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| The '''membership function''' of a [[fuzzy set]] is a generalization of the [[indicator function]] in classical [[Set (mathematics)|sets]]. In [[fuzzy logic]], it represents the [[degree of truth]] as an extension of [[Valuation (logic)|valuation]]. Degrees of truth are often confused with [[probability|probabilities]], although they are conceptually distinct, because fuzzy truth represents membership in vaguely defined sets, not likelihood of some event or condition. Membership functions were introduced by [[Lotfi Asker Zadeh|Zadeh]] in the first paper on fuzzy sets (1965).
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| == Definition ==
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| For any set <math>X</math>, a membership function on <math>X</math> is any function from <math>X</math> to the real unit interval [0,1].
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| Membership functions on <math>X</math> represent [[fuzzy set|fuzzy subsets]] of <math>X</math>. The membership function which represents a fuzzy set <math>\tilde A</math> is usually denoted by <math>\mu_A.</math> For an element <math>x</math> of <math>X</math>, the value <math>\mu_A(x)</math> is called the ''membership degree'' of <math>x</math> in the fuzzy set <math>\tilde A.</math> The membership degree <math>\mu_{A}(x)</math> quantifies the grade of membership of the element <math>x</math> to the fuzzy set <math>\tilde A.</math> The value 0 means that <math>x</math> is not a member of the fuzzy set; the value 1 means that <math>x</math> is fully a member of the fuzzy set. The values between 0 and 1 characterize fuzzy members, which belong to the fuzzy set only partially.
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| <center>[[File:Fuzzy crisp.svg]]</center>
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| <center>Membership function of a fuzzy set</center>
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| Sometimes,<ref>First in Goguen (1967).</ref> a more general definition is used, where membership functions take values in an arbitrary fixed algebra or structure <math>L</math>; usually it is required that <math>L</math> be at least a [[poset]] or [[lattice (order)|lattice]]. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions.
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| <!-- this is only true for so-called normal fuzzy sets:
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| The following holds for the functional values of the membership function <math>\mu_{A}(x)</math>
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| <center><math>
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| \mu_{A}(x)\ge0\quad\forall\quad x\in\mathbf{X}</math>
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| <br>
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| <math>\sup_{x\in X}[\mu_{A}(x)]=1</math></center>-->
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| == Capacity ==
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| One application of membership functions is as capacities in [[decision theory]].
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| In decision theory, a capacity is defined as a function, <math>\nu</math> from '''S''', the set of subsets of some set, into <math>[0,1]</math>, such that <math>\nu</math> is set-wise monotone and is normalized (i.e. <math>\nu(\empty) = 0, \nu(\Omega)=1).</math> Clearly this is a generalization of a [[probability measure]], where the [[probability axiom]] of countability is weakened. A capacity is used as a subjective measure of the likelihood of an event, and the "[[expected value]]" of an outcome given a certain capacity can be found by taking the [[Choquet integral]] over the capacity.
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| == See also ==
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| * [[Defuzzification]]
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| * [[Fuzzy measure theory]]
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| * [[Fuzzy set operations]]
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| * [[Rough set]]
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| == References ==
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| {{reflist}}
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| == Bibliography ==
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| *Zadeh L.A., 1965, "Fuzzy sets". ''Information and Control'' '''8''': 338–353. [http://www-bisc.cs.berkeley.edu/zadeh/papers/Fuzzy%20Sets-1965.pdf]
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| *Goguen J.A, 1967, "''L''-fuzzy sets". ''Journal of Mathematical Analysis and Applications'' '''18''': 145–174
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| ==External links==
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| *[http://pami.uwaterloo.ca/tizhoosh/set.htm Fuzzy Image Processing]
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| [[Category:Fuzzy logic]]
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My name is Mahalia Courtney. I life in Manchester (United States).
my page - Fifa 15 Coin Generator - http://cspstab.org/elggcspstab/blog/view/31639/fifa-coin-generator,