A-Level Maths: Proof

Proof opens the A-Level Mathematics specification for a reason: every technique that follows rests on knowing what actually constitutes a valid mathematical argument. This quiz covers the four proof methods the prescribed content names, with the two classic contradiction arguments examiners return to again and again.

The method questions separate the four approaches cleanly: proof by deduction as the general algebraic argument (tested through the classic "sum of two odd integers is even" proof, where the working with 2m+1 and 2n+1 must be recognised in full), proof by exhaustion and its defining constraint — it only works when the cases form a finite, checkable set — and disproof by counter-example, tested twice: once with the famous polynomial n² + n + 41, which produces primes for every n from 0 to 39 and then fails at n = 40 (41²) and n = 41 (41 × 43), and once with the simplest counter-example in mathematics, the even prime 2 against "all primes are odd".

Proof by contradiction gets the deepest treatment, as befits its place in the full A level: the structural questions pin down its two necessary components — assuming the negation, deriving a logical impossibility — and the two canonical arguments are tested directly: the irrationality of √2 through its even-odd contradiction, and Euclid's infinitude of primes through the construction that multiplies a supposed complete list and adds one.

A strategy question rounds out the set, asking which method most directly proves x² − 4x + 5 > 0 for all real x — completing the square, the deductive route that avoids case-checking entirely.

Every explanation names the logical structure at work, which is exactly what "state which method" and "complete the proof" exam questions reward.

• Distinguish deduction, exhaustion, counter-example and contradiction as proof strategies
• Construct the standard deductive proof for parity statements
• Disprove universal claims with a single counter-example, including n²+n+41
• Reproduce the contradiction arguments for the irrationality of √2 and the infinitude of primes
• Choose the most direct proof method for a given statement

Topic scope follows section A (Proof) of the DfE's prescribed AS and A level mathematics subject content — 100% common across all exam boards: proof by deduction, proof by exhaustion, disproof by counter-example, and (A level) proof by contradiction including the irrationality of √2 and the infinity of primes.