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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 ''k''-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 0.

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>

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 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.

: <math>\begin{matrix}

: <math>\begin{matrix}

(\ )     & = & 0 & = & \mbox{false} \\

(~)       & = & 0 & = & \mbox{false} \\

(x)      & = & \tilde{x} & = & x' \\

(x)      & = & \tilde{x} & = & x' \\

(x, y)    & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\

(x, y)    & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\