# negation

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

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

Note 2. $\mathrm{\neg}P$ may be expressed by implication^{} as

$$P\to \u22cf$$ |

where $\u22cf$ means any contradictory statement.

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

$$\mathrm{\neg}(P\vee Q)\equiv \mathrm{\neg}P\wedge \mathrm{\neg}Q,\mathrm{\neg}(P\wedge Q)\equiv \mathrm{\neg}P\vee \mathrm{\neg}Q.$$ |

Analogical results concern the quantifier^{} statements:

$$\mathrm{\neg}(\exists x)P(x)\equiv (\forall x)\mathrm{\neg}P(x),\mathrm{\neg}(\forall x)P(x)\equiv (\exists x)\mathrm{\neg}P(x).$$ |

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:
$\ne ,\u2288,\notin ,\ncong ,\nparallel ,\nmid $.

Title | negation |
---|---|

Canonical name | Negation |

Date of creation | 2015-04-25 17:44:13 |

Last modified on | 2015-04-25 17:44:13 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 03B05 |

Synonym | logical not |

Related topic | SetMembership |