Fuzzy sets are defined as sets that contain elements having varying degrees of membership values. Given A and B are two fuzzy sets, here are the main properties of those fuzzy sets:
Commutativity :-
- (A ∪ B) = (B ∪ A)
- (A ∩ B) = (B ∩ A)
Associativity :-
- (A ∪ B) ∪ C = A ∪ (B ∪ C)
- (A ∩ B) ∩ C = A ∩ (B ∩ C)
Distributivity :-
- A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
- A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Idempotent :-
- A ∪ A = A
- A ∩ A = A
Identity :-
- A ∪ Φ = A => A ∪ X = X
- A ∩ Φ = Φ => A ∩ X = A
Note: (1) Universal Set ‘X’ has elements with unity membership value.
(2) Null Set ‘Φ’ has all elements with zero membership value.
Transitivity :-
- If A ⊆ B, B ⊆ C, then A ⊆ C
Involution :-
- (Ac)c = A
De morgan Property :-
- (A ∪ B)c = Ac ∩ Bc
- (A ∩ B)c = Ac ∪ Bc
Note: A ∪ Ac ≠ X ; A ∩ Ac ≠ Φ
Since fuzzy sets can overlap “law of excluded middle” and “law of contradiction” does not hold good.