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FUZZY LOGICFuzzy Set (Value)Let X be a universe of discourse of a fuzzy variable and x be its elementsOne or more fuzzy sets (or values) Ai can be defined over XExample: Fuzzy variable: Age Universe of discourse: 0 – 120 years Fuzzy values: Child, Young, OldA fuzzy set A is characterized by a membership function µA(x) that associates each element x with a degree of membership value in AThe value of membership is between 0 and 1 and it represents the degree to which an element x belongs to the fuzzy set A
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FUZZY LOGICFuzzy Set RepresentationFuzzy Set A = (a1, a2, … an) ai = µA(xi) xi = an element of X X = universe of discourseFor clearer representation A = (a1/x1, a2/x2, …, an/xn)Example: Tall = (0/5’, 0.25/5.5’, 0.9/5.75’, 1/6’, 1/7’, …)
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FUZZY LOGICFuzzy Sets OperationsIntersection (A B)In classical set theory the intersection of two sets containsthose elements that are common to bothIn fuzzy set theory, the value of those elements in theintersection: µA B(x) = min [µA(x), µB(x)]e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6) Short = (1/5, 0.8/5.25, 0.5/5.5, 0.1/5.75, 0/6) Tall Short = (0/5, 0.1/5.25, 0.5/5.5, 0.1/5.75, 0/6) = Medium
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FUZZY LOGICFuzzy Sets OperationsUnion (A B)In classical set theory the union of two sets contains thoseelements that are in any one of the two setsIn fuzzy set theory, the value of those elements in the union: µA B(x) = max [µA(x), µB(x)]e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6) Short = (1/5, 0.8/5.25, 0.5/5.5, 0.1/5.75) Tall Short = (1/5, 0.8/5.25, 0.5/5.5, 0.8/5.75, 1/6) = not Medium
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FUZZY LOGICFuzzy Sets OperationsComplement (A)In fuzzy set theory, the value of complement of A is: µ A(x) = 1 - µA(x)e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6) Tall = (1/5, 0.9/5.25, 0.5/5.5, 0.2/5.75, 0/6)
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FUZZY LOGICFuzzy RelationsFuzzy relation between two universes U and V is defined as: µR (u, v) = µAxB (u, v) = min [µA (u), µB (v)]i.e. we take the minimum of the memberships of the twoelements which are to be related
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FUZZY RULESApproximate ReasoningExample: Let there be a fuzzy associative matrix M for therule: if A then Be.g. If Temperature is normal then Speed is mediumLet A = [0/100, 0.5/125, 1/150, 0.5/175, 0/200] B = [0/10, 0.6/20, 1/30, 0.6/40, 0/50]
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FUZZY LOGICComposition of Fuzzy RelationsNow we need a operator which allows us to infer somethingabout B, given Acurrent“Composition” is such an operator
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FUZZY LOGICComposition of Fuzzy RelationsLet there be three universes U, V and WLet R be the relation that relates elements from U to V e.g. R= 0.6 0.8 0.7 0.9And let S be the relation between V and W e.g. S= 0.3 0.1 0.2 0.8
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FUZZY LOGICComposition of Fuzzy RelationsWith the help of an operation called “composition” we can findthe relation T that maps elements of U to WBy max-min rule T = R S = maxvV { min(R(u, v), S(v, w)) } 0.6 0.8 0.3 0.1 = 0.3 0.8 0.7 0.9 0.2 0.8 0.3 0.8Where element (1,1) is obtained by max{min(0.6, 0.3), min(0.8, 0.2)} = 0.3Note that S R = 0.3 0.3 R S 0.7 0.8
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FUZZY LOGICComposition of Fuzzy RelationsR = R(u, v) v1 v2 u1 0.6 0.8 u2 0.7 0.9 V v2 0.9 0.8 v1 0.6 0.7 u1 u2 U
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FUZZY LOGICComposition of Fuzzy RelationsS = S(v, w) w1 w2 v1 0.3 0.1 v2 0.2 0.8 V v2 0.8 0.2 v1 0.1 0.3 w1 w2 W
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FUZZY LOGICComposition of Fuzzy RelationsT = R S = maxvV { min(R(u, v), S(v, w)) } 0.6 0.8 0.3 0.1 = 0.3 0.8 0.7 0.9 0.2 0.8 0.3 0.8 V v2 0.9 0.8 0.2 v1 0.8 0.1 0.3 0.6 0.7 u1 u2 U w1 w2 W
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FUZZY LOGICComposition of Fuzzy RelationsT = R S = maxvV { min(R(u, v), S(v, w)) } 0.6 0.8 0.3 0.1 = 0.3 0.8 0.7 0.9 0.2 0.8 0.3 0.8 V v2 v1 u1 u2 w1 0.3 0.3 U 0.8 w2 0.8 W
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