Editing Fuzzy set (section)

===Distance and similarity===
For any fuzzy set <math>A</math> the membership function <math>\mu_A: U \to [0,1]</math> can be regarded as a family <math>\mu_A = (\mu_A(x))_{x \in U} \in [0,1]^U</math>. The latter is a [[metric space]] with several metrics <math>d</math> known. A metric can be derived from a [[Norm (mathematics)|norm]] (vector norm) <math>\|\,\|</math> via
:<math>d(\alpha,\beta) = \| \alpha - \beta \|</math>.
For instance, if <math>U</math> is finite, i.e. <math>U = \{x_1, x_2, ... x_n\}</math>, such a metric may be defined by: 
:<math>d(\alpha,\beta) := \max \{ |\alpha(x_i) - \beta(x_i)| : i=1, ..., n \}</math> where <math>\alpha</math> and <math>\beta</math> are sequences of real numbers between 0 and 1. 
For infinite <math>U</math>, the maximum can be replaced by a supremum.
Because fuzzy sets are unambiguously defined by their membership function, this metric can be used to measure distances between fuzzy sets on the same universe:
:<math>d(A,B) := d(\mu_A,\mu_B)</math>,
which becomes in the above sample:
:<math>d(A,B) = \max \{ |\mu_A(x_i) - \mu_B(x_i)| : i=1,...,n \}</math>.
Again for infinite <math>U</math> the maximum must be replaced by a supremum. Other distances (like the canonical 2-norm) may diverge, if infinite fuzzy sets are too different, e.g., <math>\varnothing</math> and <math>U</math>.

Similarity measures (here denoted by <math>S</math>) may then be derived from the distance, e.g. after a proposal by Koczy:
:<math>S = 1 / (1 + d(A,B))</math> if <math>d(A,B)</math> is finite, <math>0</math> else, 
or after Williams and Steele:
:<math>S = \exp(-\alpha{d(A,B)})</math> if <math>d(A,B)</math> is finite, <math>0</math> else
where <math>\alpha > 0</math> is a steepness parameter and <math>\exp(x) = e^x</math>.{{Cn|date=December 2024}}