contradiction Contradiction 2006-06-26 05:25:15 2007-06-14 03:46:18 Definition proof by contradiction reductio ad absurdum % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A \emph{contradiction} occurs when the statements $p$ and $\neg p$ are shown to be true simultaneously. This concept appears most often in a \emph{proof by contradiction} (also known as \emph{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 $\neg q$ and attempt to obtain a statement of the form $\neg r$, 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; \PMlinkname{i.e.}{Ie}, a proof by contradiction occurs within a proof by contradiction. Some mathematicians prefer to use a direct proof whenever possible, as such \PMlinkescapetext{arguments} 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 $p \implies q$, using the technique of proof by contradiction gives an additional hypothesis with which to work.