A contradiction occurs when the statements and 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 , one may assume and attempt to obtain a statement of the form , where 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 , using the technique of proof by contradiction gives an additional hypothesis with which to work.
|Date of creation||2013-03-22 16:02:48|
|Last modified on||2013-03-22 16:02:48|
|Last modified by||Wkbj79 (1863)|
|Defines||proof by contradiction|
|Defines||reductio ad absurdum|