negation

In logics and mathematics, (from Latin negare ‘to deny’) is the unary operation “$\lnot$” which swaps the truth value of any operand to the 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:   $\neq,\,\nsubseteq,\,\notin,\,\ncong,\,\nparallel,\,\nmid$.

Title negation Negation 2015-04-25 17:44:13 2015-04-25 17:44:13 pahio (2872) pahio (2872) 8 pahio (2872) Definition msc 03B05 logical not SetMembership