Re: Why alpha-cut?

Geraldo =?iso-8859-1?Q?Xex=E9o?= (xexeo@cos.ufrj.br)
Sat, 30 Jan 1999 19:51:44 +0100 (MET)

Francesco Carriero wrote:

> IMHO, the usefulness of alpha-cuts shows up mainly in the field of fuzzy sets
> theory, and stems from the theorem that states the equivalence between any fuzzy set
> and the OR of all its distinct alpha cuts. For instance, this equivalence allows to
> handle problems like fuzzy numbers and fuzzy arithmetic resorting to well known
> tools of standard maths (i.e., the interval arithmetic)

Francesco Carriero has a point. We are broadly using the concept of alpha-cut here,
while alpha-cuts formally are defined over fuzzy sets, not for rules. But I believe
the previous discussion stands, since we apply the treesholds and alpha-cuts to
the resulting fuzzy set, making them totally indistinct..

Nguyen and Walker (A First Course in Fuzzy Logic) discuss the importance of the
concept of alpha-cut. As Francesco noted, knowing all the alpha-cuts of A is the same
of knowing A. Klir and Yuan (Fuzzy sets and Fuzzy Logic) say that this is the
principal role os alpha-cuts (and strong alpha-cuts), and since "all alpha-cuts and
strong alpha-cuts
of any fuzzy set form two distinct family of nested crisp sets" we are allowed to
extend
properties of crisp sets to fuzzy sets.

Jerry

--
Geraldo Xexéo, D.Sc. (DCC/IM/UFRJ e PESC/COPPE/UFRJ)
mailto:xexeo@cos.ufrj.br http://www.cos.ufrj.br/~xexeo
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