Mathematical foundations of fuzzy sets

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 method...

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Detalles Bibliográficos
Otros Autores: Wu, Hsien-Chung, author (author)
Formato: Libro electrónico
Idioma:Inglés
Publicado: West Sussex, England : John Wiley & Sons Ltd [2023]
Materias:
Ver en Biblioteca Universitat Ramon Llull:https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009724220906719
Tabla de Contenidos:
  • Cover
  • Title Page
  • Copyright
  • Contents
  • Preface
  • Chapter 1 Mathematical Analysis
  • 1.1 Infimum and Supremum
  • 1.2 Limit Inferior and Limit Superior
  • 1.3 Semi‐Continuity
  • 1.4 Miscellaneous
  • Chapter 2 Fuzzy Sets
  • 2.1 Membership Functions
  • 2.2 α‐level Sets
  • 2.3 Types of Fuzzy Sets
  • Chapter 3 Set Operations of Fuzzy Sets
  • 3.1 Complement of Fuzzy Sets
  • 3.2 Intersection of Fuzzy Sets
  • 3.3 Union of Fuzzy Sets
  • 3.4 Inductive and Direct Definitions
  • 3.5 α‐Level Sets of Intersection and Union
  • 3.6 Mixed Set Operations
  • Chapter 4 Generalized Extension Principle
  • 4.1 Extension Principle Based on the Euclidean Space
  • 4.2 Extension Principle Based on the Product Spaces
  • 4.3 Extension Principle Based on the Triangular Norms
  • 4.4 Generalized Extension Principle
  • Chapter 5 Generating Fuzzy Sets
  • 5.1 Families of Sets
  • 5.2 Nested Families
  • 5.3 Generating Fuzzy Sets from Nested Families
  • 5.4 Generating Fuzzy Sets Based on the Expression in the Decomposition Theorem
  • 5.4.1 The Ordinary Situation
  • 5.4.2 Based on One Function
  • 5.4.3 Based on Two Functions
  • 5.5 Generating Fuzzy Intervals
  • 5.6 Uniqueness of Construction
  • Chapter 6 Fuzzification of Crisp Functions
  • 6.1 Fuzzification Using the Extension Principle
  • 6.2 Fuzzification Using the Expression in the Decomposition Theorem
  • 6.2.1 Nested Family Using α‐Level Sets
  • 6.2.2 Nested Family Using Endpoints
  • 6.2.3 Non‐Nested Family Using Endpoints
  • 6.3 The Relationships between EP and DT
  • 6.3.1 The Equivalences
  • 6.3.2 The Fuzziness
  • 6.4 Differentiation of Fuzzy Functions
  • 6.4.1 Defined on Open Intervals
  • 6.4.2 Fuzzification of Differentiable Functions Using the Extension Principle
  • 6.4.3 Fuzzification of Differentiable Functions Using the Expression in the Decomposition Theorem
  • 6.5 Integrals of Fuzzy Functions.
  • 6.5.1 Lebesgue Integrals on a Measurable Set
  • 6.5.2 Fuzzy Riemann Integrals Using the Expression in the Decomposition Theorem
  • 6.5.3 Fuzzy Riemann Integrals Using the Extension Principle
  • Chapter 7 Arithmetics of Fuzzy Sets
  • 7.1 Arithmetics of Fuzzy Sets in R
  • 7.1.1 Arithmetics of Fuzzy Intervals
  • 7.1.2 Arithmetics Using EP and DT
  • 7.1.2.1 Addition of Fuzzy Intervals
  • 7.1.2.2 Difference of Fuzzy Intervals
  • 7.1.2.3 Multiplication of Fuzzy Intervals
  • 7.2 Arithmetics of Fuzzy Vectors
  • 7.2.1 Arithmetics Using the Extension Principle
  • 7.2.2 Arithmetics Using the Expression in the Decomposition Theorem
  • 7.3 Difference of Vectors of Fuzzy Intervals
  • 7.3.1 α‐Level Sets of A˜⊖EPB˜
  • 7.3.2 α‐Level Sets of A˜⊖DT⋄B˜
  • 7.3.3 α‐Level Sets of A˜⊖DT⋆B˜
  • 7.3.4 α‐Level Sets of A˜⊖DT†B˜
  • 7.3.5 The Equivalences and Fuzziness
  • 7.4 Addition of Vectors of Fuzzy Intervals
  • 7.4.1 α‐Level Sets of A⊕EPB˜
  • 7.4.2 α‐Level Sets of A⊕DTB˜
  • 7.5 Arithmetic Operations Using Compatibility and Associativity
  • 7.5.1 Compatibility
  • 7.5.2 Associativity
  • 7.5.3 Computational Procedure
  • 7.6 Binary Operations
  • 7.6.1 First Type of Binary Operation
  • 7.6.2 Second Type of Binary Operation
  • 7.6.3 Third Type of Binary Operation
  • 7.6.4 Existence and Equivalence
  • 7.6.5 Equivalent Arithmetic Operations on Fuzzy Sets in R
  • 7.6.6 Equivalent Additions of Fuzzy Sets in Rm
  • 7.7 Hausdorff Differences
  • 7.7.1 Fair Hausdorff Difference
  • 7.7.2 Composite Hausdorff Difference
  • 7.7.3 Complete Composite Hausdorff Difference
  • 7.8 Applications and Conclusions
  • 7.8.1 Gradual Numbers
  • 7.8.2 Fuzzy Linear Systems
  • 7.8.3 Summary and Conclusion
  • Chapter 8 Inner Product of Fuzzy Vectors
  • 8.1 The First Type of Inner Product
  • 8.1.1 Using the Extension Principle
  • 8.1.2 Using the Expression in the Decomposition Theorem.
  • 8.1.2.1 The Inner Product A˜⊛DT⋄B˜
  • 8.1.2.2 The Inner Product A˜⊛DT⋆B˜
  • 8.1.2.3 The Inner Product A˜⊛DT†B˜
  • 8.1.3 The Equivalences and Fuzziness
  • 8.2 The Second Type of Inner Product
  • 8.2.1 Using the Extension Principle
  • 8.2.2 Using the Expression in the Decomposition Theorem
  • 8.2.3 Comparison of Fuzziness
  • Chapter 9 Gradual Elements and Gradual Sets
  • 9.1 Gradual Elements and Gradual Sets
  • 9.2 Fuzzification Using Gradual Numbers
  • 9.3 Elements and Subsets of Fuzzy Intervals
  • 9.4 Set Operations Using Gradual Elements
  • 9.4.1 Complement Set
  • 9.4.2 Intersection and Union
  • 9.4.3 Associativity
  • 9.4.4 Equivalence with the Conventional Situation
  • 9.5 Arithmetics Using Gradual Numbers
  • Chapter 10 Duality in Fuzzy Sets
  • 10.1 Lower and Upper Level Sets
  • 10.2 Dual Fuzzy Sets
  • 10.3 Dual Extension Principle
  • 10.4 Dual Arithmetics of Fuzzy Sets
  • 10.5 Representation Theorem for Dual‐Fuzzified Function
  • Bibliography
  • Mathematical Notations
  • Index
  • EULA.