Re: The MEANING of fuzzy AND and OR.

ca314159@bestweb.net
Sun, 21 Jun 1998 15:02:07 +0200 (MET DST)

In article <199806081809.10370.snidil.ifi.uio.no@ifi.uio.no>,
fuzzy-mail@dbai.tuwien.ac.at wrote:
>
>
> The proliferation of email letters concerning AND and OR
> (one of these letters is reproduced at the end of this letter),
> shows that some clarification is necessary concerning the precise
> meaning of a fuzzy
> label containing one of these connectives. Such a meaning is
> associated with the idea of a `composite experiment'.
>

The example you give of 'composite experiments' uses disjoint
(and apparently crisp) sets with a subjective determination of
set membership as a probability that is normalized on the
number of these disjoint sets. These don't sound like true
fuzzy sets ?

In this case (A AND B) = 0 because the two sets are completely
distinct in terms of whether a sample will wind up in one or
the other. The fact that you probabilistically assign membership
does not change this essential point that each sample winds up
in only one set. Hence these sets are disjoint.

This is not as general a case as was being considered in which
the sets were not necessarily disjoint. That is, one "calue"
(crisp value?) sample might simultaneously be a member of A and B to
various degrees. In this case (A AND B) >=0 and we say that these
sets are not necessarily orthogonal but that there is a degenerate
case where they can be. This degenerate case is when there is
certainty in the disjointness of the sets. The measure of disjointness
of these sets is a measure of their distinctness, orthogonality or
degree of overlap, which can be in the range:

max(A+B-1,0) <= (A AND B) <= min(A,B) [1]

if A and B are not disjoint and A + B <=2.

My initial question was why does fuzzy theory only
use the upper bound of the expression [1] above ?
Why does fuzzy theory ignore the lower bound
max(A+B-1,0) which is significant when the membership
grades are both > .5 which can happen in non-disjoint sets ?

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