Proof by Contradiction in Depth

Proof by contradiction — also known as reductio ad absurdum — is one of the most elegant and powerful techniques in mathematics. The method works by assuming the opposite of what you want to prove and then showing that this assumption leads to a logical impossibility (a contradiction). Since the assumption produces a contradiction, it must be false, and therefore the original statement must be true.

The AQA A-Level Mathematics specification (7357) explicitly lists proof by contradiction as a required technique. The classic examples — the irrationality of √2 and the infinitude of primes — are staple exam and interview questions.

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The Logical Structure

The logical structure of proof by contradiction is as follows:

1. State the proposition P that you wish to prove.
2. Assume ¬P (the negation of P).
3. Derive a sequence of logical consequences from ¬P.
4. Arrive at a contradiction — a statement that is logically impossible (e.g., both X and ¬X are true, or a known fact is violated).
5. Conclude that since ¬P leads to a contradiction, ¬P is false. Therefore P is true.