A: Proof — AQA A-Level Mathematics
Test yourself on A: Proof with AQA A-Level practice questions.
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A: Proof explained
Proof is a logical argument that starts from agreed assumptions and reaches a conclusion by valid steps.
Read the full explanation
Deduction uses algebra or known results to show a statement is always true; exhaustion checks every possible case; a single counter example disproves a universal claim; contradiction assumes the opposite and derives an impossibility. You must be able to write proofs clearly, justify each step, and apply these methods to unfamiliar statements. For example, to prove √2 is irrational, assume it equals a fraction in lowest terms, derive that both numerator and denominator are even, contradicting lowest terms. To prove infinitely many primes, assume a finite list, multiply them and add 1, then show the new number has a prime factor not in the list.
Your focus
- Construct a proof by deduction from given assumptions to a conclusion.
- Use proof by exhaustion to verify a statement for all possible cases.
- Disprove a universal statement by providing a valid counter example.
Show all 4 objectives
- Apply proof by contradiction, including the irrationality of √2 and the infinity of primes, to unfamiliar statements.
A: Proof exam tips
Marking Points
- Start from the given assumptions or definitions and make each logical step explicit, so the argument can be followed without gaps.
- Use deduction correctly, for example by expanding brackets, factorising, or completing the square to reach the required conclusion.
- For exhaustion, identify all possible cases and check each one, showing that no case is omitted.
- For disproof by counter example, provide a specific example that satisfies the conditions but makes the statement false.
- For contradiction, state the negation of the result, derive a contradiction with a known fact or assumption, and conclude the original statement must be true.
- In the irrationality of √2, use the fact that if a square is even then the number is even, and that a fraction can be written in lowest terms.
- In the infinity of primes, construct a number that is not divisible by any prime in the assumed finite list, then use the fundamental theorem of arithmetic to obtain a contradiction.
Examiner Tips
- 💡Write in full sentences and use mathematical notation correctly; do not rely on unexplained symbols.
- 💡For contradiction proofs, clearly label the assumption and the point where the contradiction arises.
- 💡When asked to disprove, give a counter example that is easy to verify and clearly satisfies the conditions.
Common Mistakes
- Assuming what you are trying to prove. Correction: begin from known facts or the negation for contradiction, and only use the conclusion at the end.
- In proof by exhaustion, missing a case, such as not checking all residue classes modulo n. Correction: list all possible cases systematically before checking.
- In proof by contradiction, failing to use the negation correctly, for example assuming √2 is rational but not that the fraction is in lowest terms. Correction: state the full negation, including all necessary conditions.