Editing Fuzzy set (section)

==Entropy==
A measure ''d'' of fuzziness for fuzzy sets of universe <math>U</math> should fulfill the following conditions for all <math>x \in U</math>:
#<math>d(A) = 0</math> if <math>A</math> is a crisp set: <math>\mu_A(x) \in \{0,\,1\}</math>
#<math>d(A)</math> has a unique maximum iff <math>\forall x \in U: \mu_A(x) = 0.5</math>
#<math>\forall x \in U:(\mu_A(x) \leq  \mu_B(x) \leq 0.5) \or (\mu_A(x) \geq \mu_B(x) \geq 0.5)</math>
::<math>\Rightarrow d(A) \leq d(B)</math>,
::which means that ''B'' is "crisper" than ''A''.
#<math>d(\neg{A}) = d(A)</math>
In this case <math>d(A)</math> is called the '''entropy''' of the fuzzy set ''A''.

For '''finite''' <math>U = \{x_1, x_2, ... x_n\}</math> the entropy of a fuzzy set <math>A</math> is given by
:<math>d(A) = H(A) + H(\neg{A})</math>,
::<math>H(A) = -k \sum_{i=1}^n \mu_A(x_i) \ln \mu_A(x_i)</math>
or just
:<math>d(A) = -k \sum_{i=1}^n S(\mu_A(x_i))</math>
where <math>S(x) = H_e(x)</math> is [[Binary entropy function|Shannon's function]] (natural entropy function)
:<math>S(\alpha) = -\alpha \ln \alpha - (1-\alpha) \ln (1-\alpha),\ \alpha \in [0,1]</math>
and <math>k</math> is a constant depending on the measure unit and the logarithm base used (here we have used the natural base [[e (mathematical constant)|e]]).
The physical interpretation of ''k'' is the [[Boltzmann constant]] ''k''<sup>''B''</sup>.

Let <math>A</math> be a fuzzy set with a '''continuous''' membership function (fuzzy variable). Then 
:<math>H(A) = -k \int_{- \infty}^\infty \operatorname{Cr} \lbrace A \geq t \rbrace \ln \operatorname{Cr} \lbrace A \geq t \rbrace \,dt</math>
and its entropy is 
:<math>d(A) = -k \int_{- \infty}^\infty S(\operatorname{Cr} \lbrace A \geq t \rbrace )\,dt.</math><ref>{{cite journal|doi=10.1016/0165-0114(92)90239-Z|title=Entropy, distance measure and similarity measure of fuzzy sets and their relations|journal=Fuzzy Sets and Systems|volume=52|issue=3|pages=305–318|year=1992|last1=Xuecheng|first1=Liu}}</ref><ref>{{cite journal|doi=10.1186/s40467-015-0029-5|title=Fuzzy cross-entropy|journal=Journal of Uncertainty Analysis and Applications|volume=3|year=2015|last1=Li|first1=Xiang|doi-access=free}}</ref>