Proof

What proof covers

The unit that sets out what counts as an argument in this subject, and how a claim about infinitely many cases can be settled in half a page. It is two lessons long, but its methods are examinable across both Pure papers of 9MA0, and the marks go to the reasoning rather than to the final line.

The main ideas

• What a proof is: stated assumptions, checkable steps, and a conclusion that settles every case at once.
• Proof by deduction, with completing the square as the standard route to a claim about the sign of a quadratic.
• Proof by exhaustion, where the list of cases has to be shown complete before any case is checked.
• Disproof by counter example, hunted where the claim looks weakest and then demonstrated in full.
• Proof by contradiction: assume the opposite, reason correctly to an impossibility, and close the loop in words.
• The two contradiction proofs the specification names, on the irrationality of root 2 and on the supply of primes.

The results it turns on

let the numbers be 2m + 1 and 2n + 1, with m and n integers

naming the family a proof about odd numbers starts from

a(x + p)² + q, with a > 0 and q > 0

the completed square that settles a sign for every value of x

one counter example

all it takes to disprove a claim made about every member of a set

assume √2 = a/b in lowest terms

the opening line whose lowest-terms clause carries the contradiction

assume a complete list of primes, then form N = p₁p₂ … p_k + 1

the opening of the second named contradiction proof

Where it usually goes wrong

• The shape of the claim chooses the method. Finitely many cases invites exhaustion, and a claim about all the integers or all the reals needs deduction instead.
• In the primes proof, N itself need not be prime. The claim is only that its prime factors are missing from the list, and 30 031 is the examiner's stock counterexample to the stronger sentence.
• A disproof needs the failure shown, not asserted. Give the value, evaluate the expression, and factorise or otherwise demonstrate what goes wrong.
• The closing sentence is marked. Both the family statement at the start and the sentence answering the claim in its own words carry marks of their own.

Where to start

Take the structure lesson first, since deduction and exhaustion set the layout everything else uses. Disproof and contradiction follow, and the two named proofs are worth rehearsing as arguments rather than as memorised scripts.