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

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]

 

Leave a Comment

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Adblocker detected! Please consider reading this notice.

We've detected that you are using AdBlock Plus or some other adblocking software which is preventing the page from fully loading.

We don't have any banner, Flash, animation, obnoxious sound, or popup ad. We do not implement these annoying types of ads!

We need fund to operate the site, and almost all of it comes from our online advertising.

Please add electricalvoice.com to your ad blocking whitelist or disable your adblocking software.

×