Proof by contradiction

How it is used

Typical openings are 'Suppose, for contradiction, that…' and 'Assume the opposite'. The argument ends with 'This is a contradiction' or a symbol such as ↯. Classic examples prove that √2 is irrational and that there are infinitely many primes.

Example sentences

1. We use proof by contradiction: suppose √2 = p/q in lowest terms.

2. Proof by contradiction is often the easiest way to show that something does not exist.

Common collocations

• use proof by contradiction
• suppose for contradiction
• reach a contradiction
• assume the opposite

Related terms

• proofA logical argument that shows, step by step from accepted facts, that a mathematical statement is always true.
• counterexampleA single example that shows a general statement is false; for the claim 'all prime numbers are odd', the number 2 is a counterexample.
• QEDShort for the Latin 'quod erat demonstrandum', meaning 'which was to be shown'; written at the end of a proof to show that it is complete.
• proof by inductionA method of proving a statement for all positive integers: show it is true for the first case, then show that if it holds for n, it also holds for n + 1.

More terms: Proofs and Mathematical Writing

• theorem
• let x be
• lemma
• without loss of generality
• it follows that