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Fuzzy mathematics
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<h2 id="definition">Definition</h2> <p>A fuzzy subset <i>A</i> of a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> <i>X</i> is defined by a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> <i>A</i>: <i>X</i> → <i>L</i>, where <i>L</i> is typically the <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> [0, 1]. This function is called the membership function of the fuzzy subset and assigns to each element <i>x</i> in <i>X</i> a degree of membership <i>A</i>(<i>x</i>) in the fuzzy set <i>A</i>. </p><p>In classical set theory, a subset of <i>X</i> can be represented by an <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> (also known as a <a href="/facts/Indicator_function/QEuh04NM">characteristic function</a>), which maps elements to either 0 or 1, indicating non-membership or full membership, respectively. Fuzzy subsets generalize this concept by allowing any real value between 0 and 1, thereby enabling partial membership. </p><p>More generally, the codomain <i>L</i> of the membership function can be replaced with any <a href="/facts/Complete_lattice/pQbphWeR">complete lattice</a>, resulting in the broader framework of L-fuzzy sets.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <h2 id="fuzzification">Fuzzification</h2> <p>The evolution of the fuzzification of mathematical concepts can be broken down into three stages:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <ol><li>straightforward fuzzification during the sixties and seventies,</li> <li>the explosion of the possible choices in the generalization process during the eighties,</li> <li>the standardization, axiomatization, and <i>L</i>-fuzzification in the nineties.</li></ol> <p>Usually, a fuzzification of mathematical concepts is based on a generalization of these concepts from characteristic functions to membership functions. Let <i>A</i> and <i>B</i> be two fuzzy subsets of <i>X</i>. The <i>intersection</i> <i>A</i> ∩ <i>B</i> and <i>union</i> <i>A</i> ∪ <i>B</i> are defined as follows: (<i>A</i> ∩ <i>B</i>)(<i>x</i>) = min(<i>A</i>(<i>x</i>), <i>B</i>(<i>x</i>)), (<i>A</i> ∪ <i>B</i>)(<i>x</i>) = max(<i>A</i>(<i>x</i>), <i>B</i>(<i>x</i>)) for all <i>x</i> in <i>X</i>. Instead of min and max one can use <a href="/facts/T-norm/oYR3JgIy">t-norm</a> and t-conorm, respectively;<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> for example, min(<i>a</i>, <i>b</i>) can be replaced by multiplication <i>ab</i>. A straightforward fuzzification is usually based on min and max operations because in this case more properties of <a href="/facts/Traditional_mathematics/ak01PhpL">traditional mathematics</a> can be extended to the fuzzy case. </p><p>An important generalization principle used in fuzzification of algebraic operations is a closure property. Let * be a <a href="/facts/Binary_operation/CYtow0bz">binary operation</a> on <i>X</i>. The closure property for a fuzzy subset <i>A</i> of <i>X</i> is that for all <i>x</i>, <i>y</i> in <i>X</i>, <i>A</i>(<i>x</i>*<i>y</i>) ≥ min(<i>A</i>(<i>x</i>), <i>A</i>(<i>y</i>)). Let (<i>G</i>, *) be a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> and <i>A</i> a fuzzy subset of <i>G</i>. Then <i>A</i> is a <i>fuzzy subgroup</i> of <i>G</i> if for all <i>x</i>, <i>y</i> in <i>G</i>, <i>A</i>(<i>x</i>*<i>y</i>−1) ≥ min(<i>A</i>(<i>x</i>), <i>A</i>(<i>y</i>−1)). </p><p>A similar generalization principle is used, for example, for fuzzification of the <a href="/facts/Transitive_relation/9r90y50r">transitivity</a> property. Let <i>R</i> be a fuzzy relation on <i>X</i>, i.e. <i>R</i> is a fuzzy subset of <i>X</i> × <i>X</i>. Then <i>R</i> is (fuzzy-)transitive if for all <i>x</i>, <i>y</i>, <i>z</i> in <i>X</i>, <i>R</i>(<i>x</i>, <i>z</i>) ≥ min(<i>R</i>(<i>x</i>, <i>y</i>), <i>R</i>(<i>y</i>, <i>z</i>)). </p> <h2 id="fuzzy-analogues">Fuzzy analogues</h2> <p>Fuzzy subgroupoids and fuzzy subgroups were introduced in 1971 by A. Rosenfeld.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p><p>Analogues of other mathematical subjects have been translated to fuzzy mathematics, such as fuzzy field theory and fuzzy <a href="/facts/Galois_theory/54wscJfj">Galois theory</a>,<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> fuzzy topology,<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> fuzzy geometry,<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a><a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> fuzzy orderings,<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> and fuzzy graphs.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a><a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a><a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Fuzzy_measure_theory/5DUI2Nh6">Fuzzy measure theory</a></li> <li><a href="/facts/Fuzzy_subalgebra/LUsgYHtT">Fuzzy subalgebra</a></li> <li><a href="/facts/Monoidal_t-norm_logic/pouM465m">Monoidal t-norm logic</a></li> <li><a href="/facts/Possibility_theory/XKHDEpbJ">Possibility theory</a></li> <li><a href="/facts/T-norm/oYR3JgIy">T-norm</a></li></ul> <h2 id="external-links">External links</h2> <ul><li>Zadeh, L.A. <a href="http://www.scholarpedia.org/article/Fuzzy_Logic">Fuzzy Logic</a> - article at <a href="/facts/Scholarpedia/h2r7IY7J">Scholarpedia</a></li> <li>Hajek, P. <a href="http://plato.stanford.edu/entries/logic-fuzzy/">Fuzzy Logic</a> - article at <a href="/facts/Stanford_Encyclopedia_of_Philosophy/LDuL5HW4">Stanford Encyclopedia of Philosophy</a></li> <li>Navara, M. <a href="http://www.scholarpedia.org/article/Triangular_Norms_and_Conorms">Triangular Norms and Conorms</a> - article at <a href="/facts/Scholarpedia/h2r7IY7J">Scholarpedia</a></li> <li>Dubois, D., Prade H. <a href="http://www.scholarpedia.org/article/Possibility_Theory">Possibility Theory</a> - article at <a href="/facts/Scholarpedia/h2r7IY7J">Scholarpedia</a></li> <li>Center <a href="http://www.niftystats.com/ref.php?cid=C84b541099">for</a> Mathematics of Uncertainty <a href="http://www.creighton.edu/ccas/fuzzymath/">Fuzzy Math Research</a> <a href="https://web.archive.org/web/20090629104833/http://www.creighton.edu/ccas/fuzzymath/">Archived</a> 2009-06-29 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a> - Web site hosted at Creighton University</li> <li>Seising, R. <a href="https://www.springer.com/engineering/computational+intelligence+and+complexity/book/978-3-540-71794-2">[1]</a> Book on the history of the mathematical theory of Fuzzy Sets: The Fuzzification of Systems. The Genesis of Fuzzy Set Theory and Its Initial Applications -- Developments up to the 1970s (Studies in Fuzziness and Soft Computing, Vol. 216) Berlin, New York, [et al.]: Springer 2007.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Zadeh, L. A. (1965) "Fuzzy sets", Information and Control, 8, 338–353. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Goguen, J. (1967) "L-fuzzy sets", J. Math. Anal. Appl., 18, 145–174. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Kerre, E.E., Mordeson, J.N. (2005) "A historical overview of fuzzy mathematics", New Mathematics and Natural Computation, 1, 1-26. <a href="/wiki/New_Mathematics_and_Natural_Computation" target="_blank">/wiki/New_Mathematics_and_Natural_Computation</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Klement, E.P., Mesiar, R., Pap, E. (2000) Triangular Norms. Dordrecht, Kluwer. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Rosenfeld, A. (1971) "Fuzzy groups", J. Math. Anal. Appl., 35, 512-517. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Mordeson, J.N., Malik, D.S., Kuroli, N. (2003) Fuzzy Semigroups. Studies in Fuzziness and Soft Computing, vol. 131, Springer-Verlag <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Mordeson, J.N., Bhutani, K.R., Rosenfeld, A. (2005) Fuzzy Group Theory. Studies in Fuzziness and Soft Computing, vol. 182. Springer-Verlag. <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Mordeson, J.N., Malik, D.S (1998) Fuzzy Commutative Algebra. World Scientific. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Chang, C.L. (1968) "Fuzzy topological spaces", J. Math. Anal. Appl., 24, 182—190. <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Liu, Y.-M., Luo, M.-K. (1997) Fuzzy Topology. Advances in Fuzzy Systems - Applications and Theory, vol. 9, World Scientific, Singapore. <a href="/wiki/Liu_Yingming" target="_blank">/wiki/Liu_Yingming</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Poston, Tim, "Fuzzy Geometry". <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Buckley, J.J., Eslami, E. (1997) "Fuzzy plane geometry I: Points and lines". Fuzzy Sets and Systems, 86, 179-187. <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Ghosh, D., Chakraborty, D. (2012) "Analytical fuzzy plane geometry I". Fuzzy Sets and Systems, 209, 66-83. <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Chakraborty, D. and Ghosh, D. (2014) "Analytical fuzzy plane geometry II". Fuzzy Sets and Systems, 243, 84–109. <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Zadeh L.A. (1971) "Similarity relations and fuzzy orderings". Inform. Sci., 3, 177–200. <a href="/wiki/Inform._Sci." target="_blank">/wiki/Inform._Sci.</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Kaufmann, A. (1973). Introduction a la théorie des sous-ensembles flows. Paris. Masson. <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>A. Rosenfeld, A. (1975) "Fuzzy graphs". In: Zadeh, L.A., Fu, K.S., Tanaka, K., Shimura, M. (eds.), Fuzzy Sets and their Applications to Cognitive and Decision Processes, Academic Press, New York, ISBN 978-0-12-775260-0, pp. 77–95. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Yeh, R.T., Bang, S.Y. (1975) "Fuzzy graphs, fuzzy relations and their applications to cluster analysis". In: Zadeh, L.A., Fu, K.S., Tanaka, K., Shimura, M. (eds.), Fuzzy Sets and their Applications to Cognitive and Decision Processes, Academic Press, New York, ISBN 978-0-12-775260-0, pp. 125–149. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> </ol>