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A ''proposition about things in the universe'', more simply, a ''proposition'', is the same thing as an indicator function, that is, a function of the form <math>f : X \to \underline\mathbb{B}.</math>  The convenience of this seemingly redundant usage is that it permits one to refer to an indicator function without having to specify right away, as a part of its only available designation, exactly what set it indicates, even though a proposition is always an indicator function of some subset of the universe, and even though one probably or eventually wants to know which one.

A "proposition about things in the universe", for short, a "proposition", is the same thing as an indicator function, that is, a function of the form f : U �> B. The convenience of this seemingly redundant usage is that it permits one to refer to an indicator function without having to specify right away, as a part of its only available designation, exactly what set it indicates, even though a proposition is always an indicator function of some subset of the universe, and even though one probably or eventually wants to know which one.

 

According to the stated understandings, a proposition is a function that indicates a set, in the sense that a function associates values with the elements of a domain, some which values can be interpreted to mark out for special consideration a subset of that domain.  The way in which an indicator function is imagined to "indicate" a set can be expressed in terms of the following concepts.

According to the stated understandings, a proposition is a function that indicates a set, in the sense that a function associates values with the elements of a domain, some which values can be interpreted to mark out for special consideration a subset of that domain.  The way in which an indicator function is imagined to "indicate" a set can be expressed in terms of the following concepts.

The "fiber" of a codomain element y in Y under a function f : X -> Y is the subset of the domain X that is mapped onto y, in short, it is f^(-1)(y) c X.

The ''fiber'' of a codomain element <math>y \in Y\!</math> under a function <math>f : X \to Y</math> is the subset of the domain <math>X\!</math> that is mapped onto <math>y,\!</math> in short, it is <math>f^{-1} (y) \subseteq X.</math>

In other language that is often used, the fiber of y under f is called the "antecedent set", the "inverse image", the "level set", or the "pre-image" of y under f.  All of these equivalent concepts are defined as follows:

In other language that is often used, the fiber of <math>y\!</math> under <math>f\!</math> is called the ''antecedent set'', the ''inverse image'', the ''level set'', or the ''pre-image'' of <math>y\!</math> under <math>f.\!</math> All of these equivalent concepts are defined as follows:

Fiber of y under f = f^(-1)(y) = {x in X : f(x) = y}.

| <math>\operatorname{Fiber~of}~ y ~\operatorname{under}~ f \ = \ f^{-1} (y) \ = \ \{ x \in X : f(x) = y \}.</math>

In the special case where f is the indicator function f_Q of the set Q c X, the fiber of %1% under f_Q is just the set Q back again:

In the special case where f is the indicator function f_Q of the set Q c X, the fiber of %1% under f_Q is just the set Q back again: