Editing Fuzzy set (section)

== Fuzzy number ==
{{main|Fuzzy number}}
A '''fuzzy number'''<ref name="semanticscholar.org">{{cite journal | doi=10.1016/S0165-0114(99)80004-9 | title=Fuzzy sets as a basis for a theory of possibility | date=1999 | last1=Zadeh | first1=L.A. | journal=Fuzzy Sets and Systems | volume=100 | pages=9–34 }}</ref> is a fuzzy set that satisfies all the following conditions:
* A is normalised;
* A is a convex set;
* The membership function <math>\mu_{A}(x)</math> achieves the value 1 at least once;
* The membership function <math>\mu_{A}(x)</math> is at least segmentally continuous.

If these conditions are not satisfied, then A is not a '''fuzzy number'''. The core of this fuzzy number is a [[Singleton (mathematics)|singleton]]; its location is:
:: <math> \, C(A) = x^* : \mu_A(x^*)=1</math>

Fuzzy numbers can be likened to the [[funfair]] game "guess your weight," where someone guesses the contestant's weight, with closer guesses being more correct, and where the guesser "wins" if he or she guesses near enough to the contestant's weight, with the actual weight being completely correct (mapping to 1 by the membership function).

The kernel <math>K(A) = \operatorname{Kern}(A)</math> of a fuzzy interval <math>A</math> is defined as the 'inner' part, without the 'outbound' parts where the membership value is constant ad infinitum. In other words, the smallest subset of <math>\R</math> where <math>\mu_A(x)</math> is constant outside of it, is defined as the kernel.

However, there are other concepts of fuzzy numbers and intervals as some authors do not insist on convexity.