# Mathematical Foundations of Fuzzy Sets

1. Edition February 2023

416 Pages, Hardcover*Wiley & Sons Ltd*

**978-1-119-98152-7**

Mathematical Foundations of Fuzzy Sets

Introduce yourself to the foundations of fuzzy logic with this easy-to-use guide

Many fields studied are defined by imprecise information or high degrees of uncertainty. When this uncertainty derives from randomness, traditional probabilistic statistical methods are adequate to address it; more everyday forms of vagueness and imprecision, however, require the toolkit associated with 'fuzzy sets' and 'fuzzy logic'. Engineering and mathematical fields related to artificial intelligence, operations research and decision theory are now strongly driven by fuzzy set theory.

Mathematical Foundations of Fuzzy Sets introduces readers to the theoretical background and practical techniques required to apply fuzzy logic to engineering and mathematical problems. It introduces the mathematical foundations of fuzzy sets as well as the current cutting edge of fuzzy-set operations and arithmetic, offering a rounded introduction to this essential field of applied mathematics. The result can be used either as a textbook or as an invaluable reference for working researchers and professionals.

Mathematical Foundations of Fuzzy Sets offers thereader:

* Detailed coverage of set operations, fuzzification of crisp operations, and more

* Logical structure in which each chapter builds carefully on previous results

* Intuitive structure, divided into 'basic' and 'advanced' sections, to facilitate use in one- or two-semester courses

Mathematical Foundations of Fuzzy Sets is essential for graduate students and academics in engineering and applied mathematics, particularly those doing work in artificial intelligence, decision theory, operations research, and related fields.

1 Mathematical Analysis 1

1.1 Infimum and Supremum 1

1.2 Limit Inferior and Limit Superior 3

1.3 Semi-Continuity 11

1.4 Miscellaneous 19

2 Fuzzy Sets 23

2.1 Membership Functions 23

2.2 alpha-level Sets 24

2.3 Types of Fuzzy Sets 34

3 Set Operations of Fuzzy Sets 43

3.1 Complement of Fuzzy Sets 43

3.2 Intersection of Fuzzy Sets 44

3.3 Union of Fuzzy Sets 51

3.4 Inductive and Direct Definitions 56

3.5 alpha-Level Sets of Intersection and Union 61

3.6 Mixed Set Operations 65

4 Generalized Extension Principle 69

4.1 Extension Principle Based on the Euclidean Space 69

4.2 Extension Principle Based on the Product Spaces 75

4.3 Extension Principle Based on the Triangular Norms 84

4.4 Generalized Extension Principle 92

5 Generating Fuzzy Sets 109

5.1 Families of Sets 110

5.2 Nested Families 112

5.3 Generating Fuzzy Sets from Nested Families 119

5.4 Generating Fuzzy Sets Based on the Expression in the Decomposition

Theorem 123

5.4.1 The Ordinary Situation 123

5.4.2 Based on One Function 129

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5.4.3 Based on Two Functions 140

5.5 Generating Fuzzy Intervals 150

5.6 Uniqueness of Construction 160

6 Fuzzification of Crisp Functions 173

6.1 Fuzzification Using the Extension Principle 173

6.2 Fuzzification Using the Expression in the Decomposition Theorem 176

6.2.1 Nested Family Using alpha-Level Sets 177

6.2.2 Nested Family Using Endpoints 181

6.2.3 Non-Nested Family Using Endpoints 184

6.3 The Relationships between EP and DT 187

6.3.1 The Equivalences 187

6.3.2 The Fuzziness 191

6.4 Differentiation of Fuzzy Functions 196

6.4.1 Defined on Open Intervals 196

6.4.2 Fuzzification of Differentiable Functions Using the Extension Principle 197

6.4.3 Fuzzification of Differentiable Functions Using the Expression in the

Decomposition Theorem 198

6.5 Integrals of Fuzzy Functions 201

6.5.1 Lebesgue Integrals on a Measurable Set 201

6.5.2 Fuzzy Riemann Integrals Using the Expression in the Decomposition

Theorem 203

6.5.3 Fuzzy Riemann Integrals Using the Extension Principle 207

7 Arithmetics of Fuzzy Sets 211

7.1 Arithmetics of Fuzzy Sets in R 211

7.1.1 Arithmetics of Fuzzy Intervals 214

7.1.2 Arithmetics Using EP and DT 220

7.1.2.1 Addition of Fuzzy Intervals 220

7.1.2.2 Difference of Fuzzy Intervals 222

7.1.2.3 Multiplication of Fuzzy Intervals 224

7.2 Arithmetics of Fuzzy Vectors 227

7.2.1 Arithmetics Using the Extension Principle 230

7.2.2 Arithmetics Using the Expression in the Decomposition Theorem 230

7.3 Difference of Vectors of Fuzzy Intervals 235

7.3.1 alpha-Level Sets of AOEP

B 235

7.3.2 alpha-Level Sets of A Oo

DT

B 237

7.3.3 alpha-Level Sets of A Oo

DT

B 239

7.3.4 alpha-Level Sets of A Oi

DT

B 241

7.3.5 The Equivalences and Fuzziness 243

7.4 Addition of Vectors of Fuzzy Intervals 244

7.4.1 alpha-Level Sets of A oplus EP

B 244

7.4.2 alpha-Level Sets of A oplus DT

B 246

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7.5 Arithmetic Operations Using Compatibility and Associativity 249

7.5.1 Compatibility 250

7.5.2 Associativity 255

7.5.3 Computational Procedure 264

7.6 Binary Operations 268

7.6.1 First Type of Binary Operation 269

7.6.2 Second Type of Binary Operation 273

7.6.3 Third Type of Binary Operation 274

7.6.4 Existence and Equivalence 277

7.6.5 Equivalent Arithmetic Operations on Fuzzy Sets in R 282

7.6.6 Equivalent Additions of Fuzzy Sets in Rm 289

7.7 Hausdorff Differences 294

7.7.1 Fair Hausdorff Difference 294

7.7.2 Composite Hausdorff Difference 299

7.7.3 Complete Composite Hausdorff Difference 304

7.8 Applications and Conclusions 312

7.8.1 Gradual Numbers 312

7.8.2 Fuzzy Linear Systems 313

7.8.3 Summary and Conclusion 315

8 Inner Product of Fuzzy Vectors 317

8.1 The First Type of Inner Product 317

8.1.1 Using the Extension Principle 318

8.1.2 Using the Expression in the Decomposition Theorem 322

8.1.2.1 The Inner Product A Oo

DT

B 323

8.1.2.2 The Inner Product A Oo

DT

B 325

8.1.2.3 The Inner Product A Oi

DT

B 327

8.1.3 The Equivalences and Fuzziness 329

8.2 The Second Type of Inner Product 330

8.2.1 Using the Extension Principle 333

8.2.2 Using the Expression in the Decomposition Theorem 335

8.2.3 Comparison of Fuzziness 338

9 Gradual Elements and Gradual Sets 343

9.1 Gradual Elements and Gradual Sets 343

9.2 Fuzzification Using Gradual Numbers 347

9.3 Elements and Subsets of Fuzzy Intervals 348

9.4 Set Operations Using Gradual Elements 351

9.4.1 Complement Set 351

9.4.2 Intersection and Union 353

9.4.3 Associativity 359

9.4.4 Equivalence with the Conventional Situation 363

9.5 Arithmetics Using Gradual Numbers 364

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10 Duality in Fuzzy Sets 373

10.1 Lower and Upper Level Sets 373

10.2 Dual Fuzzy Sets 376

10.3 Dual Extension Principle 378

10.4 Dual Arithmetics of Fuzzy Sets 380

10.5 Representation Theorem for Dual-Fuzzified Function 385

Bibliography 389

Mathematical Notations 397

Index 401