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negation

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

\textbf{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).

\textbf{Note 2.}\, $\lnot P$ may be expressed by implication as
$$P\to\curlywedge$$
where $\curlywedge$ means any contradictory statement.

\textbf{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.

\textbf{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:\,\,
 $\neq,\, \nsubseteq,\, \notin,\, \ncong,\, \nparallel,\, \nmid$.
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