Proof by contradiction

In logic and mathematics proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. It is a particular kind of the more general form of argument known as reductio ad absurdum. In logic and mathematics proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. It is a particular kind of the more general form of argument known as reductio ad absurdum. G. H. Hardy described proof by contradiction as 'one of a mathematician's finest weapons', saying 'It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.' Proof by contradiction is based on the law of noncontradiction as first formalized as a metaphysical principle by Aristotle. Noncontradiction is also a theorem in propositional logic. This states that an assertion or mathematical statement cannot be both true and false. That is, a proposition Q and its negation ¬ {displaystyle lnot } Q ('not-Q') cannot both be true. In a proof by contradiction, it is shown that the denial of the statement being proved results in such a contradiction. It has the form of a reductio ad absurdum argument. If P is the proposition to be proved: