Changes

| <math>\operatorname{If}~ f : X \to \underline\mathbb{B},</math>

| <math>\operatorname{If}~ f : X \to \underline\mathbb{B},</math>

|-

|-

| <math>\operatorname{then}~ [| f |] ~=~ f^{-1} (\underline{1}) ~=~ \{ x \in X : f(x) = \underline{1}.</math>

| <math>\operatorname{then}~ [| f |] ~=~ f^{-1} (\underline{1}) ~=~ \{ x \in X : f(x) = \underline{1} \}.</math>

|}

|}

The definition of a fiber, in either the general or the boolean case, is a purely nominal convenience for referring to the antecedent subset, the inverse image under a function, or the pre-image of a functional value.

The definition of a fiber, in either the general or the boolean case, is a purely nominal convenience for referring to the antecedent subset, the inverse image under a function, or the pre-image of a functional value.

The definition of an operator on propositions, signified by framing the signs of propositions with fiber bars or ground signs, remains a purely notational device, and yet the notion of a fiber in a logical context serves to raise an interesting point.  By way of illustration, it is legitimate to rewrite the above definition in the following form:

The definition of an operator on propositions, signified by framing the signs of propositions with fiber bars or ground signs, remains a purely notational device, and yet the notion of a fiber in a logical context serves to raise an interesting point.  By way of illustration, it is legitimate to rewrite the above definition in the following form:

If f : X -> %B%,

| <math>\operatorname{If}~ f : X \to \underline\mathbb{B},</math>

| <math>\operatorname{then}~ [| f |] ~=~ f^{-1} (\underline{1}) ~=~ \{ x \in X : f(x) \}.</math>

The set-builder frame <math>\{ x \in X : \underline{~~~} \}</math> requires a sentence to fill in the blank, as with the sentence <math>^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}</math> that serves to fill the frame in the first definition of a logical fiber. And what is a sentence but the expression of a proposition, in other words, the name of an indicator function?  As it happens, the sign <math>^{\backprime\backprime} f(x) ^{\prime\prime}</math> and the sentence <math>^{\backprime\backprime} f(x) = \underline{1} ^{\prime\prime}</math> represent the very same value to this context, for all <math>x\!</math> in <math>X,\!</math> that is, they are equal in their truth or falsity to any reasonable interpreter of signs or sentences in this context, and so either one of them can be tendered for the other, in effect, exchanged for the other, within this frame.

 

The set-builder frame "{x in X : ... }" requires a sentence to

fill in the blank, as with the sentence "f(x) = %1%" that serves

to fill the frame in the initial definition of a logical fiber.

And what is a sentence but the expression of a proposition, in

other words, the name of an indicator function?  As it happens,

the sign "f(x)" and the sentence "f(x) = %1%" represent the very

same value to this context, for all x in X, that is, they are equal

in their truth or falsity to any reasonable interpreter of signs or

sentences in this context, and so either one of them can be tendered

for the other, in effect, exchanged for the other, within this frame.

The "fiber" of a codomain element y ? Y under a function f : X �> Y is the subset of the domain X that is mapped onto y, in short, f�1(y) ? X.  In other language that is often used, the fiber of y under f is called the "antecedent set", "inverse image", "level set", or "pre�image" of y under f.  All of these equivalent concepts are defined as follows:

The "fiber" of a codomain element y ? Y under a function f : X �> Y is the subset of the domain X that is mapped onto y, in short, f�1(y) ? X.  In other language that is often used, the fiber of y under f is called the "antecedent set", "inverse image", "level set", or "pre�image" of y under f.  All of these equivalent concepts are defined as follows: