Changes

By way of general definition, the ''fiber'' of a function <math>f : X \to Y</math> at a given value <math>y\!</math> of its co-domain <math>Y\!</math> is the ''antecedent'' (also known as the ''inverse image'' or ''pre-image'') of <math>y\!</math> under <math>f.\!</math>  This is a subset, possibly empty, of the domain <math>X,\!</math> notated as <math>f^{-1}(y) \subseteq X.</math>

By way of general definition, the ''fiber'' of a function <math>f : X \to Y</math> at a given value <math>y\!</math> of its co-domain <math>Y\!</math> is the ''antecedent'' (also known as the ''inverse image'' or ''pre-image'') of <math>y\!</math> under <math>f.\!</math>  This is a subset, possibly empty, of the domain <math>X,\!</math> notated as <math>f^{-1}(y) \subseteq X.</math>

In particular, if ''f'' is a proposition ''f'' : ''X'' &rarr; '''B''', then we think of ''f''^&minus;1(''y'') as the subset of ''X'' that is ''indicated'' by the proposition ''f''.  Whenever we ''assert'' a proposition ''f'' : ''X'' &rarr; '''B''', we are saying that what it indicates is all that happens to be the case in the relevant universe of discourse ''X''.  Because the special case of the fiber of truth is used so often in logical contexts, we will sometimes use the notation <nowiki>[|</nowiki>''f''<nowiki>|]</nowiki> = ''f''^&minus;1(1).

In particular, if <math>f\!</math> is a proposition <math>f : X \to \mathbb{B},</math> then the fiber of truth <math>f^{-1}(1)\!</math> is the subset of <math>X\!</math> that is ''indicated'' by the proposition <math>f.\!</math> Whenever we ''assert'' a proposition <math>f : X \to \mathbb{B},</math> we are saying that what it indicates is all that happens to be the case in the relevant universe of discourse <math>X.\!</math> Because the fiber of truth is used so often in logical contexts, it is convenient to define the more compact notation <math>[| f |] = f^{-1}(1).\!</math>

Using this panoply of notions and notations, we may treat the fiber of truth of each proposition ''f'' : '''B'''³ &rarr; '''B''' as if it were a relational data table of the shape {(''p'', ''q'', ''r'')} &sube; '''B'''³, where the (''p'', ''q'', ''r'') are bit vectors indicated by the proposition ''f''.

Using this panoply of notions and notations, we may treat the fiber of truth of each proposition <math>f : \mathbb{B}^3 \to \mathbb{B}</math> as if it were a relational data table of the shape <math>\{ (p, q, r) \} \subseteq \mathbb{B}^3,</math> where the triples <math>(p, q, r)\!</math> are bit-tuples indicated by the proposition <math>f.\!</math>

Thus we obtain the following four relational data tables for the propositions that we are looking at in Example 2.

Thus we obtain the following four relational data tables for the propositions that we are looking at in Example&nbsp;2.

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