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Mathematical Foundation of Fuzzy Sets

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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.

Author(s): Hsien-Chung Wu
Publisher: Wiley
Year: 2023

Language: English
Pages: 417
City: Hoboken

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