Changes

</ol>

</ol>

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 <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 <math>f : X \to \underline\mathbb{B}.</math>  The convenience of this seemingly redundant usage is that it allows one to refer to an indicator function without having to specify right away, as a part of its designated subscript, exactly what set it indicates, even though a proposition always indicates some subset of its designated universe, and even though one will probably or eventually want to know exactly what subset that is.

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 of 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 <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 <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:

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 <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: