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

Cordero" <ocordero@cti.espol.edu.ec> wrote:

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

--Nick Smith http://www.princeton.edu/~njsmith replace `nospam' with `princeton' to reply

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