Closure and interior - Revision history

Vassia Atanassova: images

2009-04-17T17:33:56Z

images

← Older revision		Revision as of 20:33, 17 April 2009
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			[[Image:IFS-closure.png\|right\|thumb\|250px\|Closure]]
			[[Image:IFS-interior.png\|right\|thumb\|250px\|Interior]]

	'''Closure and interior''' are two topological [[Operators over intuitionistic fuzzy sets\|operators]], defined over [[intuitionistic fuzzy sets]], as follows.		'''Closure and interior''' are two topological [[Operators over intuitionistic fuzzy sets\|operators]], defined over [[intuitionistic fuzzy sets]], as follows.

Krassimir Atanassov at 16:33, 15 April 2009

2009-04-15T16:33:01Z

← Older revision		Revision as of 19:33, 15 April 2009
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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]]

Vassia Atanassova: /* Open and closed IFSs */

2009-04-15T12:14:25Z

Open and closed IFSs

← Older revision		Revision as of 15:14, 15 April 2009
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	== Open and closed IFSs ==		== 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>.		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.		The following statement is valid: If <math>A</math> is an open (''closed'') IFS, then <math>A</math> is closed (''open''), too.

Vassia Atanassova at 12:13, 15 April 2009

2009-04-15T12:13:30Z

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 * <math>\Diamond (C(A)) \ = \ C(\Diamond (A))</math>
 * <math>\Diamond (I(A)) \ = \ I(\Diamond (A))</math>
 [[Category:Intuitionistic fuzzy sets]]

Vassia Atanassova: New page: '''Closure and interior''' are two topological operators, defined over intuitionistic fuzzy sets, as follows. Let $A \subset E$ ...

2009-04-15T11:58:41Z

New page: '''Closure and interior''' are two topological operators, defined over intuitionistic fuzzy sets, as follows. Let <math>A \subset E</math>...

New page

'''Closure and interior''' are two topological [[Operators over intuitionistic fuzzy sets|operators]], defined over [[intuitionistic fuzzy sets]], as follows.

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>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>

are respectively called '''closure''' and '''interior'''.

The following basic statements are valid:
* <math>C(A)</math> and <math>I(A)</math> are intuitionistic fuzzy sets.
* <math>I(A) \subset A \subset C(A)</math>
* <math>C(C(A)) \ = \ C(A)</math>
* <math>C(I(A)) \ = \ I(A)</math>
* <math>I(C(A)) \ = \ C(A)</math>
* <math>I(I(A)) \ = \ I(A)</math>

When [[operations over intuitionistic fuzzy sets|operations]] and [[relations over intuitionistic fuzzy sets|relations]] are applied over the closure and interior operators, the following valid statements can be formulated:
* <math>C(A \cap B) \ = \ C(A) \cap C(B)</math>
* <math>C(A \cup B) \ \subset \ C(A) \cup C(B)</math>
* <math>I(A \cap B) \ \supset \ I(A) \cap I(B)</math>
* <math>I(A \cup B) \ = \ I(A) \cup I(B)</math>
* <math>\overline{I(\overline{A})} \ = \ C(A)</math>

Further, when the modal operators [[necessity and possibility]] are applied, it holds that:
* <math>\Box (C(A)) \ = \ C(\Box(A))</math>
* <math>\Box (I(A)) \ = \ I(\Box(A))</math>
* <math>\Diamond (C(A)) \ = \ C(\Diamond (A))</math>
* <math>\Diamond (I(A)) \ = \ I(\Diamond (A))</math>

[[Category:Intuitionistic fuzzy sets]]

Closure and interior - Revision history

Vassia Atanassova: images

Krassimir Atanassov at 16:33, 15 April 2009

Vassia Atanassova: /* Open and closed IFSs */

Vassia Atanassova at 12:13, 15 April 2009

Vassia Atanassova: New page: '''Closure and interior''' are two topological operators, defined over intuitionistic fuzzy sets, as follows. Let A \subset E...

Vassia Atanassova: New page: '''Closure and interior''' are two topological operators, defined over intuitionistic fuzzy sets, as follows. Let $A \subset E$ ...