Editing Fuzzy set (section)

===Fuzzy set operations===
{{main|Fuzzy set operations}}
Although the complement of a fuzzy set has a single most common definition, the other main operations, union and intersection, do have some ambiguity.
* For a given fuzzy set <math>A</math>, its '''complement''' <math>\neg{A}</math> (sometimes denoted as <math>A^c</math> or <math>cA</math>) is defined by the following membership function:
::<math>\forall x \in U: \mu_{\neg{A}}(x) = 1 - \mu_A(x)</math>.
* Let t be a [[t-norm]], and s the corresponding s-norm (aka t-conorm). Given a pair of fuzzy sets <math>A, B</math>, their '''intersection''' <math>A\cap{B}</math> is defined by:
::<math>\forall x \in U: \mu_{A\cap{B}}(x) = t(\mu_A(x),\mu_B(x))</math>,
:and their '''union''' <math>A\cup{B}</math> is defined by:
::<math>\forall x \in U: \mu_{A\cup{B}}(x) = s(\mu_A(x),\mu_B(x))</math>.

By the definition of the t-norm, we see that the union and intersection are [[commutative]], [[monotonic]], [[associative]], and have both a [[absorbing element|null]] and an [[identity element]]. For the intersection, these are ∅ and ''U'', respectively, while for the union, these are reversed. However, the union of a fuzzy set and its complement may not result in the full universe ''U'', and the intersection of them may not give the empty set ∅. Since the intersection and union are associative, it is natural to define the intersection and union of a finite [[Indexed family|family]] of fuzzy sets recursively. It is noteworthy that the generally accepted standard operators for the union and intersection of fuzzy sets are the max and min operators:
* <math>\forall x \in U: \mu_{A\cup{B}}(x) = \max(\mu_A(x),\mu_B(x))</math> and <math>\mu_{A\cap{B}}(x) = \min(\mu_A(x),\mu_B(x))</math>.<ref name="BGFuzzy">{{cite journal|last1=Bellman|first1=Richard|last2=Giertz|first2=Magnus|date=1973|title=On the analytic formalism of the theory of fuzzy sets|journal=Information Sciences|volume=5|pages=149–156|doi=10.1016/0020-0255(73)90009-1}}</ref>

* If the standard negator <math>n(\alpha) = 1 - \alpha, \alpha \in [0, 1]</math> is replaced by another [[t-norm#Non-standard negators|strong negator]], the fuzzy set difference (defined below) may be generalized by
::<math>\forall x \in U: \mu_{\neg{A}}(x) = n(\mu_A(x)).</math>
* The triple of fuzzy intersection, union and complement form a '''De Morgan Triplet'''. That is, [[De Morgan's laws]] extend to this triple.
:Examples for fuzzy intersection/union pairs with standard negator can be derived from samples provided in the article about [[t-norm]]s.

:The fuzzy intersection is not [[Idempotence|idempotent]] in general, because the standard t-norm {{math|min}} is the only one which has this property. Indeed, if the arithmetic multiplication is used as the t-norm, the resulting fuzzy intersection operation is not idempotent. That is, iteratively taking the intersection of a fuzzy set with itself is not trivial. It instead defines the '''''m''-th power''' of a fuzzy set, which can be canonically generalized for non-[[integer]] exponents in the following way:

* For any fuzzy set <math>A</math> and <math>\nu \in \R^+</math> the &nu;-th power of <math>A</math> is defined by the membership function:
::<math>\forall x \in U: \mu_{A^{\nu}}(x) = \mu_{A}(x)^{\nu}.</math>
The case of exponent two is special enough to be given a name.
* For any fuzzy set <math>A</math> the '''concentration''' <math>CON(A) = A^2</math> is defined
::<math>\forall x \in U: \mu_{CON(A)}(x) = \mu_{A^2}(x) = \mu_{A}(x)^2.</math>
Taking <math>0^0 = 1</math>, we have <math>A^0 = U</math> and <math>A^1 = A.</math>

* Given fuzzy sets <math>A, B</math>, the fuzzy set '''difference''' <math>A \setminus B</math>, also denoted <math> A - B</math>, may be defined straightforwardly via the membership function:
::<math>\forall x \in U: \mu_{A\setminus{B}}(x) = t(\mu_A(x),n(\mu_B(x))),</math>
:which means <math>A \setminus B = A \cap \neg{B}</math>, e. g.:
::<math>\forall x \in U: \mu_{A\setminus{B}}(x) = \min(\mu_A(x),1 - \mu_B(x)).</math><ref name="Vemuri2014">N.R. Vemuri, A.S. Hareesh, M.S. Srinath: [http://www.math.sk/fsta2014/presentations/VemuriHareeshSrinath.pdf Set Difference and Symmetric Difference of Fuzzy Sets], in: Fuzzy Sets Theory and Applications 2014, Liptovský Ján, Slovak Republic</ref>

:Another proposal for a set difference could be: 
::<math>\forall x \in U: \mu_{A-{B}}(x) = \mu_A(x) - t(\mu_A(x),\mu_B(x)).</math><ref name="Vemuri2014" />

* Proposals for symmetric fuzzy set differences have been made by Dubois and Prade (1980), either by taking the [[absolute value]], giving
::<math>\forall x \in U: \mu_{A \triangle B}(x) = |\mu_A(x) - \mu_B(x)|,</math>
:or by using a combination of just {{math|max}}, {{math|min}}, and standard negation, giving
::<math>\forall x \in U: \mu_{A \triangle B}(x) = \max(\min(\mu_A(x), 1 - \mu_B(x)), \min(\mu_B(x), 1 - \mu_A(x))).</math><ref name="Vemuri2014" />

:Axioms for definition of generalized symmetric differences analogous to those for t-norms, t-conorms, and negators have been proposed by Vemur et al. (2014) with predecessors by Alsina et al. (2005) and Bedregal et al. (2009).<ref name="Vemuri2014" />

* In contrast to crisp sets, averaging operations can also be defined for fuzzy sets.