Understand and be able to use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion.

In particular, learners should use methods of proof including proof by deduction and proof by exhaustion.

Proof by deduction Proof by exhaustion

Understand and be able to use the logical connectives \(≡\), \(\implies\), \(\iff\).

Learners should be familiar with the language associated with the logical connectives: “congruence”, “if..... then” and “if and only if” (or “iff”).

Be able to show disproof by counter example.

Learners should understand that this means that, given a statement of the form “if \(P(x)\) is true then \(Q(x)\) is true”, finding a single \(x\) for which \(P(x)\) is true but \(Q(x)\) is false is to offer a disproof by counter example.

Questions requiring proof will be set on content with which the learner is expected to be familiar e.g. through study of GCSE (9-1) or AS Level Mathematics.

Learners are expected to understand and be able to use terms such as “integer”, “real”, “rational” and “irrational”.

Disproof by counter example

Understand and be able to use proof by contradiction.

In particular, learners should understand a proof of the irrationality of \(\sqrt{2}\) and the infinity of primes.

Questions requiring proof by contradiction will be set on content with which the learner is expected to be familiar e.g. through study of GCSE (9-1), AS or A Level Mathematics.

Proof by contradiction