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Proof By Contradiction

  Proof by contradiction, also known as reductio ad absurdum (Latin, ‘reduction to an absurdity’), is one of the most commonly used methods of proof in mathematics. To prove a result using this method, you assume that the opposite is true, and then derive a contradiction, that is, a proof of both a result and its opposite. This means that the original assumption is false, on the grounds that a contradiction could not be derived from true assumptions, and therefore that the theorem that you wish to prove must indeed be true. The proof method relies on the ‘law of the excluded middle’, that something cannot simultaneously be and not be. Many of the results obtained by this method could also be proved directly, but proof by contradiction is often the quickest means to an end (and mathematicians generally look for the shortest proof, for the sake of elegance and, possibly, out of laziness). The proofs of a large number of existence results (theorems that assert that some kind of object exists) in many areas of mathematics (set theory and logic, for example) are carried out using proof by contradiction, and it is precisely this use which is challenged by the school of intuitionism, which insists on constructive proofs where the object whose existence is asserted is, in at least a theoretical sense, constructed. SMcL  



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