This site is supported by donations to The OEIS Foundation.

# Logical negation

Logical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false and a value of false when its operand is true.

A truth table for ${\displaystyle \mathrm {NOT} ~p,}$ also written ${\displaystyle \lnot p,}$ appears below:

 ${\displaystyle p}$ ${\displaystyle \lnot p}$ ${\displaystyle \mathrm {F} }$ ${\displaystyle \mathrm {T} }$ ${\displaystyle \mathrm {T} }$ ${\displaystyle \mathrm {F} }$

The negation of a proposition ${\displaystyle p}$ may be found notated in various ways in various contexts of application, often merely for typographical convenience.  Among these variants are the following:

 ${\displaystyle {\text{Notation}}}$ ${\displaystyle {\text{Vocalization}}}$ ${\displaystyle {\bar {p}}}$ ${\displaystyle p}$ bar ${\displaystyle {\tilde {p}}}$ ${\displaystyle p}$ tilde ${\displaystyle p'}$ ${\displaystyle p}$ prime ${\displaystyle p}$ complement ${\displaystyle !p}$ bang ${\displaystyle p}$

A logical graph for ${\displaystyle \lnot p}$ is shown below:

The traversal string of this graph is ${\displaystyle {\texttt {(}}p{\texttt {)}}.}$  The proposition ${\displaystyle \lnot p}$ may be taken as a Boolean function ${\displaystyle f(p)}$ having the abstract type ${\displaystyle f:\mathbb {B} \to \mathbb {B} ,}$ where ${\displaystyle \mathbb {B} =\{0,1\}}$ is interpreted in such a way that ${\displaystyle 0}$ means ${\displaystyle \mathrm {false} }$ and ${\displaystyle 1}$ means ${\displaystyle \mathrm {true} .}$

A Venn diagram for ${\displaystyle \lnot p}$ indicates the region where ${\displaystyle \lnot p}$ is true by means of a distinctive color or shading.  In this case the region is a single cell, as shown below:

## Document history

Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.