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Closure and interior: Difference between revisions

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Let <math>A \subset E</math> be an IFS. Then,
Let <math>A \subset E</math> be an IFS. Then,


<math>C(A) = \lbrace \langle x, \max_{y \in E} \mu_A(y), \min_{y \in E} \nu_A(y) \rangle \ | \ x \in E \rbrace</math>
<math>C(A) = \lbrace \langle x, \sup_{y \in E} \mu_A(y), \inf_{y \in E} \nu_A(y) \rangle \ | \ x \in E \rbrace</math>


<math>I(A) = \lbrace \langle x, \min_{y \in E} \mu_A(y), \max_{y \in E} \nu_A(y) \rangle \ | \ x \in E \rbrace</math>
<math>I(A) = \lbrace \langle x, \inf_{y \in E} \mu_A(y), \sup_{y \in E} \nu_A(y) \rangle \ | \ x \in E \rbrace</math>


are respectively called '''closure''' and '''interior'''.
are respectively called '''closure''' and '''interior'''.
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* <math>\Diamond (C(A)) \ = \ C(\Diamond (A))</math>
* <math>\Diamond (C(A)) \ = \ C(\Diamond (A))</math>
* <math>\Diamond (I(A)) \ = \ I(\Diamond (A))</math>
* <math>\Diamond (I(A)) \ = \ I(\Diamond (A))</math>
== Open and closed IFSs ==
An IFS <math>A</math> is called '''open''' in <math>E</math> if <math>A = I(A)</math>. <math>A</math> is called '''closed''' in <math>E</math> if <math>A = C(A)</math>.
The following statement is valid: If <math>A</math> is an open (''closed'') IFS, then <math>A</math> is closed (''open''), too.


If <math>A</math> and <math>B</math> are intuitionistic fuzzy sets over <math>E</math>, the following statements hold about them:
If <math>A</math> and <math>B</math> are intuitionistic fuzzy sets over <math>E</math>, the following statements hold about them:
* If <math>A \subset B</math>, then <math>I(A) \subset I(B)</math> and <math>C(A) \subset C(B)</math>.
* If <math>A \subset B</math>, then <math>I(A) \subset I(B)</math> and <math>C(A) \subset C(B)</math>.
* If <math>B</math> is open and <math>B \subset A</math>, then <math>B \subset I(A)</math>.
* If <math>B</math> is closed and <math>A \subset B</math>, then <math>C(A) \subset B</math>.


[[Category:Intuitionistic fuzzy sets]]
[[Category:Intuitionistic fuzzy sets]]

Revision as of 18:33, 15 April 2009

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

Let [math]\displaystyle{ A \subset E }[/math] be an IFS. Then,

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

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

are respectively called closure and interior.

The following basic statements are valid:

  • [math]\displaystyle{ C(A) }[/math] and [math]\displaystyle{ I(A) }[/math] are intuitionistic fuzzy sets.
  • [math]\displaystyle{ I(A) \subset A \subset C(A) }[/math]
  • [math]\displaystyle{ C(C(A)) \ = \ C(A) }[/math]
  • [math]\displaystyle{ C(I(A)) \ = \ I(A) }[/math]
  • [math]\displaystyle{ I(C(A)) \ = \ C(A) }[/math]
  • [math]\displaystyle{ I(I(A)) \ = \ I(A) }[/math]

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

  • [math]\displaystyle{ C(A \cap B) \ = \ C(A) \cap C(B) }[/math]
  • [math]\displaystyle{ C(A \cup B) \ \subset \ C(A) \cup C(B) }[/math]
  • [math]\displaystyle{ I(A \cap B) \ \supset \ I(A) \cap I(B) }[/math]
  • [math]\displaystyle{ I(A \cup B) \ = \ I(A) \cup I(B) }[/math]
  • [math]\displaystyle{ \overline{I(\overline{A})} \ = \ C(A) }[/math]

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

  • [math]\displaystyle{ \Box (C(A)) \ = \ C(\Box(A)) }[/math]
  • [math]\displaystyle{ \Box (I(A)) \ = \ I(\Box(A)) }[/math]
  • [math]\displaystyle{ \Diamond (C(A)) \ = \ C(\Diamond (A)) }[/math]
  • [math]\displaystyle{ \Diamond (I(A)) \ = \ I(\Diamond (A)) }[/math]

If [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] are intuitionistic fuzzy sets over [math]\displaystyle{ E }[/math], the following statements hold about them:

  • If [math]\displaystyle{ A \subset B }[/math], then [math]\displaystyle{ I(A) \subset I(B) }[/math] and [math]\displaystyle{ C(A) \subset C(B) }[/math].
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