## Operations on Fuzzy Set

**1. Subset**

A ⊂ B ↔ μ_{A}(x) ≤ μ_{B}(x), ∀ x ∈ X

**2. Complement**

A^{c} ↔ μ_{Ac}(x) = 1 − μ_{A}(x), ∀ x ∈ X

**3. Superset**

A ⊃ B ↔ μ_{A}(x) ≥ μ_{B}(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 ↔ μ_{A∩ B}(x) = μ_{A}(x) ∧ μ_{B}(x), ∀ x ∈ X

∧ is a T-norm operator, can be minimum of μ_{A}(x) and μ_{B}(x).

**5. Union**

A ∪ B ↔ μ_{A ∪ B}(x) = μ_{A}(x) ∨ μ_{B}(x), ∀ x ∈ X

∨ is a S-norm operator, we can take it as a maximum of μ_{A}(x) and μ_{B}(x).

**6. Law of excluded middle**

A ∪ A^{c} ≠ U or A ∪ A^{c} ⊂ U

**7. Law of Contradiction**

A ∩ A^{c} ≠ *Φ* or A ∩ A^{c} ⊃ *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**

(A^{c})^{c} = A

**14. De-Morgan’s law**

(A ∩ B)^{c} = A^{c} ∪ B^{c}

(A ∪ B)^{c} = A^{c} ∩ B^{c}

**15. Boundary Conditions**

A ∪ φ = A and A ∪ X = X

A ∩ φ = φ and A ∩ X = A

**16. Difference**

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

**17. Absolute Difference**

A Δ B ↔ μ_{A Δ B}(x) = [μ_{A}(x) − μ_{B}(x)], ∀ x ∈ X

**18. λ-sum**

A + λB ↔ μ_{A + λB}(x) = |λμ_{A}(x) + (1−λ) μ_{B}(x)|, ∀ x ∈ X

λ ∈ [0, 1]