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Minimal negation operator (view source)
Revision as of 10:56, 24 August 2009
, 10:56, 24 August 2009add exposition
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<math>\begin{matrix}
<math>\begin{matrix}
\texttt{(~)}
\texttt{()}
& = & 0
& = & 0
& = & \operatorname{false}
& = & \operatorname{false}
& = & x^\prime y z \lor x y^\prime z \lor x y z^\prime
& = & x^\prime y z \lor x y^\prime z \lor x y z^\prime
\end{matrix}</math>
\end{matrix}</math>
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Expressing the general case of <math>\nu_k\!</math> in terms of familiar operations is facilitated by making a preliminary definition.
'''Definition.''' Let the function <math>\lnot_j : \mathbb{B}^k \to \mathbb{B},</math> for each integer <math>j \in [1, k],</math> be defined by the following equation:
{| align="center" cellpadding="8" width="90%"
| <math>\lnot_j (x_1, \ldots, x_j, \ldots, x_k) ~=~ x_1 \land \ldots \land x_{j-1} \land \lnot x_j \land x_{j+1} \land \ldots \land x_k.</math>
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Then <math>\nu_k : \mathbb{B}^k \to \mathbb{B}</math> is defined by the following equation:
{| align="center" cellpadding="8" width="90%"
| <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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