Mathematical Foundations of Fuzzy Sets

Preface ix 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 𝛼-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 𝛼-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 Trim Size: 170mm x 244mm Single Column Tight Wu981527 ftoc.tex V1 - 10/14/2022 2:05pm Page vi [1] [1] [1] [1] vi Contents 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 𝛼-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 𝛼-Level Sets of 𝐀̃⊖EP 𝐁̃ 235 7.3.2 𝛼-Level Sets of 𝐀̃ ⊖⋄ DT 𝐁̃ 237 7.3.3 𝛼-Level Sets of 𝐀̃ ⊖⋆ DT 𝐁̃ 239 7.3.4 𝛼-Level Sets of 𝐀̃ ⊖† DT 𝐁̃ 241 7.3.5 The Equivalences and Fuzziness 243 7.4 Addition of Vectors of Fuzzy Intervals 244 7.4.1 𝛼-Level Sets of 𝐀̃⊕EP 𝐁̃ 244 7.4.2 𝛼-Level Sets of 𝐀̃⊕DT 𝐁̃ 246 Trim Size: 170mm x 244mm Single Column Tight Wu981527 ftoc.tex V1 - 10/14/2022 2:05pm Page vii [1] [1] [1] [1] Contents vii 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 𝐀̃ ⊛⋄ DT 𝐁̃ 323 8.1.2.2 The Inner Product 𝐀̃ ⊛⋆ DT 𝐁̃ 325 8.1.2.3 The Inner Product 𝐀̃ ⊛† DT 𝐁̃ 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 Trim Size: 170mm x 244mm Single Column Tight Wu981527 ftoc.tex V1 - 10/14/2022 2:05pm Page viii [1] [1] [1] [1] viii Contents 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