# Operations on Fuzzy Set

## Operations on Fuzzy Set

1. Subset

A ⊂ B ↔ μA(x) ≤ μB(x),      ∀ x ∈ X

2. Complement

Ac ↔ μAc(x) = 1 − μA(x),      ∀ x ∈ X

3. Superset

A ⊃ B ↔ μA(x) ≥ μB(x),      ∀ x ∈ X

The characteristic function will never exceed beyond the value of 1. It’s value varies between 0 and 1 always.

4. Intersection

A ∩ B ↔ μA∩ B(x) = μA(x) ∧ μB(x),      ∀ x ∈ X

∧ is a T-norm operator, can be minimum of μA(x) and μB(x).

5. Union

A ∪ B ↔ μA ∪ B(x) = μA(x) ∨ μB(x),      ∀ x ∈ X

∨ is a S-norm operator, we can take it as a maximum of μA(x) and μB(x).

6. Law of excluded middle

A ∪ Ac ≠ U or A ∪ Ac ⊂ U

A ∩ Ac ≠ Φ or A ∩ Ac ⊃ Φ

8. Idempotency

A ∪ A = A   and   A ∩ A = A

9. Commutative

A ∪ B = B ∪ A  and   A ∩ B = B ∩ A

10. Associative

A ∪ (B ∪ C) = (A ∪ B) ∪ C and

A ∩ (B ∩ C) = (A ∩ B) ∩ C

11. Absorption

A ∪ (A ∩ B) = A ∩ (A ∪ B) = A

12. Distribution

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

13. Double Negation or Involution

(Ac)c = A

14. De-Morgan’s  law

(A ∩ B)c = Ac ∪ Bc

(A ∪ B)c = Ac ∩ Bc

15. Boundary Conditions

A ∪ φ = A and A ∪ X = X

A ∩ φ = φ and A ∩ X = A

16. Difference

A −B ↔ μA −B(x) = max[{μA(x) − μB(x)}, 0],      ∀ x ∈ X

17. Absolute Difference

A Δ B ↔ μA Δ B(x) = [μA(x) − μB(x)],      ∀ x ∈ X

18. λ-sum

A + λB ↔ μA + λB(x) = |λμA(x) + (1−λ) μB(x)|,      ∀ x ∈ X

λ ∈ [0, 1]

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