In logics and mathematics, negation (from Latin negare ‘to deny’) is the unary operation “$\lnot$” which swaps the truth value of any operand to the opposite truth value. So, if the statement $P$ is true then its negated statement $\lnot P$ is false, and vice versa.

Note 1. The negated statement $\lnot P$ (by Heyting) has been denoted also with $-P$ (Peano), $\sim\!P$ (Russell), $\overline{P}$ (Hilbert) and $NP$ (by the Polish notation).

Note 2. $\lnot P$ may be expressed by implication as

$P\to\curlywedge$ |

where $\curlywedge$ means any contradictory statement.

Note 3. The negation of logical or and logical and give the results

$\lnot(P\lor Q)\equiv\lnot P\land\lnot Q,\qquad\lnot(P\land Q)\equiv\lnot P\lor% \lnot Q.$ |

Analogical results concern the quantifier statements:

$\lnot(\exists x)P(x)\equiv(\forall x)\lnot P(x),\qquad\lnot(\forall x)P(x)% \equiv(\exists x)\lnot P(x).$ |

These all are known as de Morgan’s laws.

Note 4. Many mathematical relation statements, expressed with such special relation symbols as $=,\,\subseteq,\,\in,\,\cong,\,\parallel,\,\mid$, are negated by using in the symbol an additional cross line: $\neq,\,\nsubseteq,\,\notin,\,\ncong,\,\nparallel,\,\nmid$.