7.4PureStretch

Mathematical Proof

A proof is a watertight chain of logic showing a statement is always true — not just for a few examples. This lesson covers the language of proof and the most common method at this level: proof by deduction.

30 min Video by Zeeshan Zamurred Algebraic Methods

Edexcel AS Level Maths: 7.4 Mathematical ProofWatch the full walkthrough before the notes below.

Open on YouTube

What you'll be able to do

Understand what counts as a rigorous proof
Use correct proof notation and structure
Prove results by deduction (direct proof)
Prove an identity by manipulating one side

What a proof must do

A proof must show a statement holds in $every$ case, using known facts and logical steps. Checking examples is never a proof — it only takes one counterexample to disprove a claim, but no number of examples can prove one.

Tip — Examples build intuition but never prove a general statement. A proof argues for all cases at once.

Proof by deduction

In a direct (deductive) proof you start from known facts and definitions and reason step by step to the result. Algebra is the usual engine: represent the general case with letters, then manipulate.

even = 2 n, odd = 2 n + 1 (n \in Z)

Standard representations that make general number proofs possible.

1Let the integers be

n

and

n + 1

2Their sum is

n + (n + 1) = 2 n + 1

2 n + 1

has the form of an odd number, for every integer

n

Answerhence the sum is always odd ∎

Proving identities

To prove an identity (true for all values, shown with $\equiv$ ), take the more complicated side and manipulate it until it matches the other side. Never move terms across the $\equiv$ as if solving an equation.

1Expand the left side:

(x^{2} + 6 x + 9) - (x^{2} - 6 x + 9)

2Simplify:

12 x

3This equals the right-hand side for all

x

Answeridentity proved ∎

Tip — For identities, work one side down to the other — do NOT rearrange across the ≡ sign.

Formula recap

2 n, 2 n + 1

Even and odd integers.

\equiv means “identically equal” (true for all values)

Identity vs equation.

■

Marks the end of a proof.

Common mistakes to avoid

Proving a general statement by checking a few examples.

Use a general case (e.g. 2n + 1) so the argument covers every value.

Rearranging an identity across the ≡ sign like an equation.

Manipulate one side only until it matches the other.

Key takeaways

A proof shows a statement is true in every case, not just examples.
Deduction: start from known facts and reason to the result.
Represent general numbers (2n even, 2n + 1 odd) to argue for all cases.
Prove identities by transforming one side into the other.

Test yourself

Ready to lock in Mathematical Proof? Pick a mode and earn XP & Dobloons.