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Chapter 2 The Operations of Fuzzy Set

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Outline Standard operations of fuzzy set Fuzzy complement Fuzzy union Fuzzy intersection Other operations in fuzzy set Disjunctive sum Difference Distance Cartesian product T-norms and t-conorms

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Standard operation of fuzzy set Complement 3

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Standard operation of fuzzy set Union

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Standard operation of fuzzy set Intersection

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Fuzzy complement C:[0,1] [0,1]

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Fuzzy complement

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Axioms C1 and C2 called “axiomatic skeleton ” are fundamental requisites to be a complement function, i.e., for any function C:[0,1] [0,1] that satisfies axioms C1 and C2 is called a fuzzy complement. Additional requirements

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Fuzzy complement Example 1 : Standard function Axiom C1 Axiom C2 Axiom C3 Axiom C4

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Fuzzy complement Example 2 : Axiom C1 Axiom C2 XAxiom C3 XAxiom C4

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Fuzzy complement Example 3: Axiom C1 Axiom C2 Axiom C3 XAxiom C4

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Fuzzy complement Example 4: Yager’s function Axiom C1 Axiom C2 Axiom C3 Axiom C4

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Fuzzy complement Fuzzy partition If m subsets are defined in X, m-tuple (A 1, A 2,…,A m ) holding the following conditions is called a fuzzy partition.

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Fuzzy union

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Axioms U1,U2,U3 and U4 called “axiomatic skeleton ” are fundamental requisites to be a union function, i.e., for any function U:[0,1]X[0,1] [0,1] that satisfies axioms U1,U2,U3 and U4 is called a fuzzy union. Additional requirements

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Fuzzy union Example 1 : Standard function Axiom U1 Axiom U2 Axiom U3 Axiom U4 Axiom U5 Axiom U6

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Fuzzy union Example 2: Yager’s function Axiom U1 Axiom U2 Axiom U3 Axiom U4 Axiom U5 XAxiom U6

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Fuzzy union

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Some frequently used fuzzy unions – Probabilistic sum (Algebraic Sum): – Bounded Sum (Bold union): – Drastic Sum: – Hamacher’s Sum

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Fuzzy union

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Fuzzy intersection

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Axioms I1,I2,I3 and I4 called “axiomatic skeleton ” are fundamental requisites to be a intersection function, i.e., for any function I:[0,1]X[0,1] [0,1] that satisfies axioms I1,I2,I3 and I4 is called a fuzzy intersection. Additional requirements

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Fuzzy intersection Example 1 : Standard function Axiom I1 Axiom I2 Axiom I3 Axiom I4 Axiom I5 Axiom I6

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Fuzzy intersection Example 2: Yager’s function Axiom I1 Axiom I2 Axiom I3 Axiom I4 Axiom I5 XAxiom I6

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Fuzzy intersection

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Some frequently used fuzzy intersections – Probabilistic product (Algebraic product): – Bounded product (Bold intersection): – Drastic product : – Hamacher’s product

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Fuzzy intersection

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Other operations Disjunctive sum (exclusive OR)

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Other operations

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Disjoint sum (elimination of common area)

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Other operations Difference Crisp set Fuzzy set : Simple difference By using standard complement and intersection operations. Fuzzy set : Bounded difference

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Other operations Example Simple difference

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Other operations Example Bounded difference

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Other operations Distance and difference

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Other operations Distance Hamming distance Relative Hamming distance

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Other operations Euclidean distance Relative Euclidean distance Minkowski distance (w=1-> Hamming and w=2-> Euclidean)

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Other operations Cartesian product Power Cartesian product

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Other operations Example: – A = { (x1, 0.2), (x2, 0.5), (x3, 1) } – B = { (y1, 0.3), (y2, 0.9) }

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t-norms and t-conorms (s-norms)

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Duality of t-norms and t-conorms Applying complements DeMorgan’s law

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