# contradiction

A *contradiction ^{}* occurs when the statements $p$ and $\mathrm{\neg}p$ are shown to be true simultaneously. This concept appears most often in a

*proof by contradiction*(also known as

^{}*reductio ad absurdum*), which is proving a statement by supposing its negation

^{}^{}is true and logically deducing an absurd statement. That is, in attempting to prove $q$, one may assume $\mathrm{\neg}q$ and attempt to obtain a statement of the form $\mathrm{\neg}r$, where $r$ is a statement that is assumed or known to be true.

Proofs by contradiction can become confusing. This is especially the case when such proofs are nested; i.e. (http://planetmath.org/Ie), a proof by contradiction occurs within a proof by contradiction. Some mathematicians prefer to use a direct proof whenever possible, as such are easier to follow in general. A small minority of mathematicians go so far as to reject proof by contradiction as a valid proof technique. It should be pointed out that something good can be said for proof by contradiction: If one wants to prove a statement of the form $p\u27f9q$, using the technique of proof by contradiction gives an additional hypothesis^{} with which to work.

Title | contradiction |
---|---|

Canonical name | Contradiction |

Date of creation | 2013-03-22 16:02:48 |

Last modified on | 2013-03-22 16:02:48 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 9 |

Author | Wkbj79 (1863) |

Entry type | Definition |

Classification | msc 03F07 |

Classification | msc 03B05 |

Related topic | ContradictoryStatement |

Defines | proof by contradiction |

Defines | reductio ad absurdum |