Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; use methods of proof, including:

Proof by deduction

e.g. using completion of the square, prove that \(n^2-6n+10\) is positive for all values of \(n\) or, for example, differentiation from first principles for small positive integer powers of \(x\) or proving results for arithmetic and geometric series. This is the most commonly used method of proof throughout this specification.

Proof by exhaustion

This involves trying all the options. Suppose \(x\) and \(y\) are odd integers less than \(7\). Prove that their sum is divisible by \(2\).

Disproof by counter example

e.g. show that the statement "\(n^2-n+1\) is a prime number for all values of \(n\)" is untrue.

Proof by contradiction (including proof of the irrationality of √2 and the infinity of primes, and application to unfamiliar proofs).

Proof by deduction Proof by exhaustion Disproof by counter example Proof by contradiction