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Minimal negation operator (view source)
Revision as of 13:08, 24 August 2009
, 13:08, 24 August 2009rewrite
| <math>\nu_k (x_1, \ldots, x_k) ~=~ \lnot_1 (x_1, \ldots, x_k) \lor \ldots \lor \lnot_j (x_1, \ldots, x_k) \lor \ldots \lor \lnot_k (x_1, \ldots, x_k).</math>
| <math>\nu_k (x_1, \ldots, x_k) ~=~ \lnot_1 (x_1, \ldots, x_k) \lor \ldots \lor \lnot_j (x_1, \ldots, x_k) \lor \ldots \lor \lnot_k (x_1, \ldots, x_k).</math>
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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>
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>
==Truth tables==
==Truth tables==