1. Fuzzy Logic C. Alternate Fuzzy Logic F1F1

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 2 F22F2 Yager Class3 F23F2

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  and b  , 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) 44 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  and b  , 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) 55 Sum-Product Inferencing 55

8. Interesting Theorems u(a,b)  max(a,b) 66 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 77