Changes

: In mathematics generally, the ''[[fiber (mathematics)|fiber]]'' of a point <math>y \in Y</math> under a function <math>f : X \to Y</math> is defined as the inverse image <math>f^{-1}(y) \subseteq X.</math>

: In mathematics generally, the ''[[fiber (mathematics)|fiber]]'' of a point <math>y \in Y</math> under a function <math>f : X \to Y</math> is defined as the inverse image <math>f^{-1}(y) \subseteq X.</math>

* In the case of a boolean function ''f''&nbsp;:&nbsp;'''B'''^''k''&nbsp;→&nbsp;'''B''', there are just two fibers:

: In the case of a boolean function <math>f : \mathbb{B}^k \to \mathbb{B},</math> there are just two fibers:

** The fiber of 0 under ''f'', defined as <math>f^{-1}(0)</math>, is the set of points where ''f'' is 0.

: The fiber of <math>0\!</math> under <math>f,\!</math> defined as <math>f^{-1}(0),\!</math> is the set of points where the value of <math>f\!</math> is <math>0.\!</math>

** The fiber of 1 under ''f'', defined as <math>f^{-1}(1)</math>, is the set of points where ''f'' is 1.

: The fiber of <math>1\!</math> under <math>f,\!</math> defined as <math>f^{-1}(1),\!</math> is the set of points where the value of <math>f\!</math> is <math>1.\!</math>

* When 1 is interpreted as the logical value ''true'', then <math>f^{-1}(1)\!</math> is called the '''fiber&nbsp;of&nbsp;truth''' in the proposition ''f''.  Frequent mention of this fiber makes it useful to have a shorter way of referring to it.  This leads to the definition of the notation <math>[|f|] = f^{-1}(1)\!</math> for the fiber of truth in the proposition ''f''.

: When <math>1\!</math> is interpreted as the logical value <math>\operatorname{true},</math> then <math>f^{-1}(1)\!</math> is called the ''fiber of truth'' in the proposition <math>f.\!</math> Frequent mention of this fiber makes it useful to have a shorter way of referring to it.  This leads to the definition of the notation <math>[|f|] = f^{-1}(1)\!</math> for the fiber of truth in the proposition <math>f.\!</math>

* A '''singular boolean function''' ''s''&nbsp;:&nbsp;'''B'''^''k''&nbsp;→&nbsp;'''B''' is a boolean function whose fiber of 1 is a single point of '''B'''^''k''.

* A '''singular boolean function''' ''s''&nbsp;:&nbsp;'''B'''^''k''&nbsp;→&nbsp;'''B''' is a boolean function whose fiber of 1 is a single point of '''B'''^''k''.