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Minimal negation operator

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In logic and mathematics, the minimal negation operator \nu\! is a multigrade operator (\nu_{k})_{k \in \mathbb{N}} where each \nu_{k}\! is a k-ary boolean function defined in such a way that \nu_{k}(x_1, \ldots , x_k) = 1 if and only if exactly one of the arguments x_{j} is 0.

In contexts where the initial letter \nu\! is understood, the mno's can be indicated by argument lists in parentheses. The first four members of this family of operators are shown below, with paraphrases in a couple of other notations, where tildes and primes, respectively, indicate logical negation.

\begin{matrix}
(\ )      & = & 0 & = & \mbox{false} \\
(x)       & = & \tilde{x} & = & x' \\
(x, y)    & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\
(x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz'
\end{matrix}

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

The minimal negation operator (mno) has a legion of aliases: logical boundary operator, limen operator, threshold operator, or least action operator, to name but a few. The rationale for these names is visible in the Venn diagrams of the corresponding operations on sets.

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