common Multiples

(3)

Chinese Remainder Theorem

(1)

Factoring Integers

(1)

Primality Testing

(2)

The Euler φ-Function

(1)

Applications of Number Theory

(8)

Hash Functions

(4)

Linear Congruential Pseudorandom Number Generators

(2)

Classical Cryptography

(2)

RSA Cryptography

(4)

Generating Large Primes

(1)

Base Expansions

(2)

Matrices

(6)

Zero-One Matrices

(3)

Computations and Explorations

(4)

Exercises/Projects

(4)

Mathematical Reasoning

(18)

Methods of Proof

(5)

Mathematical Induction

104

(4)

Recursive and Iterative Definitions

108

(3)

Computations and Explorations

111

(4)

Exercises/Projects

115

(2)

Counting

117

(30)

Relevant Maple Functions

118

(2)

More Combinatorial Functions

120

(8)

Binomial Coefficients

120

(3)

Multinomial Coefficients

123

(3)

Stirling Numbers

126

(2)

Permutations

128

(4)

Partitions of Integers

130

(2)

Discrete Probability

132

(1)

Permutations

133

(3)

Computations and Explorations

136

(8)

Exercises/Projects

144

(3)

Counting

147

(36)

Recurrence Relations

147

(3)

Towers of Hanoi Problem

148

(2)

Solving Recurrences with Maple

150

(14)

Inhomogeneous Recurrence Relations

157

(2)

Maple's Recurrence Solver

159

(4)

Divide and Conquer Relations

163

(1)

Inclusion - Exclusion

164

(5)

Generating Functions

169

(4)

Computations and Explorations

173

(7)

Exercises/Projects

180

(3)

Relations

183

(34)

An Introduction to Relations in Maple

183

(2)

Determining Properties of Relations using Maple

185

(2)

n-ary Relations in Maple

187

(3)

Representing Relations as Digraphs and Zero-One Matrices

190

(5)

Representing Relations Using Directed Graphs

190

(2)

Representing Relations Using Zero-One Matrices

192

(3)

Computing Closures of Relations

195

(4)

Reflexive Closure

195

(1)

Symmetric Closure

196

(1)

Transitive Closure

196

(3)

Equivalence Relations

199

(2)

Partial Ordering and Minimal Elements

201

(4)

Lattices

205

(1)

Covering Relations

206

(2)

Hasse Diagrams

208

(4)

Computations and Explorations

212

(3)

Exercises/Projects

215

(2)

Graphs

217

(56)

Getting Started with Graphs

217

(12)

Simple Graphs

217

(2)

Visualizing Graphs in Maple

219

(6)

Directed Graphs

225

(4)

Simple Computations on Graphs

229

(3)

Degrees of Vertices

230

(2)

Constructing Special Graphs

232

(14)

Bipartite Graphs

237

(3)

Subgraphs, Unions and Complements

240

(6)

Representing Graphs, and Graph Isomorphism

246

(5)

Representing Graphs

246

(2)

Graphs Isomorphism

248

(3)

Connectivity

251

(4)

Euler and Hamilton Paths

255

(3)

Euler Circuits

255

(3)

Shortest Path Problems

258

(3)

Planar Graphs and Graph Coloring

261

(2)

Planar Graphs

261

(1)

Graph Colorings

261

(2)

Flows

263

(2)

Computations and Explorations

265

(6)

Exercises/Projects

271

(2)

Trees

273

(50)

Introduction to Trees

273

(7)

Rooted Trees

276

(4)

Application of Trees

280

(8)

Binary Insertion

281

(5)

Huffman Coding

286

(2)

Tree Traversal

288

(8)

Infix, Prefix and Postfix Notation

292

(4)

Trees and Sorting

296

(4)

Bubble Sort

297

(1)

Merge Sort

298

(2)

Spanning Trees

300

(9)

Backtracking

303

(6)

Minimum Spanning Trees

309

(5)

Computations and Explorations

314

(6)

Additional Exercises

320

(3)

Boolean Algebra

323

(26)

Boolean Functions

323

(7)

A Boolean Evaluator

326

(1)

Representing Boolean Functions

327

(1)

Verifying Boolean Identities

327

(1)

Duality

328

(1)

Disjunctive Normal Form

329

(1)

Representing Boolean Functions

330

(3)

Minimization of Boolean Expressions and Circuits

333

(3)

Don't Care Conditions

336

(3)

Computations and Explorations

339

(6)

Exercises/Projects

345

(4)

Modeling Computation

349

(38)

Introduction

349

(1)

Stacks

350

(3)

Finite-State Machines with Output

353

(3)

Finite-State Machines with No Output

356

(5)

Deterministic Finite-State Machine Simulation

361

(2)

Nondeterministic Finite Automata

363

(7)

DFA to NFA

370

(4)

Converting Regular Expressions to/from Finite Automata

374

(4)

Turing Machines

378

(6)

Computations and Explorations

384

(3)

Exercises/Projects

387

(1)

Index

Exploring Discrete Mathematics With Maple

9780070541283

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Table of Contents