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 $p⟹q$, using the technique of proof by contradiction gives an additional hypothesis with which to work.