Mathematical Foundations of Fuzzy SetsIntroduce yourself to the foundations of fuzzy logic with this easy-to-use guideMany 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 coursesMathematical 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.

Preface ix1 Mathematical Analysis 11.1 Infimum and Supremum 11.2 Limit Inferior and Limit Superior 31.3 Semi-Continuity 111.4 Miscellaneous 192 Fuzzy Sets 232.1 Membership Functions 232.2 alpha-level Sets 242.3 Types of Fuzzy Sets 343 Set Operations of Fuzzy Sets 433.1 Complement of Fuzzy Sets 433.2 Intersection of Fuzzy Sets 443.3 Union of Fuzzy Sets 513.4 Inductive and Direct Definitions 563.5 alpha-Level Sets of Intersection and Union 613.6 Mixed Set Operations 654 Generalized Extension Principle 694.1 Extension Principle Based on the Euclidean Space 694.2 Extension Principle Based on the Product Spaces 754.3 Extension Principle Based on the Triangular Norms 844.4 Generalized Extension Principle 925 Generating Fuzzy Sets 1095.1 Families of Sets 1105.2 Nested Families 1125.3 Generating Fuzzy Sets from Nested Families 1195.4 Generating Fuzzy Sets Based on the Expression in the DecompositionTheorem 1235.4.1 The Ordinary Situation 1235.4.2 Based on One Function 129Trim Size: 170mm x 244mm Single Column Tight Wu981527 ftoc.tex V1 - 10/14/2022 2:05pm Page vi[1][1] [1][1]vi Contents5.4.3 Based on Two Functions 1405.5 Generating Fuzzy Intervals 1505.6 Uniqueness of Construction 1606 Fuzzification of Crisp Functions 1736.1 Fuzzification Using the Extension Principle 1736.2 Fuzzification Using the Expression in the Decomposition Theorem 1766.2.1 Nested Family Using alpha-Level Sets 1776.2.2 Nested Family Using Endpoints 1816.2.3 Non-Nested Family Using Endpoints 1846.3 The Relationships between EP and DT 1876.3.1 The Equivalences 1876.3.2 The Fuzziness 1916.4 Differentiation of Fuzzy Functions 1966.4.1 Defined on Open Intervals 1966.4.2 Fuzzification of Differentiable Functions Using the Extension Principle 1976.4.3 Fuzzification of Differentiable Functions Using the Expression in theDecomposition Theorem 1986.5 Integrals of Fuzzy Functions 2016.5.1 Lebesgue Integrals on a Measurable Set 2016.5.2 Fuzzy Riemann Integrals Using the Expression in the DecompositionTheorem 2036.5.3 Fuzzy Riemann Integrals Using the Extension Principle 2077 Arithmetics of Fuzzy Sets 2117.1 Arithmetics of Fuzzy Sets in R 2117.1.1 Arithmetics of Fuzzy Intervals 2147.1.2 Arithmetics Using EP and DT 2207.1.2.1 Addition of Fuzzy Intervals 2207.1.2.2 Difference of Fuzzy Intervals 2227.1.2.3 Multiplication of Fuzzy Intervals 2247.2 Arithmetics of Fuzzy Vectors 2277.2.1 Arithmetics Using the Extension Principle 2307.2.2 Arithmetics Using the Expression in the Decomposition Theorem 2307.3 Difference of Vectors of Fuzzy Intervals 2357.3.1 alpha-Level Sets of AOEPB 2357.3.2 alpha-Level Sets of A OoDTB 2377.3.3 alpha-Level Sets of A OoDTB 2397.3.4 alpha-Level Sets of A OiDTB 2417.3.5 The Equivalences and Fuzziness 2437.4 Addition of Vectors of Fuzzy Intervals 2447.4.1 alpha-Level Sets of A oplus EPB 2447.4.2 alpha-Level Sets of A oplus DTB 246Trim Size: 170mm x 244mm Single Column Tight Wu981527 ftoc.tex V1 - 10/14/2022 2:05pm Page vii[1][1] [1][1]Contents vii7.5 Arithmetic Operations Using Compatibility and Associativity 2497.5.1 Compatibility 2507.5.2 Associativity 2557.5.3 Computational Procedure 2647.6 Binary Operations 2687.6.1 First Type of Binary Operation 2697.6.2 Second Type of Binary Operation 2737.6.3 Third Type of Binary Operation 2747.6.4 Existence and Equivalence 2777.6.5 Equivalent Arithmetic Operations on Fuzzy Sets in R 2827.6.6 Equivalent Additions of Fuzzy Sets in Rm 2897.7 Hausdorff Differences 2947.7.1 Fair Hausdorff Difference 2947.7.2 Composite Hausdorff Difference 2997.7.3 Complete Composite Hausdorff Difference 3047.8 Applications and Conclusions 3127.8.1 Gradual Numbers 3127.8.2 Fuzzy Linear Systems 3137.8.3 Summary and Conclusion 3158 Inner Product of Fuzzy Vectors 3178.1 The First Type of Inner Product 3178.1.1 Using the Extension Principle 3188.1.2 Using the Expression in the Decomposition Theorem 3228.1.2.1 The Inner Product A OoDTB 3238.1.2.2 The Inner Product A OoDTB 3258.1.2.3 The Inner Product A OiDTB 3278.1.3 The Equivalences and Fuzziness 3298.2 The Second Type of Inner Product 3308.2.1 Using the Extension Principle 3338.2.2 Using the Expression in the Decomposition Theorem 3358.2.3 Comparison of Fuzziness 3389 Gradual Elements and Gradual Sets 3439.1 Gradual Elements and Gradual Sets 3439.2 Fuzzification Using Gradual Numbers 3479.3 Elements and Subsets of Fuzzy Intervals 3489.4 Set Operations Using Gradual Elements 3519.4.1 Complement Set 3519.4.2 Intersection and Union 3539.4.3 Associativity 3599.4.4 Equivalence with the Conventional Situation 3639.5 Arithmetics Using Gradual Numbers 364Trim Size: 170mm x 244mm Single Column Tight Wu981527 ftoc.tex V1 - 10/14/2022 2:05pm Page viii[1][1] [1][1]viii Contents10 Duality in Fuzzy Sets 37310.1 Lower and Upper Level Sets 37310.2 Dual Fuzzy Sets 37610.3 Dual Extension Principle 37810.4 Dual Arithmetics of Fuzzy Sets 38010.5 Representation Theorem for Dual-Fuzzified Function 385Bibliography 389Mathematical Notations 397Index 401