Necessity and possibility - Revision history

Vassia Atanassova: /* Propositions about necessity and possibility */

2017-05-11T16:14:09Z

Propositions about necessity and possibility

← Older revision		Revision as of 19:14, 11 May 2017
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	\begin{array}{r l} & \\		\begin{array}{r l} & \\
	& \Box (A \cap B) \ = \\		& \Box (A \cap B) \ = \\
	= & \Box \lbrace \langle x, min(\mu_A(x), \mu_B(x)), max(\nu_A(x), \nu_B(x)) \rangle \ \| \ x \in E \rbrace \\		= & \Box \lbrace \langle x, \min(\mu_A(x), \mu_B(x)), \max(\nu_A(x), \nu_B(x)) \rangle \ \| \ x \in E \rbrace \\
	= & \lbrace \langle x, min(\mu_A(x), \mu_B(x)), 1 - min(\mu_A(x), \mu_B(x)) \rangle \ \| \ x \in E \rbrace \\		= & \lbrace \langle x, \min(\mu_A(x), \mu_B(x)), 1 - \min(\mu_A(x), \mu_B(x)) \rangle \ \| \ x \in E \rbrace \\
	= & \lbrace \langle x, min(\mu_A(x), \mu_B(x)), max(1 - \mu_A(x), 1 - \mu_B(x)) \rangle \ \| \ x \in E \rbrace \\		= & \lbrace \langle x, \min(\mu_A(x), \mu_B(x)), \max(1 - \mu_A(x), 1 - \mu_B(x)) \rangle \ \| \ x \in E \rbrace \\
	= & \lbrace \langle x, \mu_A(x), 1 - \mu_A(x) \rangle \ \| \ x \in E \rbrace \cap \lbrace \langle x, \mu_B(x), 1 - \mu_B(x) \rangle \ \| \ x \in E \rbrace \\		= & \lbrace \langle x, \mu_A(x), 1 - \mu_A(x) \rangle \ \| \ x \in E \rbrace \cap \lbrace \langle x, \mu_B(x), 1 - \mu_B(x) \rangle \ \| \ x \in E \rbrace \\
	= & \Box A \cap \Box B		= & \Box A \cap \Box B

Vassia Atanassova: + proof / format

2009-04-14T11:43:59Z

+ proof / format

← Older revision		Revision as of 14:43, 14 April 2009
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	[[Image:IFS-necessity-possibility.gif\|right\|thumb\|250px\|[[Geometrical interpretations of intuitionistic fuzzy sets\|~~Geometrical~~ interpretation]] of necessity and possibility with a point]]		[[Image:IFS-necessity-possibility.gif\|right\|thumb\|250px\|The most common [[Geometrical interpretations of intuitionistic fuzzy sets\|geometrical interpretation]] of necessity and possibility with a point]]
	'''Necessity and possibility''' in the context of [[intuitionistic fuzzy sets]] are two modal [[Operators over intuitionistic fuzzy sets\|operators]] defined as follows:		'''Necessity and possibility''' in the context of [[intuitionistic fuzzy sets]] are two modal [[Operators over intuitionistic fuzzy sets\|operators]] defined as follows:

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	<math> \Box A = A = \Diamond A </math>.		<math> \Box A = A = \Diamond A </math>.
	</div>		</div>

			These operators are meaningless in the case of fuzzy sets, hence, this is a demonstration that intuitionistic fuzzy sets are proper extensions of the ordinary fuzzy sets. Both operators were defined in May 1983 by [[Krassimir Atanassov]].

	== Propositions about necessity and possibility ==		== Propositions about necessity and possibility ==
	[[Image:IFS-necessity-possibility-with-segments.gif\|right\|thumb\|~~300px~~\|A possible, though rarely used, geometrical interpretation of necessity and possibility with segments]]		[[Image:IFS-necessity-possibility-with-segments.gif\|right\|thumb\|250px\|A possible, though rarely used, geometrical interpretation of necessity and possibility with segments]]
	[[Image:IFS-necessity-possibility-equilateral-triangle.gif\|right\|thumb\|~~300px~~\|Another possible geometrical interpretation, within the equilateral triangle]]		[[Image:IFS-necessity-possibility-equilateral-triangle.gif\|right\|thumb\|250px\|Another possible geometrical interpretation, within the equilateral triangle]]

	For every intuitionistic fuzzy set the following statements are valid:<ref>Proposition 1.42, page 61 from [[Intuitionistic Fuzzy Sets: Theory and Applications]], [[Krassimir Atanassov]], Springer, 1999.</ref>		For every intuitionistic fuzzy set the following statements are valid:<ref>Proposition 1.42, page 61 from [[Intuitionistic Fuzzy Sets: Theory and Applications]], [[Krassimir Atanassov]], Springer, 1999.</ref>
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	\overline{\Diamond (\overline{A} + \overline{B})} & = & \Box A . \Box B \\		\overline{\Diamond (\overline{A} + \overline{B})} & = & \Box A . \Box B \\
	\overline{\Diamond (\overline{A} . \overline{B})} & = & \Box A + \Box B \\		\overline{\Diamond (\overline{A} . \overline{B})} & = & \Box A + \Box B \\
			\end{array}
			</math></div>

			''Proof of the first statement:''

			<div align="center"><math>
			\begin{array}{r l} & \\
			& \Box (A \cap B) \ = \\
			= & \Box \lbrace \langle x, min(\mu_A(x), \mu_B(x)), max(\nu_A(x), \nu_B(x)) \rangle \ \| \ x \in E \rbrace \\
			= & \lbrace \langle x, min(\mu_A(x), \mu_B(x)), 1 - min(\mu_A(x), \mu_B(x)) \rangle \ \| \ x \in E \rbrace \\
			= & \lbrace \langle x, min(\mu_A(x), \mu_B(x)), max(1 - \mu_A(x), 1 - \mu_B(x)) \rangle \ \| \ x \in E \rbrace \\
			= & \lbrace \langle x, \mu_A(x), 1 - \mu_A(x) \rangle \ \| \ x \in E \rbrace \cap \lbrace \langle x, \mu_B(x), 1 - \mu_B(x) \rangle \ \| \ x \in E \rbrace \\
			= & \Box A \cap \Box B
	\end{array}		\end{array}
	</math></div>		</math></div>

Vassia Atanassova: /* Propositions about necessity and possibility */ +image

2009-04-13T20:03:52Z

Propositions about necessity and possibility: +image

← Older revision		Revision as of 23:03, 13 April 2009
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	== Propositions about necessity and possibility ==		== Propositions about necessity and possibility ==
	[[Image:IFS-necessity-possibility-with-segments.gif\|right\|thumb\|300px\|A possible, though rarely used, geometrical interpretation of necessity and possibility with segments]]		[[Image:IFS-necessity-possibility-with-segments.gif\|right\|thumb\|300px\|A possible, though rarely used, geometrical interpretation of necessity and possibility with segments]]
			[[Image:IFS-necessity-possibility-equilateral-triangle.gif\|right\|thumb\|300px\|Another possible geometrical interpretation, within the equilateral triangle]]

	For every intuitionistic fuzzy set the following statements are valid:<ref>Proposition 1.42, page 61 from [[Intuitionistic Fuzzy Sets: Theory and Applications]], [[Krassimir Atanassov]], Springer, 1999.</ref>		For every intuitionistic fuzzy set the following statements are valid:<ref>Proposition 1.42, page 61 from [[Intuitionistic Fuzzy Sets: Theory and Applications]], [[Krassimir Atanassov]], Springer, 1999.</ref>

Vassia Atanassova at 19:51, 13 April 2009

2009-04-13T19:51:25Z

← Older revision		Revision as of 22:51, 13 April 2009
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	'''Necessity and possibility''' in the context of [[intuitionistic fuzzy sets]] are two modal [[Operators over intuitionistic fuzzy sets\|operators]] defined as follows:		'''Necessity and possibility''' in the context of [[intuitionistic fuzzy sets]] are two modal [[Operators over intuitionistic fuzzy sets\|operators]] defined as follows:

	Let <math>E</math> be a fixed universe and <math>A \subset E</math> be a given set. Let functions <math>\mu_A, \nu_A \ : \ E \ \rightarrow [0,1]</math> determine the degrees of [[membership]] and [[non-membership]]. Then,		Let <math>E</math> be a fixed universe and <math>A \subset E</math> be a given set. Let functions <math>\mu_A, \nu_A \ : \ E \ \rightarrow [0,1]</math> determine the degrees of [[membership]] and [[non-membership]]. Then, the sets

	<div align="center">		<div align="center">
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	</div>		</div>

	are called, respectively, necessity and possibility.		are called, respectively, necessity and possibility operators.

	When <math>A</math> is a proper IFS, i.e. there exists an element <math>x \in E</math> for which <math>\mu_A(x) > 0</math>, then		When <math>A</math> is a proper IFS, i.e. there exists an element <math>x \in E</math> for which <math>\mu_A(x) > 0</math>, then
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	== Propositions about necessity and possibility ==		== Propositions about necessity and possibility ==
	[[Image:IFS-necessity-possibility-with-segments.gif\|right\|thumb\|300px\|~~Geometrical~~ interpretation of necessity and possibility with segments]]		[[Image:IFS-necessity-possibility-with-segments.gif\|right\|thumb\|300px\|A possible, though rarely used, geometrical interpretation of necessity and possibility with segments]]

	For every intuitionistic fuzzy set the following statements are valid:<ref>Proposition 1.42, page 61 from [[Intuitionistic Fuzzy Sets: Theory and Applications]], [[Krassimir Atanassov]], Springer, 1999.</ref>		For every intuitionistic fuzzy set the following statements are valid:<ref>Proposition 1.42, page 61 from [[Intuitionistic Fuzzy Sets: Theory and Applications]], [[Krassimir Atanassov]], Springer, 1999.</ref>

Vassia Atanassova: + image

2009-04-13T17:46:51Z

+ image

← Older revision		Revision as of 20:46, 13 April 2009
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	[[Image:IFS-necessity-possibility.gif\|right\|thumb\|250px\|[[Geometrical interpretations of intuitionistic fuzzy sets\|Geometrical interpretation]] of necessity and possibility ]]		[[Image:IFS-necessity-possibility.gif\|right\|thumb\|250px\|[[Geometrical interpretations of intuitionistic fuzzy sets\|Geometrical interpretation]] of necessity and possibility with a point]]
	'''Necessity and possibility''' in the context of [[intuitionistic fuzzy sets]] are two modal [[Operators over intuitionistic fuzzy sets\|operators]] defined as follows:		'''Necessity and possibility''' in the context of [[intuitionistic fuzzy sets]] are two modal [[Operators over intuitionistic fuzzy sets\|operators]] defined as follows:

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	== Propositions about necessity and possibility ==		== Propositions about necessity and possibility ==
			[[Image:IFS-necessity-possibility-with-segments.gif\|right\|thumb\|300px\|Geometrical interpretation of necessity and possibility with segments]]

	For every intuitionistic fuzzy set the following statements are valid:<ref>Proposition 1.42, page 61 from [[Intuitionistic Fuzzy Sets: Theory and Applications]], [[Krassimir Atanassov]], Springer, 1999.</ref>		For every intuitionistic fuzzy set the following statements are valid:<ref>Proposition 1.42, page 61 from [[Intuitionistic Fuzzy Sets: Theory and Applications]], [[Krassimir Atanassov]], Springer, 1999.</ref>

Vassia Atanassova at 16:29, 13 April 2009

2009-04-13T16:29:01Z

@@ Line 1: / Line 1: @@
 '''Necessity and possibility''' in the context of [[intuitionistic fuzzy sets]] are two modal [[Operators over intuitionistic fuzzy sets|operators]] defined as follows:
@@ Line 19: / Line 20: @@
 </div>
-On the other hand, for every [[fuzzy set]], i.e. intuitionistic fuzzy set with <math>(\forall x \in E)(\pi_A(x) = 0)</math> it holds that
+Obviously, for every [[fuzzy set]], i.e. intuitionistic fuzzy set with <math>(\forall x \in E)(\pi_A(x) = 0)</math> it holds that
 <div align="center">
@@ Line 25: / Line 26: @@
 </div>
-[[Category:intuitionistic fuzzy sets]]
+== Propositions about necessity and possibility ==

Vassia Atanassova: New page: '''Necessity and possibility''' in the context of intuitionistic fuzzy sets are two modal operators defined as follows: Let $E$ ...

2009-04-13T13:22:26Z

New page: '''Necessity and possibility''' in the context of intuitionistic fuzzy sets are two modal operators defined as follows: Let <math>E</math>...

New page

'''Necessity and possibility''' in the context of [[intuitionistic fuzzy sets]] are two modal [[Operators over intuitionistic fuzzy sets|operators]] defined as follows:

Let <math>E</math> be a fixed universe and <math>A \subset E</math> be a given set. Let functions <math>\mu_A, \nu_A \ : \ E \ \rightarrow [0,1]</math> determine the degrees of [[membership]] and [[non-membership]]. Then,

<div align="center">
<math> \Box A = \lbrace \langle x, \mu_A(x), 1 - \mu_A(x) \rbrace \ | \ x \in E \rbrace </math>

<math> \Diamond A = \lbrace \langle x, 1 - \nu_A(x), \nu_A(x) \rbrace \ | \ x \in E \rbrace </math>
</div>

are called, respectively, necessity and possibility.

When <math>A</math> is a proper IFS, i.e. there exists an element <math>x \in E</math> for which <math>\mu_A(x) > 0</math>, then

<div align="center">
<math> \Box A \subset A \subset \Diamond A </math>

<math> \Box A \ne A \ne \Diamond A </math>.
</div>

On the other hand, for every [[fuzzy set]], i.e. intuitionistic fuzzy set with <math>(\forall x \in E)(\pi_A(x) = 0)</math> it holds that

<div align="center">
<math> \Box A = A = \Diamond A </math>.
</div>

[[Category:intuitionistic fuzzy sets]]