9780130457974

Combining a careful selection of topics with coverage of theirgenuineapplications in computer science, this book, more than any other in this field, is clearly and concisely written, presenting the basic ideas of discrete mathematical structures in a manner that is understandable.Limiting its scope and depth of topics to those that readers can actually utilize, this book covers first the fundamentals, then follows with logic, counting, relations and digraphs, functions, order relations and structures, trees, graph theory, semigroups and groups, languages and finite-state machines, and groups and coding.With its comprehensive appendices and index, this book can be an excellent reference work for mathematicians and those in the field of computer science.

Preface

viii

A Word to Students

xii

1 Fundamentals

1

(49)

1.1 Sets and Subsets

2

(3)

1.2 Operations on Sets

5

(8)

1.3 Sequences

13

(7)

1.4 Division in the Integers

20

(12)

1.5 Matrices

32

(9)

1.6 Mathematical Structures

41

(9)

2 Logic

50

(28)

2.1 Propositions and Logical Operations

51

(6)

2.2 Conditional Statements

57

(5)

2.3 Methods of Proof

62

(5)

2.4 Mathematical Induction

67

(11)

3 Counting

78

(32)

3.1 Permutations

79

(4)

3.2 Combinations

83

(5)

3.3 Pigeonhole Principle

88

(3)

3.4 Elements of Probability

91

(9)

3.5 Recurrence Relations

100

(10)

4 Relations and Digraphs

110

(58)

4.1 Product Sets and Partitions

111

(4)

4.2 Relations and Digraphs

115

(8)

4.3 Paths in Relations and Digraphs

123

(6)

4.4 Properties of Relations

129

(7)

4.5 Equivalence Relations

136

(4)

4.6 Computer Representation of Relations and Digraphs

140

(7)

4.7 Operations on Relations

147

(10)

4.8 Transitive Closure and Warshall's Algorithm

157

(11)

5 Functions

168

(32)

5.1 Functions

169

(9)

5.2 Functions for Computer Science

178

(5)

5.3 Growth of Functions

183

(5)

5.4 Permutation Functions

188

(12)

6 Order Relations and Structures

200

(54)

6.1 Partially Ordered Sets

201

(10)

6.2 Extremal Elements of Partially Ordered Sets

211

(5)

6.3 Lattices

216

(10)

6.4 Finite Boolean Algebras

226

(7)

6.5 Functions on Boolean Algebras

233

(4)

6.6 Circuit Design

237

(17)

7 Trees

254

(36)

7.1 Trees

254

(5)

7.2 Labeled Trees

259

(5)

7.3 Tree Searching

264

(9)

7.4 Undirected Trees

273

(7)

7.5 Minimal Spanning Trees

280

(10)

8 Topics in Graph Theory

290

(39)

8.1 Graphs

291

(5)

8.2 Euler Paths and Circuits

296

(8)

8.3 Hamiltonian Paths and Circuits

304

(3)

8.4 Transport Networks

307

(8)

8.5 Matching Problems

315

(5)

8.6 Coloring Graphs

320

(9)

9 Semigroups and Groups

329

(43)

9.1 Binary Operations Revisited

330

(4)

9.2 Semigroups

334

(7)

9.3 Products and Quotients of Semigroups

341

(6)

9.4 Groups

347

(11)

9.5 Products and Quotients of Groups

358

(5)

9.6 Other Mathematical Structures

363

(9)

10 Languages and Finite-State Machines

372

(44)

10.1 Languages

373

(8)

10.2 Representations of Special Grammars and Languages

381

(9)

10.3 Finite-State Machines

390

(6)

10.4 Monoids, Machines, and Languages

396

(5)

10.5 Machines and Regular Languages

401

(6)

10.6 Simplification of Machines

407

(9)

11 Groups and Coding

416

(27)

11.1 Coding of Binary Information and Error Detection

417

(11)

11.2 Decoding and Error Correction

428

(8)

11.3 Public Key Cryptology

436

(7)

Appendix A: Algorithms and Pseudocode

443

(11)

Appendix B: Additional Experiments in Discrete Mathematics

454

(5)

Answers to Odd-Numbered Exercises

459

(38)

Answers to Chapter Self-Tests

497

Glossary

G-1

Index

I-1

Photo Credits

P-1

0521016983

List of Figures and Tables

ix

Acknowledgments

xiii

Abbreviations

xvii

1 THE POLITICS OF TAXATION

1

(37)

The Problem in Perspective

5

(4)

The Argument

9

(14)

Alternative Explanations

23

(9)

Plan of the Book: A "Nested" Approach to Comparative Analysis

32

(6)

2 THE TAX STATE IN COMPARATIVE PERSPECTIVE

38

(30)

The Tax State Defined

39

(4)

Toward a Typology of the Tax State

43

(17)

Measurement and Classification

60

(7)

Conclusion

67

(1)

3 CRITICAL JUNCTURE: DEFINING NATIONAL POLITICAL COMMUNITY

68

(38)

Cleavages: Race and Region

70

(8)

Two Definitions of National Political Community

78

(11)

Influence on Class Relations

89

(15)

Conclusion

104

(2)

4 THE RISE OF THE MODERN TAX STATE IN BRAZIL AND SOUTH AFRICA

106

(67)

Legacies of Pre-Modern Taxation

107

(6)

The Context of 20th-Century State-Building

113

(4)

Patterns of Taxation and State Development (1900-1975)

117

(5)

Three Episodes of Politics and Taxation

122

(45)

Implications: Weathering Economic and Political Crises (1975-1990)

167

(4)

Conclusion

171

(2)

5 SHADOWS OF THE PAST: TAX REFORM IN AN ERA OF GLOBALIZATION AND DEMOCRATIZATION

173

(64)

Pressures for Tax Reform from Home and Abroad

177

(8)

The Reconstruction of Racial and Regional Identities

185

(10)

Class Relations and Political Strategies

195

(18)

Patterns of Tax Reform Compared

213

(22)

Conclusion

235

(2)

6 POLITICAL COMMUNITY AND TAXATION BEYOND BRAZIL AND SOUTH AFRICA

237

(34)

Case Selection

238

(1)

Estimating the Impact of National Political Community on Taxation

239

(25)

Alternative Explanations

264

(5)

Conclusion

269

(2)

7 CONCLUSION

271

(13)

The Influence of Identity Politics and Political Community

274

(3)

Implications for Brazil and South Africa in the 21st Century

277

(7)

APPENDIX COMPARATIVE-HISTORICAL ANALYSIS

284

(7)

References

291

(18)

Index

309

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Discrete mathematics is an interesting course to teach and to study at the freshman and sophomore level for several reasons. Its content is mathematics, but most of its applications and more than half its students are from computer science. Thus careful motivation of topics and previews of applications are important and necessary strategies. Moreover, there are a number of substantive and diverse topics covered in the course, so a text must proceed clearly and carefully, emphasizing key ideas with suitable pedagogy. In addition, the student is often expected to develop an important new skill: the ability to write a mathematical proof. This skill is excellent training for writing good computer programs. This text can be used by students in mathematics as an introduction to the fundamental ideas of discrete mathematics, and a foundation for the development of more advanced mathematical concepts. If used in this way, the topics dealing with specific computer science applications can be ignored or selected independently as important examples. The text can also be used in a computer science or computer engineering curriculum to present the foundations of many basic computer-related concepts and provide a coherent development and common theme for these ideas. The instructor can easily develop a suitable course by referring to the chapter prerequisites which identify material needed by that chapter. Approach First, we have limited both the areas covered and the depth of coverage to what we deem prudent in a first course taught at the freshman and sophomore level. We have identified a set of topics that we feel are of genuine use in computer science and elsewhere and that can be presented in a logically coherent fashion. We have presented an introduction to these topics along with an indication of how they can be pursued in greater depth. This approach makes our text an excellent reference for upper-division courses. Second, the material has been organized and interrelated to minimize the mass of definitions and the abstraction of some of the theory. Relations and digraphs are treated as two aspects of the same fundamental mathematical idea, with a directed graph being a pictorial representation of a relation. This fundamental idea is then used as the basis of virtually all the concepts introduced in the book, including functions, partial orders, graphs, and mathematical structures. Whenever possible, each new idea introduced in the text uses previously encountered material and, in turn, is developed in such a way that it simplifies the more complex ideas that follow. What Is New in the Fifth Edition We continue to believe that this book works well in the classroom because of the unifying role played by the two key concepts: relations and digraphs. In this edition we have woven in a thread of coding in all its aspects, efficiency, effectiveness, and security. Two new sections, Other Mathematical Structures and Public Key Cryptology are the major components of this thread, but smaller related insertions begin in Chapter 1. The number of exercises for this edition has been increased by more than 25%. Whatever changes we have made, our objective has remained the same as in the first four editions:to present the basic notions of discrete mathematics and some of its applications .in a clear and concise manner that will be understandable to the student. A cryptology thread begins in Chapter 1 and presents the basic ideas of the field. The thread concludes in Public Key Cryptology. Included now is coding in all its aspects, efficiency, effectiveness, and security. A new section, Other Mathematical Structures, introduces the basic concepts of rings and fields, in particularZ p . More opportunities for students to build modeling skills are provided. Whether seen as modeling, abstraction, pattern recognition, or problem solving, the ability to see the m

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Discrete Mathematical Structures

Discrete Mathematical Structures

0130457973

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