Introduction
Fuzzy set theory generalizes classical set theory by allowing degrees of membership. The following fundamental properties hold in fuzzy set theory, just like in classical set theory. These include commutativity, associativity, distributivity, and DeMorgan’s laws.
(i) Commutativity
Statement:
A ∪ B = B ∪ A
A ∩ B = B ∩ A
Proof:
Let A and B be fuzzy sets. The membership function of union is:
μA∪B(x) = max(μA(x), μB(x))
μB∪A(x) = max(μB(x), μA(x))
As max(a, b) = max(b, a), we conclude A ∪ B = B ∪ A.
Similarly, since min(a, b) = min(b, a), A ∩ B = B ∩ A.
(ii) Associativity
Statement:
(A ∪ B) ∪ C = A ∪ (B ∪ C)
(A ∩ B) ∩ C = A ∩ (B ∩ C)
Proof:
For union:
μ(A∪B)∪C(x) = max(max(μA(x), μB(x)), μC(x))
μA∪(B∪C)(x) = max(μA(x), max(μB(x), μC(x)))
As max is associative: max(max(a,b),c) = max(a, max(b,c)), the property holds.
Similarly, for intersection using min, the associativity holds.
(iii) Distributivity
Statement:
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Proof:
Let x be any element in the universe.
μA∪(B∩C)(x) = max(μA(x), min(μB(x), μC(x)))
μ(A∪B)∩(A∪C)(x) = min(max(μA(x), μB(x)), max(μA(x), μC(x)))
These are equal due to the distributive properties of max and min operations.
Similar proof holds for intersection.
(iv) DeMorgan’s Laws
Statement:
(A ∪ B)’ = A’ ∩ B’
(A ∩ B)’ = A’ ∪ B’
Proof:
μ(A∪B)’(x) = 1 – μA∪B(x) = 1 – max(μA(x), μB(x))
μA’∩B’(x) = min(1 – μA(x), 1 – μB(x))
By identity: min(1 – a, 1 – b) = 1 – max(a, b)
So, (A ∪ B)’ = A’ ∩ B’
Similarly, (A ∩ B)’ = A’ ∪ B’
Conclusion
All these fundamental properties of classical sets also hold in fuzzy sets, with appropriate use of min and max functions instead of Boolean logic. These properties are the foundation for fuzzy logic operations used in intelligent systems, control systems, and AI applications.