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In [[logic]] and [[mathematics]], the '''minimal negation operator''' <math>\nu\!</math> is a [[multigrade operator]] <math>(\nu_k)_{k \in \mathbb{N}}</math> where each <math>\nu_k\!</math> is a <math>k\!</math>-ary [[boolean function]] defined in such a way that <math>\nu_k (x_1, \ldots , x_k) = 1</math> if and only if exactly one of the arguments <math>x_j\!</math> is <math>0.\!</math>

The '''minimal negation operator''' <math>\nu\!</math> is a [[multigrade operator]] <math>(\nu_k)_{k \in \mathbb{N}}</math> where each <math>\nu_k\!</math> is a <math>k\!</math>-ary [[boolean function]] defined in such a way that <math>\nu_k (x_1, \ldots , x_k) = 1</math> if and only if exactly one of the arguments <math>x_j\!</math> is <math>0.\!</math>

In contexts where the initial letter <math>\nu\!</math> is understood, the minimal negation operators 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.

In contexts where the initial letter <math>\nu\!</math> is understood, the minimal negation operators may be indicated by argument lists in parentheses.  In the following text, a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, <math>\texttt{(x, y, z)}</math> = <math>\nu (x, y, z).\!</math>

 

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

{| align="center" cellpadding="10" style="text-align:center"

{| align="center" cellpadding="10" style="text-align:center"

& = & \operatorname{false}

& = & \operatorname{false}

\\[6pt]

\\[6pt]

\texttt{(} x \texttt{)}

\texttt{(x)}

& = &

& = & \tilde{x}

\tilde{x}

& = & x^\prime

& = & x^\prime

\\[6pt]

\\[6pt]

\texttt{(} x, y \texttt{)}

\texttt{(x, y)}

& = &

& = & \tilde{x}y \lor x\tilde{y}

\tilde{x}y \lor x\tilde{y}

& = & x^\prime y \lor x y^\prime

& = &

x^\prime y \lor x y^\prime

\\[6pt]

\\[6pt]

\texttt{(} x, y, z \texttt{)}

\texttt{(x, y, z)}

& = &

& = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{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

& = &

x^\prime y z \lor x y^\prime z \lor x y z^\prime

\end{matrix}</math>

\end{matrix}</math>

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

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

The minimal negation operator ('''mno''') has a legion of aliases:  ''logical boundary operator'', ''[[limen|limen operator]]'', ''threshold operator'', or ''least action operator'', to name but a few.  The rationale for these names is visible in the [[venn diagram]]s of the corresponding operations on [[set]]s.

The minimal negation operator ('''mno''') has a legion of aliases:  ''logical boundary operator'', ''[[limen|limen operator]]'', ''threshold operator'', or ''least action operator'', to name but a few.  The rationale for these names is visible in the [[venn diagram]]s of the corresponding operations on [[set]]s.

To pass from these limiting examples to the general case, observe that a singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be given canonical expression as a conjunction of literals, <math>s = e_1 e_2 \ldots e_{k-1} e_k</math>.  Then the proposition <math>\nu (e_1, e_2, \ldots, e_{k-1}, e_k)</math> is <math>1\!</math> on the points adjacent to the point where <math>s\!</math> is <math>1,\!</math> and 0 everywhere else on the cube.

To pass from these limiting examples to the general case, observe that a singular proposition <math>s : \mathbb{B}^k \to \mathbb{B}</math> can be given canonical expression as a conjunction of literals, <math>s = e_1 e_2 \ldots e_{k-1} e_k</math>.  Then the proposition <math>\nu (e_1, e_2, \ldots, e_{k-1}, e_k)</math> is <math>1\!</math> on the points adjacent to the point where <math>s\!</math> is <math>1,\!</math> and 0 everywhere else on the cube.

For example, consider the case where <math>k = 3.\!</math>  Then the minimal negation operation <math>\nu (p, q, r)\!</math> &mdash; written more simply as <math>\texttt{(} p \texttt{,} q \texttt{,} r \texttt{)}</math> &mdash; has the following venn diagram:

For example, consider the case where <math>k = 3.\!</math>  Then the minimal negation operation <math>\nu (p, q, r)\!</math> &mdash; written more simply as <math>\texttt{(p, q, r)}</math> &mdash; has the following venn diagram:

{| align="center" cellpadding="10" style="text-align:center"

{| align="center" cellpadding="10" style="text-align:center"

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<p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p>

<p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p>

<p><math>\text{Figure 2.} ~~ \texttt{(} p \texttt{,} q \texttt{,} r \texttt{)}</math>

<p><math>\text{Figure 2.}~~\texttt{(p, q, r)}</math>

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For a contrasting example, the boolean function expressed by the form <math>\texttt{((} p \texttt{),(} q \texttt{),(} r \texttt{))}</math> has the following venn diagram:

For a contrasting example, the boolean function expressed by the form <math>\texttt{((p),(q),(r))}</math> has the following venn diagram:

{| align="center" cellpadding="10" style="text-align:center"

{| align="center" cellpadding="10" style="text-align:center"

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<p>[[Image:Venn Diagram ((P),(Q),(R)).jpg|500px]]</p>

<p>[[Image:Venn Diagram ((P),(Q),(R)).jpg|500px]]</p>

<p><math>\text{Figure 3.} ~~ \texttt{((} p \texttt{),(} q \texttt{),(} r \texttt{))}</math>

<p><math>\text{Figure 3.}~~\texttt{((p),(q),(r))}</math>

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