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Fuzzy Logic

Fuzzy Logic. C. Alternate Fuzzy Logic F1. General Fuzzy Complement. Axioms Boundary Conditions: c (0)=1; c (1)=0 Monotonicity: If a > b , then c(a)  c(b) Supplementary Continuity: c is a continuous function Involutive: c(c(a)) = a. General Fuzzy Complement.

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Fuzzy Logic

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  1. Fuzzy Logic C. Alternate Fuzzy Logic F1

  2. General Fuzzy Complement • Axioms Boundary Conditions: c(0)=1; c(1)=0 Monotonicity: If a > b, then c(a)  c(b) • Supplementary Continuity: c is a continuous function Involutive: c(c(a)) = a

  3. General Fuzzy Complement • Example Complements Zadeh Complement: c(a)=1-a Sugeno Class 2F2 Yager Class 3F2

  4. General Fuzzy Union • Axioms Boundary Conditions: u(0,0) = 0; u(1,1) = 1 Commutative: u(a,b) = u(b,a) Monotonic: If a andb , then u(a,b)  u( ,) Associative: u(a,u(b,c)) = u(u(a,b),c) • Supplementary u is continuous Idempodent: u(a,a)=a

  5. General Fuzzy Union • Example Zadeh Union: u(a,b) = max(a,b) Yager Class (not idempotent) 4 Sum-Product Inferencing

  6. General Fuzzy Intersection • Axioms Boundary Conditions: i(0,0) = 0; i(1,1) = 1 Commutative: i(a,b) = i(b,a) Monotonic: If a andb , then i(a,b)  i( ,) Associative: i(a,i(b,c)) = i(i(a,b),c) • Supplemtary i is continuous Idempodent: i(a,a) = a

  7. General Fuzzy Intersection • Example Zadeh Intersection: i(a,b) = min(a,b) Yager Class (Not idempotent) 5 Sum-Product Inferencing 5

  8. Interesting Theorems • u(a,b)  max(a,b) 6 • i(a,b)  min(a,b) • u(a,b) = max(a,b) is the only union operation satisfying all 4 axioms and 2 supplementary properties • i(a,b) = min(a,b) is the only intersection operation satisfying all 4 axioms and 2 supplementary properties

  9. Interesting Theorems • Recall min violates the law of excluded middle and max the law of contradiction • The whack-a-mole principle: Fuzzy set operations of union, intersection and continuous complement that satisfy the law of excluded middle and the law of contradiction are not idempotent (nor distributive).

  10. AGGREGATION • h: [0,1]n  [0,1] • Axioms 1. Boundary Conditions h(0,0,…,0)=0 h(1,1,…,1)=1 2. Monotonicity: When X > x, h(a,b,c,…,X,…) h(a,b,c,…,x,…) 3. Continuous 4. Symmetric under permutations • Generalized means 7

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