Proof — OCR A-Level Mathematics

Proof is a foundational topic in OCR A-Level Mathematics that teaches you how to establish mathematical truths with absolute certainty. Unlike other areas of maths where you might calculate an answer or solve an equation, proof requires you to construct a logical argument that demonstrates a statement is always true, under given conditions. This topic is essential because it underpins all of mathematics — every theorem you use, from Pythagoras to calculus, relies on proof. In your A-Level, you will encounter proof in pure mathematics, mechanics, and statistics, so mastering it early is crucial for success across the entire syllabus.

The OCR specification focuses on several key proof techniques: direct proof, proof by contradiction, proof by exhaustion, and disproof by counterexample. You will also need to prove statements involving numbers (e.g., odd/even, rational/irrational), algebraic identities, and geometric properties. Understanding proof develops your logical reasoning and problem-solving skills, which are highly valued in further education and careers in STEM, law, and philosophy. In exams, proof questions often appear as 4–6 mark structured questions, requiring clear, step-by-step reasoning with correct mathematical notation.

Proof is not just a topic to memorise; it's a skill to practise. You need to think like a mathematician: start from known facts, use valid logical steps, and reach a conclusion that follows inevitably. The OCR examiners look for clarity, justification of each step, and the correct use of symbols (e.g., ⇒, ⇔, ∴). By mastering proof, you will gain confidence in handling unfamiliar problems and develop a deeper appreciation for the structure of mathematics.

• →Direct proof: Assume the hypothesis is true, then use logical steps and known facts to deduce the 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 (e.g., proving √2 is irrational).
• →Proof by exhaustion: Check all possible cases individually. This is only feasible when the number of cases is finite (e.g., proving that all integers from 1 to 10 satisfy a property).
• →Disproof by counterexample: To disprove a statement, find a single example that contradicts it. For instance, to disprove 'all prime numbers are odd', the counterexample is 2.
• →Mathematical notation: Use symbols like ⇒ (implies), ⇔ (if and only if), ∴ (therefore), and ∀ (for all). Correct notation is essential for clarity and marks.

• Clear definition of variables used in the proof
• Logical sequence of algebraic manipulation
• Concise and definitive conclusion
• Correct use of logical connectives such as 'if and only if'
• Rigorous application of proof by contradiction for irrationality or infinity of primes
• Identification of a single valid counter-example for disproof

• 💡Always state your assumptions clearly at the beginning of a proof
• 💡For 'show that' questions, ensure every intermediate step is explicitly written to justify the result
• 💡When asked to disprove by counter-example, only one valid example is required
• 💡Practice the standard proofs for the irrationality of root 2 and the infinity of primes as these are explicitly mentioned
• 💡Ensure the final conclusion directly addresses the original statement
• 💡Always state your method at the start. For example, 'We will prove this by contradiction' or 'We will use direct proof'. This helps the examiner follow your reasoning and shows you understand the technique.
• 💡Justify every step. Don't assume the reader knows why you can go from one line to the next. Use phrases like 'since n is even, n = 2k for some integer k' or 'by the definition of rational numbers'.
• 💡Check your conclusion. At the end, clearly state what you have proved, e.g., 'Therefore, the sum of two even numbers is even.' This ensures you haven't missed the final mark.

• Failing to define variables clearly at the start of a proof
• Assuming the result to be proved rather than deriving it
• Using examples to 'prove' a general statement instead of a formal argument
• Incorrect use of logical connectives or symbols
• Incomplete reasoning in proof by contradiction
• Misconception: 'Proof by exhaustion means checking a few examples.' Correction: Exhaustion requires checking every possible case, not just a sample. If the statement claims something for all integers, you cannot check them all individually — you need a different method.
• Misconception: 'A counterexample must be complicated.' Correction: Counterexamples can be simple, like 2 for 'all primes are odd'. Students often overlook obvious counterexamples because they expect something more complex.
• Misconception: 'Proof by contradiction is the same as proving the contrapositive.' Correction: They are different. Proof by contradiction assumes the negation and derives any contradiction; proving the contrapositive directly proves 'if not Q then not P' instead of 'if P then Q'.

1. 1Week 1, Days 1-2: Review the definitions of even, odd, rational, irrational, and prime numbers. Practise writing algebraic expressions for these (e.g., even = 2k, odd = 2k+1).
2. 2Week 1, Days 3-4: Learn direct proof and proof by exhaustion. Work through textbook examples and attempt 5-10 practice questions. Focus on clear step-by-step reasoning.
3. 3Week 1, Days 5-7: Study proof by contradiction and disproof by counterexample. Practise proving irrationality (e.g., √2, √3) and finding counterexamples for false statements.
4. 4Week 2, Days 1-2: Attempt past paper proof questions from OCR. Time yourself and check mark schemes. Identify common mistakes and note examiner comments.
5. 5Week 2, Days 3-5: Review all techniques and create a summary sheet with key notation and example proofs. Test yourself using active recall prompts. Finally, attempt a mixed set of proof questions under exam conditions.

• 📋Direct proof of a number property: e.g., 'Prove that the sum of two odd numbers is even.' Advice: Write the numbers as 2m+1 and 2n+1, add, and factor out 2.
• 📋Proof by contradiction: e.g., 'Prove that √5 is irrational.' Advice: Assume √5 = a/b in simplest form, square both sides, deduce a and b are both divisible by 5, contradiction.
• 📋Proof by exhaustion: e.g., 'Prove that for all integers n from 1 to 5, n^2 + n + 1 is prime.' Advice: Check each n individually (1,2,3,4,5) and state the result for each.
• 📋Disproof by counterexample: e.g., 'Disprove the statement: For all real numbers x, x^2 > x.' Advice: Find a counterexample like x=0.5, then show 0.25 > 0.5 is false.