Changes

<p>Considered in functional form, <math>f_Q\!</math> is the map from <math>X\!</math> to <math>\underline\mathbb{B}</math> that is given by the following condition:</p>

<p>Considered in functional form, <math>f_Q\!</math> is the map from <math>X\!</math> to <math>\underline\mathbb{B}</math> that is given by the following condition:</p>

<p><math>f_Q (x) = \underline{1} ~\Leftrightarrow~ x \in Q.</math></p></li>

<p><math>f_Q (x) ~=~ \underline{1} ~\Leftrightarrow~ x \in Q.</math></p></li>

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With this degree of flexibility in mind, one can say that the sentence <math>^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}</math> latently connotes what the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> patently connotes.  Taken in abstraction, both syntactic entities fall into an equivalence class of signs that constitutes an abstract object, a thing of value that is ''identified by'' the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime},</math> and thus an object that might as well be ''identified with'' the value <math>f(x).\!</math>

With this degree of flexibility in mind, one can say that the sentence <math>^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}</math> latently connotes what the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> patently connotes.  Taken in abstraction, both syntactic entities fall into an equivalence class of signs that constitutes an abstract object, a thing of value that is ''identified by'' the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime},</math> and thus an object that might as well be ''identified with'' the value <math>f(x).\!</math>

The upshot of this whole discussion of evaluation is that it allows one to rewrite the definitions of indicator functions and their fibers as follows:

The upshot of this whole discussion of evaluation is that it allows us to rewrite the definitions of indicator functions and their fibers as follows:

The ''indicator function'' or the ''characteristic function'' of a set <math>Q \in X,</math>  written <math>f_Q,\!</math> is the map from <math>X\!</math> to the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}</math> that is defined in the following ways:

The "indicator function" or the "characteristic function" of a set X ? U, written "fX", is the map from U to the boolean domain B = {0, 1} that is defined in the following ways:

 

1. Considered in extensional form, fX is the subset of UxB

that is given by the following formula:

<p>Considered in extensional form, <math>f_Q\!</math> is the subset of <math>X \times \underline\mathbb{B}</math> that is given by the following formula:</p>

fX  =  {<u, v> ? UxB : v  <=>  u ? X}.

<p><math>f_Q ~=~ \{ (x, b) \in X \times \underline\mathbb{B} ~:~ b ~\Leftrightarrow~ x \in Q \}.</math></p></li>

2. Considered in functional form, fX is the map from U to B

that is given by the following condition:

<p>Considered in functional form, <math>f_Q\!</math> is the map from <math>X\!</math> to <math>\underline\mathbb{B}</math> that is given by the following condition:</p>

fX(u)  <=> u ? X.

<p><math>f_Q ~\Leftrightarrow~ x \in Q.</math></p></li>

The "fibers" of truth and falsity under a proposition f : U �> B are subsets of U that are variously described as follows:

The "fibers" of truth and falsity under a proposition f : U �> B are subsets of U that are variously described as follows: