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Necessity and possibility: Difference between revisions
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[[Image:IFS-necessity-possibility.gif|right|thumb|250px|[[Geometrical interpretations of intuitionistic fuzzy sets| | [[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| | [[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| | [[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> |
Revision as of 14:43, 14 April 2009
![](/images/thumb/f/f6/IFS-necessity-possibility.gif/250px-IFS-necessity-possibility.gif)
Necessity and possibility in the context of intuitionistic fuzzy sets are two modal operators defined as follows:
Let [math]\displaystyle{ E }[/math] be a fixed universe and [math]\displaystyle{ A \subset E }[/math] be a given set. Let functions [math]\displaystyle{ \mu_A, \nu_A \ : \ E \ \rightarrow [0,1] }[/math] determine the degrees of membership and non-membership. Then, the sets
[math]\displaystyle{ \Box A = \lbrace \langle x, \mu_A(x), 1 - \mu_A(x) \rbrace \ | \ x \in E \rbrace }[/math]
[math]\displaystyle{ \Diamond A = \lbrace \langle x, 1 - \nu_A(x), \nu_A(x) \rbrace \ | \ x \in E \rbrace }[/math]
are called, respectively, necessity and possibility operators.
When [math]\displaystyle{ A }[/math] is a proper IFS, i.e. there exists an element [math]\displaystyle{ x \in E }[/math] for which [math]\displaystyle{ \mu_A(x) \gt 0 }[/math], then
[math]\displaystyle{ \Box A \subset A \subset \Diamond A }[/math]
[math]\displaystyle{ \Box A \ne A \ne \Diamond A }[/math].
Obviously, for every fuzzy set, i.e. intuitionistic fuzzy set with [math]\displaystyle{ (\forall x \in E)(\pi_A(x) = 0) }[/math] it holds that
[math]\displaystyle{ \Box A = A = \Diamond A }[/math].
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
![](/images/thumb/3/3f/IFS-necessity-possibility-with-segments.gif/250px-IFS-necessity-possibility-with-segments.gif)
![](/images/thumb/c/ce/IFS-necessity-possibility-equilateral-triangle.gif/250px-IFS-necessity-possibility-equilateral-triangle.gif)
For every intuitionistic fuzzy set the following statements are valid:[1]
Proof of the first statement:
The following statements are also valid:[2]
Proof of the first statement:
References
- ↑ Proposition 1.42, page 61 from Intuitionistic Fuzzy Sets: Theory and Applications, Krassimir Atanassov, Springer, 1999.
- ↑ Theorem 1.43, page 62 from Intuitionistic Fuzzy Sets: Theory and Applications, Krassimir Atanassov, Springer, 1999.