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.