Knowledge organisers

Proof: A Level Maths knowledge organiser

Everything to know about proof on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Proof by deduction: Start from known facts and use logical steps to reach the statement.
Proof by exhaustion: Split into a finite number of cases and check every one.
Counter-example: One example that shows a statement is false.
Proof by contradiction: Assume the statement is false and show this leads to something impossible.

Key formulas

No formulas to learn for this topic: it is about methods and clear reasoning.

Worked example

Prove that (2n + 1)² − (2n − 1)² is a multiple of 8 for every integer n.

(2n + 1)² = 4n² + 4n + 1 and (2n − 1)² = 4n² − 4n + 1
The difference is 8n
8n = 8 × n and n is an integer

Answer: (2n + 1)² − (2n − 1)² = 8n, a multiple of 8

Common mistakes

Proof: "squares are always positive" and no concluding statement
Checking a few examples and calling it a proof
Not writing a conclusion that states what has been proved

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Write clear proofs by deduction, such as showing that the sum of any three consecutive integers is a multiple of three, setting out each logical step.
Prove statements by exhaustion, checking every possible case, such as showing that the square of any integer ends in 0, 1, 4, 5, 6 or 9.
Write proofs by contradiction, such as proving that √2 is irrational and that there are infinitely many primes, stating the assumption and the contradiction clearly.

The printable sheet

Proof knowledge organiser for A Level Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Proof knowledge organiser (A Level Maths), A4. Download the PDF.

Revise it next

Other A Level Maths topics: Algebra and functions · Coordinate geometry · Sequences and series · Trigonometry · Exponentials and logarithms · Differentiation · Integration · Numerical methods · Vectors · Sampling and data · Probability · Binomial and normal distributions · Hypothesis testing · Kinematics · Forces and Newton's laws · Moments · All A Level Maths organisers