Triadic Relations, Intentions, Fuzzy Subsets : 2

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 ℐ = [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 from X to ℬ known as the characteristic function or indicator function of S in X.

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

fS(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 definition from the boolean values in ℬ to the real values in ℐ.

This entry was posted in C.S. Peirce, Fuzzy Logic, Fuzzy Sets, Intentional Contexts, Intentional Objects, Intentionality, Intentions, Logic, Logic of Relatives, Lotfi Zadeh, Mathematics, Peirce, Relation Theory, Semiotics, Triadic Relations and tagged , , , , , , , , , , , , , , . Bookmark the permalink.

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