# 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.