How do I know which proof method to use?

Look at the statement you need to prove. If it's a simple implication, try direct proof. If direct proof is tricky, consider proof by contradiction — especially for statements like 'there is no largest prime'. If the statement involves a finite number of cases (e.g., all integers modulo something), use exhaustion. For disproving, a counterexample is usually easiest.

What is the difference between proof by contradiction and proof by contrapositive?

Proof by contradiction assumes the negation of the statement and derives any contradiction. Proof by contrapositive proves 'if P then Q' by proving 'if not Q then not P'. They are related but not the same. Contrapositive is a direct proof of the equivalent statement, while contradiction can be used for any statement.

Can I use a calculator in proof questions?

Calculators are allowed in AQA A-Level exams, but they are rarely useful for proof questions. Proofs require logical reasoning, not numerical computation. However, you might use a calculator to check a counterexample or to explore patterns, but the proof itself must be written without relying on calculator output.

How do I write a proof by exhaustion for a statement about integers?

First, identify the finite set of cases. For example, to prove something about all integers n, you might consider n mod 4 = 0,1,2,3. Then, for each case, substitute the general form (e.g., n=4k, 4k+1, etc.) into the statement and simplify to show it holds. Make sure to cover all possibilities and state that you have exhausted them.

What is a counterexample and how do I find one?

A counterexample is a specific instance that shows a statement is false. To find one, think of simple numbers (like 0,1,2, negative numbers, fractions) and test them. For example, to disprove 'all prime numbers are odd', test 2. If the statement claims something for all x, try extreme or boundary values.

Do I need to memorise all the proof techniques?

Yes, you should be familiar with direct proof, contradiction, exhaustion, and counterexample. In the exam, you may be asked to 'prove' or 'disprove' a statement, so you need to know which technique to apply. Practice each method with different types of statements to build confidence.

A: Proof

AQA

A-Level

This topic covers the fundamental structure of mathematical proof, requiring students to progress from given assumptions through logical steps to a valid conclusion. It includes specific methods such as proof by deduction, proof by exhaustion, disproof by counter-example, and proof by contradiction, with applications including the irrationality of root 2 and the infinity of primes.

Objectives

Exam Tips

Pitfalls

Key Terms

Mark Points

Topic Overview

Proof is the foundation of all mathematics. In AQA A-Level Mathematics, the topic 'A: Proof' introduces you to the rigorous methods used to establish mathematical truths beyond any doubt. You will learn how to construct logical arguments, starting from given assumptions and using valid reasoning to reach a conclusion. This skill is essential not only for higher-level mathematics but also for any field that requires clear, structured thinking.

The topic covers several key proof techniques: direct proof, proof by contradiction, proof by exhaustion, and disproof by counterexample. You will apply these methods to a variety of mathematical statements involving numbers, algebra, and geometry. Understanding proof is crucial because it underpins all other areas of mathematics — from calculus to mechanics — and helps you develop a deeper appreciation for why mathematical results are true.

Mastering proof will significantly boost your problem-solving skills and your ability to communicate mathematically. In exams, proof questions often appear in both pure mathematics and applied contexts, testing your ability to reason logically and present a clear, step-by-step argument. This topic is not just about memorising techniques; it's about learning to think like a mathematician.

Key Concepts

Core ideas you must understand for this topic

→Direct proof: Starting from known facts or assumptions, use logical steps to derive the desired conclusion. For example, proving that the sum of two even numbers is even.
→Proof by contradiction: Assume the opposite of what you want to prove, then show that this leads to a contradiction. For example, proving that √2 is irrational.
→Proof by exhaustion: Check all possible cases individually. This is useful when the statement can be split into a finite number of cases, such as proving that all integers n satisfy n² ≡ 0 or 1 mod 4.
→Disproof by counterexample: To disprove a statement, find a single example that contradicts it. For instance, to disprove 'all prime numbers are odd', use the counterexample 2.

What You Need to Demonstrate

Key skills and knowledge for this topic

Clear and logical progression of steps from assumption to conclusion
Correct use of mathematical language and symbols
Explicit statement of the method of proof being used
For proof by contradiction, clearly stating the assumption that the opposite is true
For disproof by counter-example, providing a single specific case that invalidates the statement
For proof by exhaustion, ensuring all possible cases are covered

Marking Points

Key points examiners look for in your answers

Clear and logical progression of steps from assumption to conclusion
Correct use of mathematical language and symbols
Explicit statement of the method of proof being used
For proof by contradiction, clearly stating the assumption that the opposite is true
For disproof by counter-example, providing a single specific case that invalidates the statement
For proof by exhaustion, ensuring all possible cases are covered

Examiner Tips

Expert advice for maximising your marks

💡Always define your variables clearly at the start of a proof
💡If asked to prove a statement is false, look for a counter-example first
💡In proof by contradiction, ensure the final step explicitly shows the contradiction with the initial assumption
💡Use precise mathematical terminology (e.g., 'integer', 'rational', 'even', 'odd') throughout the argument
💡Always state your method at the start of a proof. For example, 'We will prove this by contradiction' or 'We consider two cases'. This helps the examiner follow your reasoning and shows you understand the technique.
💡Write each step clearly and justify it with a reason. For instance, if you say 'n is even, so n = 2k', state that this is the definition of an even number. Avoid leaps in logic.
💡For proof by exhaustion, list all cases explicitly and check each one. Use a table if it helps. Make sure you haven't missed any cases — this is a common source of lost marks.

Common Mistakes

Pitfalls to avoid in your exam answers

Failing to state the assumption clearly in proof by contradiction
Assuming the result to be proved at the start of the argument
Using a single example as a proof for a general statement
Incomplete exhaustion in proof by exhaustion
Incorrect use of logical connectives or symbols
Misconception: 'Proof by contradiction is just guessing.' Correction: Proof by contradiction is a valid logical method. You assume the negation of the statement and derive a contradiction using rigorous reasoning, not guesswork.
Misconception: 'A single example proves a statement is true.' Correction: One example only proves a statement if it is an 'if and only if' or you are proving existence. For universal statements, one example is not enough; you need a general proof.
Misconception: 'Proof by exhaustion means checking a few cases.' Correction: Exhaustion requires checking every possible case without exception. If you miss one, the proof is incomplete.

Frequently Asked Questions

Common questions students ask about this topic

Before You Start

Prior knowledge that will help with this topic

•Basic algebra: manipulating expressions, solving equations, and working with inequalities.
•Number theory basics: understanding odd, even, prime, rational, and irrational numbers.
•Logical reasoning: familiarity with 'if...then...' statements, and the concepts of 'and', 'or', 'not'.

Key Terminology

Essential terms to know

Deductive reasoning and logical chains
Methods of proof including exhaustion and counter-example
Formal mathematical language and notation
Critical evaluation of mathematical arguments

Likely Command Words

How questions on this topic are typically asked

Prove

Show that

Disprove

Explain

Ready to test yourself?

Practice questions tailored to this topic

A: Proof

Topic Overview

Key Concepts

What You Need to Demonstrate

Marking Points

Examiner Tips

Common Mistakes

Frequently Asked Questions

Before You Start

Key Terminology

Likely Command Words

Ready to test yourself?

Related Topics in AQA A-Level Mathematics

B: Algebra and functions

C: Coordinate geometry in the ( x , y ) plane

D: Sequences and series

E: Trigonometry

Topic Synopsis

Key Concepts & Core Principles

Exam Tips & Revision Strategies

Common Misconceptions & Mistakes to Avoid

Examiner Marking Points

A: Proof

Topic Overview

Key Concepts

What You Need to Demonstrate

Marking Points

Examiner Tips

Common Mistakes

Frequently Asked Questions

How do I know which proof method to use?

What is the difference between proof by contradiction and proof by contrapositive?

Can I use a calculator in proof questions?

How do I write a proof by exhaustion for a statement about integers?

What is a counterexample and how do I find one?

Do I need to memorise all the proof techniques?

Before You Start

Key Terminology

Likely Command Words

Ready to test yourself?

Related Topics in AQA A-Level Mathematics

B: Algebra and functions

C: Coordinate geometry in the ( x , y ) plane

D: Sequences and series

E: Trigonometry