Closure and 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 $$C(A) \subset C(B)$$.