Methods of proof including proof by deduction, proof by exhaustion, disproof by counter-example and proof by contradiction, applied to statements about numbers and inequalities.

What this dot point is asking

AQA wants you to understand and use the structure of mathematical proof, including proof by deduction, proof by exhaustion, disproof by counter-example, and proof by contradiction (including the classic proofs that $\sqrt{2}$ is irrational and that there are infinitely many primes). The marks here are for rigour and clear logical layout, not just reaching the answer: each step must follow from the last, and you must state a clear conclusion.

Proof by deduction

You start from known facts or definitions and use algebra and logic to reach the required conclusion. The crucial point is generality: represent the objects with letters so the argument covers all cases at once. An even number is $2n$, an odd number is $2m + 1$, and consecutive integers are $n$ and $n + 1$. Testing one numerical example proves nothing.

Proof by exhaustion

You split the statement into a finite number of cases and verify each one. This is valid only when the cases are finite and you check them all. For example, to prove that no square number ends in $7$, note that any integer ends in one of the digits $0$ to $9$; squaring each gives final digits $0, 1, 4, 9, 6, 5, 6, 9, 4, 1$, none of which is $7$. Having checked every case, the claim is proved.

Disproof by counter-example

To disprove a general ("for all") statement, a single example where it fails is enough, and is all that is needed. For instance, the claim that $n^2 - n + 41$ is prime for all positive integers fails at $n = 41$, where it equals $41^2$, which is not prime. One counter-example settles it; you should not list several.

Proof by contradiction

You assume the opposite (the negation) of what you want to prove, then show this assumption forces an impossible result, so the original statement must be true. The layout matters: state the assumption, reason carefully, and name the contradiction explicitly.