A Modern Introduction to Fuzzy Mathematics

Preface V

Chapter 1 Introduction 1

1.1 What is vagueness? 1

1.2 Vagueness, Ambiguity, Uncertainty,… 4

1.3 Vagueness and Fuzzy Mathematics 5

Chapter 2 Fuzzy Sets and Their Operations 9

2.1 Algebras of Truth Values 9

2.1.1 Posets 9

2.1.2 Lattices 11

2.1.3 Frames 11

2.2 Zadeh’s Fuzzy Sets 12

2.3 α-cuts of Fuzzy Sets 16

2.4 Interval Valued and Type 2 Fuzzy Sets 19

2.5 Triangular Norms and Conorms 21

2.6 L-fuzzy Sets 23

2.7 “Intuitionistic” Fuzzy Sets and Their Extensions 24

2.8 The Extension Principle 28

2.9* Boolean-Valued Sets 30

2.10* Axiomatic Fuzzy Set Theory 32

Chapter 3 Fuzzy Numbers and Their Arithmetic 35

3.1 Fuzzy Numbers 35

3.1.1 Triangular Fuzzy Numbers 36

3.1.2 Trapezoidal Fuzzy Numbers 37

3.1.3 Gaussian Fuzzy Numbers 37

3.1.4 Quadratic Fuzzy Numbers 39

3.1.5 Exponential Fuzzy Numbers 41

3.1.6 LR Fuzzy Numbers 41

3.1.7 Generalized Fuzzy Numbers 42

3.2 Arithmetic of Fuzzy Numbers 43

3.2.1 Interval Arithmetic 43

3.2.2 Interval Arithmetic and α-Cuts 43

3.2.3 Fuzzy Arithmetic and the Extension Principle 44

3.2.4 Fuzzy Arithmetic of Triangular Fuzzy Numbers 45

3.2.5 Fuzzy Arithmetic of Generalized Fuzzy Numbers 45

3.2.6 Comparing Fuzzy Numbers 47

3.3 Linguistic Variables 49

3.4 Fuzzy Equations 50

3.4.1 Solving the Fuzzy Equation ? ⋅ ? + ? = ? 50

3.4.2 Solving the Fuzzy Equation ? ⋅ ?2 + ? ⋅ ? + ? = ? 53

3.5 Fuzzy Inequalities 55

3.6 Constructing Fuzzy Numbers 55

3.7 Applications of Fuzzy Numbers 57

3.7.1 Simulation of the Human Glucose Metabolism 57

3.7.2 Estimation of an Ongoing Project’s Completion Time 60

Chapter 4 Fuzzy Relations 65

4.1 Crisp Relations 65

4.2 Fuzzy Relations 69

4.3 Cartesian Product, Projections, and Cylindrical Extension 70

4.4 New Fuzzy Relations From Old Ones 72

4.5 Fuzzy Binary Relations on a Set 75

4.6 Fuzzy Orders 80

4.7 Elements of Fuzzy Graph Theory 82

4.7.1 Graphs and Hypergraphs 82

4.7.2 Fuzzy Graphs 83

4.7.3 Fuzzy Hypergraphs 87

4.8* Fuzzy Category Theory 89

4.9* Fuzzy Vectors 96

4.10 Applications 97

Chapter 5 Possibility Theory 101

5.1 Fuzzy Restrictions and Possibility Theory 101

5.2 Possibility and Necessity Measures 103

5.3 Possibility Theory 105

5.4 Possibility Theory and Probability Theory 108

5.5 An Unexpected Application of Possibility Theory 110

Chapter 6 Fuzzy Statistics 117

6.1 Random Variables 117

6.2 Fuzzy Random Variables 120

6.3 Point Estimation 123

6.3.1 The unbiased estimator 124

6.3.2 The consistent estimator 125

6.3.3 The maximum likelihood estimator 126

6.4 Fuzzy Point Estimation 127

6.5 Interval Estimation 128

6.6 Interval Estimation for Fuzzy Data 129

6.7 Hypothesis Testing 131

6.8 Fuzzy Hypothesis Testing 132

6.9 Statistical Regression 134

6.10 Fuzzy Regression 136

Chapter 7 Fuzzy Logics 141

7.1 Mathematical Logic 141

7.2 Many-Valued Logics 146

7.3 On Fuzzy Logics 151

7.4 Hájek’s Basic Many-Valued Logic 152

7.5 Łukasiewicz Fuzzy Logic 155

7.6 Product Fuzzy Logic 157

7.7 Gödel Fuzzy Logic 158

7.8 First Order Fuzzy Logics 160

7.9 Fuzzy Quantifiers 162

7.10 Approximate Reasoning 163

7.11 Application: Fuzzy Expert Systems 166

7.12* A Logic of Vagueness 171

Chapter 8 Fuzzy Computation 173

8.1 Automata, Grammars, and Machines 173

8.2 Fuzzy Languages and Grammars 178

8.3 Fuzzy Automata 181

8.4 Fuzzy Turing Machines 186

8.5 Other Fuzzy Models of Computation 190

Chapter 9 Fuzzy Abstract Algebra 195

9.1 Groups, Rings, and Fields 195

9.2 Fuzzy Groups 199

9.3 Abelian Fuzzy Subgroups 204

9.4 Fuzzy Rings and Fuzzy Fields 206

9.5 Fuzzy Vector Spaces 208

9.6 Fuzzy Normed Spaces 209

9.7 Fuzzy Lie Algebras 210

Chapter 10 Fuzzy Topology 213

10.1 Metric and Topological Spaces 213

10.2 Fuzzy Metric Spaces 218

10.3 Fuzzy Topological Spaces 221

10.4 Fuzzy Product Spaces 224

10.5 Fuzzy Separation 226

10.5.1 Separation 231

10.6 Fuzzy Nets 231

10.7 Fuzzy Compactness 232

10.8 Fuzzy Connectedness 233

10.9 Smooth Fuzzy Topological Spaces 234

10.10 Fuzzy Banach and Fuzzy Hilbert Spaces 235

10.11* Fuzzy Topological Systems 238

Chapter 11 Fuzzy Geometry 243

11.1 Fuzzy Points and Fuzzy Distance 243

11.2 Fuzzy Lines and their Properties 246

11.3 Fuzzy Circles 249

11.4 Regular Fuzzy Polygons 252

11.5 Applications in Theoretical Physics 256

Chapter 12 Fuzzy Calculus 259

12.1 Fuzzy Functions 259

12.2 Integrals of Fuzzy Functions 263

12.3 Derivatives of Fuzzy Functions 266

12.4 Fuzzy Limits of Sequences and Functions 269

12.4.1 Fuzzy Ordinary Differential Equations 272

12.4.2 Fuzzy Partial Differential Equations 277

Appendix A Fuzzy Approximation 283

A.1 Weierstrass and Stone-Weierstrass Approximation Theorems 283

A.2 Weierstrass and Stone-Weierstrass Fuzzy Analogues 284

Appendix B Chaos and Vagueness 287

B.1 Chaos Theory in a Nutshell 287

B.2 Fuzzy Chaos 289

B.3 Fuzzy Fractals 291

Works Cited 293

Subject Index 311

Name Index 323