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

Mathematical proof is the foundation of all rigorous mathematics. In this topic, you learn to construct logical arguments that establish the truth of a statement beyond any doubt. Starting from given assumptions (axioms or premises), you must proceed through a series of justified steps to reach an inevitable conclusion. This skill is essential not only for A-Level exams but also for university-level mathematics and any field requiring logical reasoning.

You will study four key methods: proof by deduction (direct logical reasoning from known facts), proof by exhaustion (checking all possible cases), disproof by counterexample (showing a statement is false with one example), and proof by contradiction (assuming the opposite and deriving an impossibility). Classic results include proving that √2 is irrational and that there are infinitely many prime numbers. These proofs exemplify the power of contradiction and are frequently examined.

Mastering proof develops your ability to think critically and communicate mathematically. In the Edexcel A-Level, proof questions appear across Pure Mathematics papers, often integrated with other topics like algebra, sequences, or number theory. You will be expected to apply these methods to unfamiliar contexts, so understanding the underlying logic is more important than memorising specific proofs.