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
7. Law of Contradiction
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]
