Minimal negation operator

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.