A contradictionMathworldPlanetmathPlanetmath occurs when the statements p and ¬p are shown to be true simultaneously. This concept appears most often in a proof by contradictionMathworldPlanetmath (also known as reductio ad absurdumMathworldPlanetmath), which is proving a statement by supposing its negationMathworldPlanetmath is true and logically deducing an absurd statement. That is, in attempting to prove q, one may assume ¬q and attempt to obtain a statement of the form ¬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. (, 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 pq, using the technique of proof by contradiction gives an additional hypothesisMathworldPlanetmathPlanetmath 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