Proof

Y13: by deduction (algebra), exhaustion, contradiction, induction (Further only).

Y13 Pure. Three key types: direct (algebraic manipulation), exhaustion (check all cases), contradiction (assume the opposite leads to a contradiction).

Worked examples

Prove sum of two consecutive odd numbers divisible by 4: (2n+1) + (2n+3) = 4n+4 = 4(n+1). Done.

Prove √2 is irrational (contradiction): assume √2 = p/q in lowest terms, show p and q both even → contradiction.

Exhaustion: prove n² + 2 is not divisible by 4 for n in 0,1,2,3 (cover all cases mod 4).

Frequently asked questions

Disproof by counterexample?

Find a single example where the statement fails. e.g. ‘n² + n + 41 is always prime’ fails at n = 40.

Induction?

Further Maths only. Standard A-Level uses deduction/exhaustion/contradiction.