Properties of Fuzzy Sets

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 :-

• (A^c)^c = A

De morgan Property :-

• (A ∪ B)^c = A^c ∩ B^c
• (A ∩ B)^c = A^c ∪ B^c

Note: A ∪ A^c ≠ X ;  A ∩ A^c ≠ Φ

Since fuzzy sets can overlap “law of excluded middle” and “law of contradiction” does not hold good.