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Closure and interior

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Closure
Interior

Closure and interior are two topological operators, defined over intuitionistic fuzzy sets, as follows.

Let A \subset E be an IFS. Then,

C(A) = \lbrace \langle x, \sup_{y \in E} \mu_A(y), \inf_{y \in E} \nu_A(y) \rangle \ | \ x \in E \rbrace

I(A) = \lbrace \langle x, \inf_{y \in E} \mu_A(y), \sup_{y \in E} \nu_A(y) \rangle \ | \ x \in E \rbrace

are respectively called closure and interior.

The following basic statements are valid:

  • C(A) and I(A) are intuitionistic fuzzy sets.
  • I(A) \subset A \subset C(A)
  • C(C(A)) \ = \ C(A)
  • C(I(A)) \ = \ I(A)
  • I(C(A)) \ = \ C(A)
  • I(I(A)) \ = \ I(A)

When operations and relations are applied over the closure and interior operators, the following valid statements can be formulated:

  • C(A \cap B) \ = \ C(A) \cap C(B)
  • C(A \cup B) \ \subset \ C(A) \cup C(B)
  • I(A \cap B) \ \supset \ I(A) \cap I(B)
  • I(A \cup B) \ = \ I(A) \cup I(B)
  • \overline{I(\overline{A})} \ = \ C(A)

Further, when the modal operators necessity and possibility are applied, it holds that:

  • \Box (C(A)) \ = \ C(\Box(A))
  • \Box (I(A)) \ = \ I(\Box(A))
  • \Diamond (C(A)) \ = \ C(\Diamond (A))
  • \Diamond (I(A)) \ = \ I(\Diamond (A))

If A and B are intuitionistic fuzzy sets over E, the following statements hold about them:

  • If A \subset B, then I(A) \subset I(B) and C(A) \subset C(B).