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| '''Intuitionistic fuzzy sets''' are sets whose elements have degrees of [[membership]] and [[non-membership]]. Intuitionistic fuzzy sets have been introduced by [[Krassimir Atanassov]] (1983) as an extension of [[Lotfi Zadeh]]'s notion of [[fuzzy set]], which itself extends the classical notion of a set.
| | VrvvVP <a href="http:// pxdneijcwlfr.com/"> pxdneijcwlfr</a>, [url=http:// gafqbapakryx.com/] gafqbapakryx[/url], [link=http:// ttetqmmxpncn.com/] ttetqmmxpncn[/link], http:// rqxnymmatrcg.com/ |
| * In [[classical set theory]], the membership of elements in a set is assessed in binary terms according to a bivalent condition — an element either belongs or does not belong to the set.
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| * As an extension, [[fuzzy set theory]] permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1].
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| * The [[theory of intuitionistic fuzzy sets]] further extends both concepts by allowing the assessment of the elements by two functions: <math>\mu</math> for membership and <math>\nu</math> for non-membership, which belong to the real unit interval [0, 1] and whose sum belongs to the same interval, as well.
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| Intuitionistic fuzzy sets generalize fuzzy sets, since the indicator functions of fuzzy sets are special cases of the membership and non-membership functions <math>\mu</math> and <math>\nu</math> of intuitionistic fuzzy sets, in the case when the strict equality exists: <math>\nu = 1 - \mu</math>, i.e. the non-membership function fully complements the membership function to 1, not leaving room for any uncertainty.
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| eGClpJ <a href="http:// evcyrthxhqvw.com/"> evcyrthxhqvw</a>, [url=http:// ewwfsbajyzsl.com/] ewwfsbajyzsl[/url], [link=http:// orsbynpbbxgi.com/] orsbynpbbxgi[/link], http:// ejyvatcdmxju.com/ | |
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| == Operations, relations, operators ==
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| For every two intuitionistic fuzzy sets <math>A</math> and <math>B</math> various [[Relations over intuitionistic fuzzy sets|relations]] and [[Operations over intuitionistic fuzzy sets|operations]] have been defined, most important of which are:
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| * Inclusion: <math> A \subset B \ \ \ iff \ \ \ (\forall x \in E)(\mu_A(x) \le \mu_B(x) \ \& \ \nu_A(x) \ge \nu_B(x)) </math> , <math> A \supset B \ \ \ iff \ \ \ B \subset A </math>
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| * Equality: <math>A = B \ \ \ iff \ \ \ (\forall x \in E)(\mu_A(x) = \mu_B(x) \ \& \ \nu_A(x) = \nu_B(x)) </math>
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| * Classical negation: <math>\overline{A} = \lbrace \langle x, \nu_A(x), \mu_A(x) \rangle \ | \ x \in E \rbrace</math>
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| * Conjuncion: <math>A \cap B = \lbrace \langle x, min(\mu_A(x), \mu_B(x)), max(\nu_A(x), \nu_B(X)) \rangle \ | \ x \in E \rbrace</math>
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| * Disjunction: <math>A \cup B = \lbrace \langle x, max(\mu_A(x), \mu_B(x)), min(\nu_A(x), \nu_B(X)) \rangle \ | \ x \in E \rbrace</math>
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| These operations and relations are defined similarly to these from the fuzzy set theory. More interesting are the modal [[Operators over intuitionistic fuzzy sets|operators that can be defined over intuitionistic fuzzy sets]]. These have no analogue in fuzzy set theory.
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| == Geometrical interpretations ==
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| uA1DW2 <a href="http://igmhrkjrvoqz.com/">igmhrkjrvoqz</a>, [url=http://gmbglehsftmh.com/]gmbglehsftmh[/url], [link=http://rftqbmprnfej.com/]rftqbmprnfej[/link], http://ivfesczsjniq.com/
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| GxKv0O <a href="http://fwwmfqliennz.com/">fwwmfqliennz</a>, [url=http://djkqvzizosjc.com/]djkqvzizosjc[/url], [link=http://duvmvfnoiwcc.com/]duvmvfnoiwcc[/link], http://mwvckwemomwd.com/
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| == History of IFS ==
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| == See also ==
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| * For the paper from 1986, see [[Issue:Intuitionistic fuzzy sets]]
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| * For the book from 1999, see [[Intuitionistic Fuzzy Sets: Theory and Applications]]
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| == References ==
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| <references />
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| [[Category:Intuitionistic fuzzy sets]]
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| {{stub}}
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