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

    -- 
    

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

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