# Re: Fuzzy * Fuzzy

From: Nick Smith (njsmith@nospam.edu)
Date: Sun Feb 11 2001 - 20:44:34 MET

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In article <000e01c09160\$93879c80\$22960ac8@cti.espol.edu.ec>, "Otto
Cordero" <ocordero@cti.espol.edu.ec> wrote:

> Dear All:
>
> A generic membership functions has the form [x1, x2] --> [y1, y2], where x1,
> x2, y1 and y2 are real numbers in most of the cases. I am interested in
> those particular cases where the axis of the membership function are not
> crisp but fuzzy (for example: if [x1, x2] or [y1, y2] are fuzzy intervals or
> if their elements are fuzzy numbers).
>
> I would appreciate if you point out any publication regarding the
> mathematical treatement of this kind of membership functions defined in
> terms of fuzzy elements.
>
> Otto Cordero
> ocordero@cti.espol.edu.ec

Hi,

An _ordinary fuzzy set_ A has a membership function of the form
A:X-->[0,1]. I.e. its membership values are reals in [0,1].

A _type 2 fuzzy set_ A has a membership function of the form
A:X-->F([0,1]), where F([0,1]) is the fuzzy power set of [0,1], i.e.
the set of all ordinary fuzzy subsets of [0,1]. I.e. its membership
values are ordinary fuzzy sets. On these, see for example:
Mizumoto and Tanaka, 1976, `Some properties of fuzzy sets of type 2',
Information and Control, 31 (4), 312--40.
Mizumoto and Tanaka, 1981, `Fuzzy sets of type 2 under algebraic
product and algebraic sum', Fuzzy Sets and Systems, 5 (3), 277--90.

A _level 2 fuzzy set_ A has a membership function of the form
A:F(X)-->[0,1], where F(X) is the fuzzy power set of the crisp set X,
i.e. the set of all ordinary fuzzy subsets of X. I.e. it is defined
within a universal set whose elements are ordinary fuzzy sets. On
these, see for example:
Zadeh, 1971, `Quantitative fuzzy semantics', Information Sciences, 3
(2), 159--76.
Gottwald, 1979, `Set theory for fuzzy sets of higher level', Fuzzy Sets
and Systems, 2 (2), 125--51.

Nick

```--
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http://www.princeton.edu/~njsmith
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