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'', ''threshold operator'', or ''least action operator'', to name but a few.

It may also be noted that <math>\texttt{(x, y)}</math> is the same function as <math>x + y\!</math> and <math>x \ne y</math>, and that the inclusive disjunctions indicated for <math>\texttt{(x, y)}</math> and for <math>\texttt{(x, y, z)}</math> may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function <math>\texttt{(x, y, z)}</math> is not the same thing as the function <math>x + y + z.\!</math>

In what follows, the boolean domain <math>\mathbb{B} = \{ 0, 1 \}</math> is interpreted so that <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}.</math>  In this context, the plus sign <math>(+)\!</math> and the summation symbol <math>(\textstyle\sum)</math> both refer to addition modulo 2.  This has the following consequences:

 

| The operation <math>x + y\!</math> is a function equivalent to the exclusive disjunction of <math>x\!</math> and <math>y,\!</math> while its fiber of 1 is the relation of inequality between <math>x\!</math> and <math>y.\!</math>

| The operation <math>\textstyle\sum_{j=1}^k x_j</math> maps the bit sequence <math>(x_1, \ldots, x_k)\!</math> to its ''parity''.

 

The following properties of the minimal negation operators <math>\nu_k : \mathbb{B}^k \to \mathbb{B}</math> may be noted:

 

| The function <math>\texttt{(x, y)}</math> is the same as that associated with the operation <math>x + y\!</math> and the relation <math>x \ne y.</math></p>

| In contrast, <math>\texttt{(x, y, z)}</math> is not identical to <math>x + y + z.\!</math>

==Truth tables==

==Truth tables==