Re: Fuzzy Sets and Triadic Relations

Peircers,

We imagine all the things we might want to talk about
in a given discussion as collected together in a set X
called a “universe of discourse”.  (In probabilistic or
statistical work it might be called a “sample space”.)

By way of shorthand notation, let us single out two
domains of values, the “boolean domain” B = {0, 1}
and the “unit interval” V = [0, 1]

In many applications, especially computational or
statistical ones, it is often useful to represent
any given subset S of X by means of a function or
mapping from X to B known as the “characteristic
function” or “indicator function” of S in X.

By way of notation, we write f_S : X -> B for
the indicator function of S in X, and define
it as follows:

f_S (x) = 1 if and only if x is an element of S.

All that is standard notions from ordinary set theory.

Fuzzy sets, more properly, fuzzy subsets of a given set X
will be defined in a way that generalizes the target values
in the previous definitions from the boolean values B to the
interval values V.

Tomorrow ...

Jon

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