Supersets & Subsets in Fuzzy Logic
Richard Herrell
I have been reading the introdutory book entitled "Fuzzy Logic",
and I have a question.
The book defines a fuzzy superset as follows:
"A is a superset of B if for each element x of B, x is more an
element of A than B."
This definition seems too crisp for fuzzy logic. Since we are
dealing with fuzzy set membership, intersection, and complements,
shouldn't we also deal with fuzzy set inclusion?
(ie: set A is 0.9 a superset of set B ).
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Example:
Element: Tall Men Extremely Tall Men
Chris 0.6 0.4
Hans 0.5 0.3
Mao 0.4 0.2
Rich 0.6 0.3
Here we can say that "Tall Men" is a superset of
"Extremely Tall Men", and conversely "Extremely Tall
Men" is a subset of "Tall Men".
but what if we add:
Shackeem 0.9 1.0
Shakeem is more of an extremely tall man than a tall one.
Now, according to the definition of superset, "Tall
Men" is not a superset of "Extremely Tall Men".
But isn't there a fuzzy middle ground here?
Couldn't we say that "Tall Men" is 4/5 or 0.8 a superset
of "Extremely Tall Men", and similarly, "Extremely Tall
Men" is 1/5 or 0.2 a superset of "Tall Men"?
Here I selected 4/5 and 1/5 because they represent the
ratio of elements that meet the corresponding definitions,
but I don't see why any monotonically increasing
function from (0,0) to (1,1) couldn't be used.
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So I'd like to offer the following "fuzzier" definition of
superset for your critique:
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Given a fn F(x) monotonically increasing, conneting the points
(0,0), and (1,1), and defined at all real numbers ( or at least
those real numbersthat will be needed) and for two Fuzzy sets
A and B,
Define Fuzy set A to be Y the superset of B,
where Y = F( #number of elements more in A than B /
#number of elements in B )
and
Fuzzy set A to be the 1-Y subset of B.
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This definition of superset and subset doesn't preclude the
old one. Just as fuzzy sets describe elements as partial members
of sets, this definition describes fuzzy sets as fuzzy subsets of
other fuzzy sets.
Furthermore, more creative definitions can be thought of using
things such as the average difference of the membership of
elements or the variance for inputs to a different function.
--Richard Herrell