Changes

|}

|}

If we think of the point <math>x = (x_1, \ldots, x_k) \in \mathbb{B}^k</math> as indicated by the boolean product <math>x_1 \cdot \ldots \cdot x_k</math> or the logical conjunction <math>x_1 \land \ldots \land x_k,</math> then the minimal negation <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> indicates the set of points in <math>\mathbb{B}^k</math> that differ from <math>x\!</math> in exactly one coordinate.  This makes <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> a discrete functional analogue of a ''point omitted neighborhood'' in analysis, more exactly, a ''point omitted distance one neighborhood''.  In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field.  It also serves to explain a variety of other names for the same concept, for example, ''logical boundary operator'', ''limen operator'', ''threshold operator'', or ''least action operator'', to name but a few.

If we think of the point <math>x = (x_1, \ldots, x_k) \in \mathbb{B}^k</math> as indicated by the boolean product <math>x_1 \cdot \ldots \cdot x_k</math> or the logical conjunction <math>x_1 \land \ldots \land x_k,</math> then the minimal negation <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> indicates the set of points in <math>\mathbb{B}^k</math> that differ from <math>x\!</math> in exactly one coordinate.  This makes <math>\texttt{(} x_1, \ldots, x_k \texttt{)}</math> a discrete functional analogue of a ''point omitted neighborhood'' in analysis, more exactly, a ''point omitted distance one neighborhood''.  In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field.  It also serves to explain a variety of other names for the same concept, for example, ''logical boundary operator'', ''limen operator'', ''least action operator'', or ''hedge operator'', to name but a few.  The rationale for these names is visible in the venn diagrams of the corresponding operations on sets.

The remainder of this discussion proceeds on the ''algebraic boolean convention'' that the plus sign <math>(+)\!</math> and the summation symbol <math>(\textstyle\sum)</math> both refer to addition modulo 2.  Unless otherwise noted, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}</math> is interpreted so that <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math>  This has the following consequences:

The remainder of this discussion proceeds on the ''algebraic boolean convention'' that the plus sign <math>(+)\!</math> and the summation symbol <math>(\textstyle\sum)</math> both refer to addition modulo 2.  Unless otherwise noted, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}</math> is interpreted so that <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math>  This has the following consequences:

\text{or}    & e_j & = & \nu (x_j),

\text{or}    & e_j & = & \nu (x_j),

\\[6pt]

\\[6pt]

\text{for}  & j  & = & 1 ~\text{to}~ k.

\text{for}  & j  & = & 1 ~\text{to}~ k.

\end{array}</math>

\end{array}</math>

|}

|}

[[Category:Combinatorics]]

[[Category:Combinatorics]]

[[Category:Computer Science]]

[[Category:Computer Science]]

[[Category:Formal Languages]]

[[Category:Formal Languages]]

[[Category:Formal Sciences]]

[[Category:Formal Sciences]]