https://ifigenia.org/index.php?title=Intuitionistic_fuzzy_sets&feed=atom&action=history
Intuitionistic fuzzy sets - Revision history
2024-03-29T11:51:10Z
Revision history for this page on the wiki
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https://ifigenia.org/index.php?title=Intuitionistic_fuzzy_sets&diff=11370&oldid=prev
Vassia Atanassova at 01:39, 9 September 2022
2022-09-09T01:39:27Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:39, 9 September 2022</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''Intuitionistic fuzzy sets''' are sets whose elements have degrees of <del style="font-weight: bold; text-decoration: none;">[[</del>membership<del style="font-weight: bold; text-decoration: none;">]] </del>and <del style="font-weight: bold; text-decoration: none;">[[</del>non-membership<del style="font-weight: bold; text-decoration: none;">]]</del>. 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. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''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. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* In <del style="font-weight: bold; text-decoration: none;">[[</del>classical set theory<del style="font-weight: bold; text-decoration: none;">]]</del>, 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. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* 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. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* As an extension, <del style="font-weight: bold; text-decoration: none;">[[</del>fuzzy set theory<del style="font-weight: bold; text-decoration: none;">]] </del>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].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* 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].</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* The <del style="font-weight: bold; text-decoration: none;">[[</del>theory of intuitionistic fuzzy sets<del style="font-weight: bold; text-decoration: none;">]] </del>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.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* 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.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
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Vassia Atanassova
https://ifigenia.org/index.php?title=Intuitionistic_fuzzy_sets&diff=7789&oldid=prev
Peter Vassilev at 13:02, 7 October 2015
2015-10-07T13:02:02Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:02, 7 October 2015</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== See also ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== See also ==</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* For the paper from 1986, see [[Issue:Intuitionistic fuzzy sets]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* For the paper from 1986, see [[Issue:<ins style="font-weight: bold; text-decoration: none;">Intuitionistic fuzzy sets|</ins>Intuitionistic fuzzy sets]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* For the book from 1999, see [[Intuitionistic Fuzzy Sets: Theory and Applications]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* For the book from 1999, see [[Intuitionistic Fuzzy Sets: Theory and Applications]]</div></td></tr>
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Peter Vassilev
https://ifigenia.org/index.php?title=Intuitionistic_fuzzy_sets&diff=7207&oldid=prev
Peter Vassilev: /* Operations, relations, operators */
2014-05-09T10:08:28Z
<p><span dir="auto"><span class="autocomment">Operations, relations, operators</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 12:08, 9 May 2014</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* 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></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* Inclusion: <math> A \subset B \ \ \ <ins style="font-weight: bold; text-decoration: none;">\text{</ins>iff<ins style="font-weight: bold; text-decoration: none;">} </ins>\ \ \ (\forall x \in E)(\mu_A(x) \le \mu_B(x) \ \& \ \nu_A(x) \ge \nu_B(x)) </math> , <math> A \supset B \ \ \ <ins style="font-weight: bold; text-decoration: none;">\text{</ins>iff<ins style="font-weight: bold; text-decoration: none;">} </ins>\ \ \ B \subset A </math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* Equality: <math>A = B \ \ \ iff \ \ \ (\forall x \in E)(\mu_A(x) = \mu_B(x) \ \& \ \nu_A(x) = \nu_B(x)) </math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* Equality: <math>A = B \ \ \ <ins style="font-weight: bold; text-decoration: none;">\text{</ins>iff<ins style="font-weight: bold; text-decoration: none;">}</ins>\ \ \ (\forall x \in E)(\mu_A(x) = \mu_B(x) \ \& \ \nu_A(x) = \nu_B(x)) </math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Classical negation: <math>\overline{A} = \lbrace \langle x, \nu_A(x), \mu_A(x) \rangle \ | \ x \in E \rbrace</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Classical negation: <math>\overline{A} = \lbrace \langle x, \nu_A(x), \mu_A(x) \rangle \ | \ x \in E \rbrace</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* 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></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* Conjuncion: <math>A \cap B = \lbrace \langle x, <ins style="font-weight: bold; text-decoration: none;">\</ins>min(\mu_A(x), \mu_B(x)), <ins style="font-weight: bold; text-decoration: none;">\</ins>max(\nu_A(x), \nu_B(X)) \rangle \ | \ x \in E \rbrace</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* 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></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* Disjunction: <math>A \cup B = \lbrace \langle x, <ins style="font-weight: bold; text-decoration: none;">\</ins>max(\mu_A(x), \mu_B(x)), <ins style="font-weight: bold; text-decoration: none;">\</ins>min(\nu_A(x), \nu_B(X)) \rangle \ | \ x \in E \rbrace</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
</table>
Peter Vassilev
https://ifigenia.org/index.php?title=Intuitionistic_fuzzy_sets&diff=3900&oldid=prev
Vassia Atanassova: Revision as of 10:17, 16 August 2009
2009-10-22T07:32:05Z
<p>Revision as of 10:17, 16 August 2009</p>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">eP6yS3 </del><a <del style="font-weight: bold; text-decoration: none;">href</del>="<del style="font-weight: bold; text-decoration: none;">http</del>:/<del style="font-weight: bold; text-decoration: none;">/byjuahovskip</del>.<del style="font-weight: bold; text-decoration: none;">com</del>/"><del style="font-weight: bold; text-decoration: none;">byjuahovskip</del></<del style="font-weight: bold; text-decoration: none;">a</del>>, [<del style="font-weight: bold; text-decoration: none;">url=http</del>://<del style="font-weight: bold; text-decoration: none;">ecfgmskdzpnm</del>.<del style="font-weight: bold; text-decoration: none;">com/</del>]<del style="font-weight: bold; text-decoration: none;">ecfgmskdzpnm</del>[/<del style="font-weight: bold; text-decoration: none;">url</del>], [<del style="font-weight: bold; text-decoration: none;">link</del>=<del style="font-weight: bold; text-decoration: none;">http:</del>//<del style="font-weight: bold; text-decoration: none;">pbepdacjdddx</del>.<del style="font-weight: bold; text-decoration: none;">com</del>/]<del style="font-weight: bold; text-decoration: none;">pbepdacjdddx</del>[/<del style="font-weight: bold; text-decoration: none;">link]</del>, <del style="font-weight: bold; text-decoration: none;">http</del>://<del style="font-weight: bold; text-decoration: none;">xzztyydjscny</del>.<del style="font-weight: bold; text-decoration: none;">com</del>/</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">'''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. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* 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. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* 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].</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* 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.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">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</ins><<ins style="font-weight: bold; text-decoration: none;">/math>, i.e. the non-membership function fully complements the membership function to 1, not leaving room for any uncertainty.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Formal definition ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Let us have a fixed universe '''''E'''''. Let '''''A''''' be </ins>a <ins style="font-weight: bold; text-decoration: none;">subset of '''''E'''''. Let us construct the set </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><div align</ins>="<ins style="font-weight: bold; text-decoration: none;">center"><math>A^* = \lbrace \langle x, \mu_A(x), \nu_A(x) \rangle \ | \ x \in E \rbrace</math></div> </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">where <math>0 \leq \mu_A(x) + \nu_A(x) \leq 1</math>. We will call the set '''''A*''''' '''intuitionistic fuzzy set''' (IFS). </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In the publications on IFS authors mainly deal with the concept of intuitionistic fuzzy set '''''A*''''' rather then with fixed set '''''A'''''. This is why, for the sake of simplicity, major publications presenting the very definition of the concept often use notation '''''A''''' instead of '''''A*'''''.<ref>Paper [[Issue</ins>:<ins style="font-weight: bold; text-decoration: none;">Intuitionistic fuzzy sets|"Intuitionistic Fuzzy Sets"]], Krassimir T. Atanassov, [[Fuzzy Sets and Systems]], North-Holland, Volume 20 (1986), pages 87-96, ISSN 0165-0114<</ins>/<ins style="font-weight: bold; text-decoration: none;">ref><ref>Book [[Intuitionistic Fuzzy Sets: Theory and Applications|"Intuitionistic Fuzzy Sets"]], Krassimir T</ins>. <ins style="font-weight: bold; text-decoration: none;">Atanassov, Series "Studies in Fuzziness and Soft Computing", Volume 35, Springer Physica-Verlag, 1999, ISBN 3-7908-1228-5<</ins>/<ins style="font-weight: bold; text-decoration: none;">ref> Mathematically, a more precise definition of the IFS is the following:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><div align="center</ins>"><ins style="font-weight: bold; text-decoration: none;"><math>A^* = \lbrace \langle x, \mu_A(x), \nu_A(x) \rangle \ | \ x \in E \ \& \ 0 \leq \mu_A(x) + \nu_A(x) \leq 1 \rbrace</math></ins></<ins style="font-weight: bold; text-decoration: none;">div> </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">but it is also more complex one and never used, as of 2008.<ref</ins>><ins style="font-weight: bold; text-decoration: none;">"25 years intuitionistic fuzzy sets</ins>, <ins style="font-weight: bold; text-decoration: none;">or: The most significant results and mistakes of mine", Krassimir Atanassov, [</ins>[<ins style="font-weight: bold; text-decoration: none;">International Workshop on Intuitionistic Fuzzy Sets and Generalized Nets/2008|Int. Workshop on IFS and GN]], 17 Oct. 2008 ([[Media:7IWIFSGN-Atanassov.pdf|presentation]], publication in press)</ref></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Functions <math>\mu_A</ins>: <ins style="font-weight: bold; text-decoration: none;">E \to [0,1]<</ins>/<ins style="font-weight: bold; text-decoration: none;">math> and <math>\nu_A: E \to [0,1]<</ins>/<ins style="font-weight: bold; text-decoration: none;">math> represent degree of [[membership]] (validity, etc</ins>.<ins style="font-weight: bold; text-decoration: none;">) and [[non-membership]</ins>] <ins style="font-weight: bold; text-decoration: none;">(non-validity, etc.). Also defined is function <math>\pi_A: E \to </ins>[<ins style="font-weight: bold; text-decoration: none;">0,1]<</ins>/<ins style="font-weight: bold; text-decoration: none;">math> through <math>\pi(x) = 1 - \mu (x) - \nu (x)</math>, corresponding to the degree of [[uncertainty|uncertainty (indeterminacy, etc.)</ins>]<ins style="font-weight: bold; text-decoration: none;">]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Obviously</ins>, <ins style="font-weight: bold; text-decoration: none;">for every ordinary [</ins>[<ins style="font-weight: bold; text-decoration: none;">fuzzy set]] <math>A</math>: <math>\pi_A(x) </ins>= <ins style="font-weight: bold; text-decoration: none;">0<</ins>/<ins style="font-weight: bold; text-decoration: none;">math> for each <math>x \in E<</ins>/<ins style="font-weight: bold; text-decoration: none;">math> and these sets have the form <math>\lbrace \langle x, \mu_{A}(x), 1-\mu_{A}(x)\rangle |x \in E \rbrace</ins>.<ins style="font-weight: bold; text-decoration: none;"></math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Properties of IFS ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Operations, relations, operators ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">For every two intuitionistic fuzzy sets <math>A</math> and <math>B<</ins>/<ins style="font-weight: bold; text-decoration: none;">math> various [[Relations over intuitionistic fuzzy sets|relations]</ins>] <ins style="font-weight: bold; text-decoration: none;">and </ins>[<ins style="font-weight: bold; text-decoration: none;">[Operations over intuitionistic fuzzy sets|operations]] have been defined, most important of which are:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Inclusion: <math> A \subset B \ \ \ iff \ \ \ (\forall x \in E)(\mu_A(x) \le \mu_B(x) \ \& \ \nu_A(x) \ge \nu_B(x)) <</ins>/<ins style="font-weight: bold; text-decoration: none;">math> , <math> A \supset B \ \ \ iff \ \ \ B \subset A </math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Equality: <math>A = B \ \ \ iff \ \ \ (\forall x \in E)(\mu_A(x) = \mu_B(x) \ \& \ \nu_A(x) = \nu_B(x)) </math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Classical negation: <math>\overline{A} = \lbrace \langle x, \nu_A(x)</ins>, <ins style="font-weight: bold; text-decoration: none;">\mu_A(x) \rangle \ | \ x \in E \rbrace</math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Conjuncion</ins>: <ins style="font-weight: bold; text-decoration: none;"><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<</ins>/<ins style="font-weight: bold; text-decoration: none;">math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* 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<</ins>/<ins style="font-weight: bold; text-decoration: none;">math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">These operations and relations are defined similarly to these from the fuzzy set theory</ins>. <ins style="font-weight: bold; text-decoration: none;">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.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Geometrical interpretations ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Relations with other concepts ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Applications of IFS ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== History of IFS ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== See also ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* For the paper from 1986, see [[Issue:Intuitionistic fuzzy sets]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* For the book from 1999, see [[Intuitionistic Fuzzy Sets: Theory and Applications]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== References ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><references </ins>/<ins style="font-weight: bold; text-decoration: none;">></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Intuitionistic fuzzy sets]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{stub}}</ins></div></td></tr>
</table>
Vassia Atanassova
https://ifigenia.org/index.php?title=Intuitionistic_fuzzy_sets&diff=3897&oldid=prev
66.17.177.171: przMhUwxPgasIYvHpzT
2009-10-21T10:26:50Z
<p>przMhUwxPgasIYvHpzT</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 12:26, 21 October 2009</td>
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66.17.177.171
https://ifigenia.org/index.php?title=Intuitionistic_fuzzy_sets&diff=3893&oldid=prev
84.52.123.19: BKabCLbRtUwxwkEgg
2009-10-21T10:19:35Z
<p>BKabCLbRtUwxwkEgg</p>
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</table>
84.52.123.19
https://ifigenia.org/index.php?title=Intuitionistic_fuzzy_sets&diff=3891&oldid=prev
194.85.242.62: drrqHyzYQGza
2009-10-21T10:16:43Z
<p>drrqHyzYQGza</p>
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</table>
194.85.242.62
https://ifigenia.org/index.php?title=Intuitionistic_fuzzy_sets&diff=3889&oldid=prev
204.62.8.253: sFgZHfzjKp
2009-10-21T10:09:39Z
<p>sFgZHfzjKp</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 12:09, 21 October 2009</td>
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</table>
204.62.8.253
https://ifigenia.org/index.php?title=Intuitionistic_fuzzy_sets&diff=3884&oldid=prev
202.12.73.8: EslrqaJDFTqB
2009-10-21T09:55:43Z
<p>EslrqaJDFTqB</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:55, 21 October 2009</td>
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</table>
202.12.73.8
https://ifigenia.org/index.php?title=Intuitionistic_fuzzy_sets&diff=3883&oldid=prev
222.124.199.111: UKTTErdLqIHYw
2009-10-21T09:53:41Z
<p>UKTTErdLqIHYw</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:53, 21 October 2009</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">'''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. </del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">ct2D1E </ins> <a href="http://<ins style="font-weight: bold; text-decoration: none;">qjlrmejgmplo</ins>.com/"><ins style="font-weight: bold; text-decoration: none;">qjlrmejgmplo</ins></a>, [url=http://<ins style="font-weight: bold; text-decoration: none;">bsjcfbhrideh</ins>.com/]<ins style="font-weight: bold; text-decoration: none;">bsjcfbhrideh</ins>[/url], [link=http://<ins style="font-weight: bold; text-decoration: none;">zewmobjmtkor</ins>.com/]<ins style="font-weight: bold; text-decoration: none;">zewmobjmtkor</ins>[/link], http://<ins style="font-weight: bold; text-decoration: none;">baankzyiqlxo</ins>.com/</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* 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. </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* 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].</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* 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.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">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.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">LrpaAf <a href="http://arqnvsydaipw.com/">arqnvsydaipw</a>, [url=http://ocevdnnssdkl.com/]ocevdnnssdkl[/url], [link=http://dvsoqcebjzxu.com/]dvsoqcebjzxu[/link], http://tvcknxjhnilt.com/</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Properties of IFS ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">kPNqXk </del> <a href="http://<del style="font-weight: bold; text-decoration: none;">ilqgjeizpeio</del>.com/"><del style="font-weight: bold; text-decoration: none;">ilqgjeizpeio</del></a>, [url=http://<del style="font-weight: bold; text-decoration: none;">kcuhacfdcddk</del>.com/]<del style="font-weight: bold; text-decoration: none;">kcuhacfdcddk</del>[/url], [link=http://<del style="font-weight: bold; text-decoration: none;">tdswbnhguzra</del>.com/]<del style="font-weight: bold; text-decoration: none;">tdswbnhguzra</del>[/link], http://<del style="font-weight: bold; text-decoration: none;">ggkwbsfuefsx</del>.com/</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Geometrical interpretations ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">JcpPpX <a href="http://hswsvgbhesvq.com/">hswsvgbhesvq</a>, [url=http://fnjaeflpmpyo.com/]fnjaeflpmpyo[/url], [link=http://jfkxczljhmhf.com/]jfkxczljhmhf[/link], http://wkdilbiwvvfa.com/</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Applications of IFS ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== History of IFS ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== See also ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* For the paper from 1986, see [[Issue:Intuitionistic fuzzy sets]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* For the book from 1999, see [[Intuitionistic Fuzzy Sets: Theory and Applications]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== References ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"><references /></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Intuitionistic fuzzy sets]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{stub}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
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