Mathematical Foundation of Fuzzy Sets

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