Proof

How it is used

You write, give, check or complete a proof. Common methods are direct proof, proof by contradiction and proof by induction. Questions often say 'prove that…' or 'show that…', and the end of a proof is marked with QED or a small square.

Example sentences

1. The proof is short, but each step needs a clear reason.

2. Can you give a proof that the square root of 2 is irrational?

Common collocations

• write a proof
• direct proof
• complete the proof
• rigorous proof

Related terms

• theoremAn important mathematical statement that has been proved to be true, such as the Pythagorean theorem.
• lemmaA smaller statement, proved on its own, that is used as a step towards proving a larger theorem.
• 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 contradictionA method of proof in which you assume the statement is false and show that this leads to something impossible, so the statement must be true.
• 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.

This word in other fields

The same English word can have a different meaning in another field:

• proof · Graphic Design

More terms: Proofs and Mathematical Writing

• let x be
• counterexample
• without loss of generality
• it follows that