Editing Fuzzy set (section)

===Other definitions===
* A fuzzy set <math>A = (U,m)</math> is '''empty''' (<math>A = \varnothing</math>) [[iff]] (if and only if)
::[[Universal quantification#notation|<math>\forall</math>]]<math> x \in U: \mu_A(x) = m(x) = 0</math>

* Two fuzzy sets <math>A</math> and <math>B</math> are '''equal''' (<math>A = B</math>) iff
::<math>\forall x \in U: \mu_A(x) = \mu_B(x)</math>

* A fuzzy set <math>A</math> is '''included''' in a fuzzy set <math>B</math> (<math>A \subseteq B</math>) iff
::<math>\forall x \in U: \mu_A(x) \le \mu_B(x)</math>

* For any fuzzy set <math>A</math>, any element <math>x \in U</math> that satisfies
::<math>\mu_A(x) = 0.5</math> 
:is called a '''crossover point'''.

* Given a fuzzy set <math>A</math>, any <math>\alpha \in [0,1]</math>, for which <math>A^{=\alpha} = \{x \in U \mid \mu_A(x) = \alpha\}</math> is not empty, is called a '''level''' of A. 
* The '''level set''' of A is the set of all levels <math>\alpha\in[0,1]</math> representing distinct cuts. It is the [[image (mathematics)|image]] of <math>\mu_A</math>:
::<math>\Lambda_A = \{\alpha \in [0,1] : A^{=\alpha} \ne \varnothing\} = \{\alpha \in [0, 1] : {}</math>[[Existential quantification|<math>\exist</math>]]<math>x \in U(\mu_A(x) = \alpha)\} = \mu_A(U)</math>

* For a fuzzy set <math>A</math>, its '''height''' is given by
::<math>\operatorname{Hgt}(A) = \sup \{\mu_A(x) \mid x \in U\} = \sup(\mu_A(U))</math>
:where <math>\sup</math> denotes the [[Infimum and supremum|supremum]], which exists because <math>\mu_A(U)</math> is non-empty and bounded above by 1. If ''U'' is finite, we can simply replace the supremum by the maximum.

* A fuzzy set <math>A</math> is said to be '''normalized''' iff
::<math>\operatorname{Hgt}(A) = 1</math>
:In the finite case, where the supremum is a maximum, this means that at least one element of the fuzzy set has full membership. A non-empty fuzzy set <math>A</math> may be normalized with result <math>\tilde{A}</math> by dividing the membership function of the fuzzy set by its height:
::<math>\forall x \in U: \mu_{\tilde{A}}(x) = \mu_A(x)/\operatorname{Hgt}(A)</math>
:Besides similarities this differs from the usual [[normalizing constant|normalization]] in that the normalizing constant is not a sum.

* For fuzzy sets <math>A</math> of real numbers <math>(U \subseteq \mathbb{R})</math> with [[bounded set|bounded]] support, the '''width''' is defined as
::<math>\operatorname{Width}(A) = \sup(\operatorname{Supp}(A)) - \inf(\operatorname{Supp}(A))</math>
:In the case when <math>\operatorname{Supp}(A)</math> is a finite set, or more generally a [[closed set]], the width is just
::<math>\operatorname{Width}(A) = \max(\operatorname{Supp}(A)) - \min(\operatorname{Supp}(A))</math>
:In the ''n''-dimensional case <math>(U \subseteq \mathbb{R}^n)</math> the above can be replaced by the ''n''-dimensional volume of <math>\operatorname{Supp}(A)</math>. 
:In general, this can be defined given any [[Measure (mathematics)|measure]] on ''U'', for instance by integration (e.g. [[Lebesgue integration]]) of <math>\operatorname{Supp}(A)</math>.

* A real fuzzy set <math>A (U \subseteq \mathbb{R})</math> is said to be '''convex''' (in the fuzzy sense, not to be confused with a crisp [[convex set]]), iff
::<math>\forall x,y \in U, \forall\lambda\in[0,1]: \mu_A(\lambda{x} + (1-\lambda)y) \ge \min(\mu_A(x),\mu_A(y))</math>.
: Without loss of generality, we may take ''x'' ≤ ''y'', which gives the equivalent formulation
::<math>\forall z \in [x,y]: \mu_A(z) \ge \min(\mu_A(x),\mu_A(y))</math>.
: This definition can be extended to one for a general [[topological space]] ''U'': we say the fuzzy set <math>A</math> is '''convex''' when, for any subset ''Z'' of ''U'', the condition 
::<math>\forall z \in Z: \mu_A(z) \ge \inf(\mu_A(\partial Z))</math>
: holds, where <math>\partial Z</math> denotes the [[Boundary (topology)|boundary]] of ''Z'' and <math>f(X) = \{f(x) \mid x \in X\}</math> denotes the [[image (mathematics)|image]] of a set ''X'' (here <math>\partial Z</math>) under a function ''f'' (here <math>\mu_A</math>).