## Minimal Negation Operators • 2

The brief description of minimal negation operators given in the previous post is enough to convey the rule of their construction.  For future reference, a slightly more formal definition is given below.

### Initial Definition

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 by the rule 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 minimal negation operators can be indicated by argument lists in parentheses.  In the discussion that follows a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, $\texttt{(x, y, z)} = \nu (x, y, z).$

The first four members of this family of operators are shown below.  The third and fourth columns give paraphrases in two other notations, where tildes and primes, respectively, indicate logical negation.

$\begin{matrix} \texttt{()} & = & \nu_0 & = & 0 & = & \mathrm{false} \\[6pt] \texttt{(x)} & = & \nu_1 (x) & = & \tilde{x} & = & x^\prime \\[6pt] \texttt{(x, y)} & = & \nu_2 (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x^\prime y \lor x y^\prime \\[6pt] \texttt{(x, y, z)} & = & \nu_3 (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x^\prime y z \lor x y^\prime z \lor x y z^\prime \end{matrix}$

### Resources

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