Proof (AS Unit 1: Pure Mathematics A) — WJEC A-Level Mathematics
Test yourself on Proof (AS Unit 1: Pure Mathematics A) with WJEC A-Level practice questions.
7 days Premium · Then free forever · No card, no charge
Proof (AS Unit 1: Pure Mathematics A) explained
Mathematical proof establishes universal truth from initial axioms through a deductive chain.
Read the full explanation
Proof by deduction starts with clear algebraic definitions: an even integer is expressed as 2k and an odd integer as 2k + 1 for k ∈ ℤ. Consecutive integers are n and n + 1. Algebraic expansion and factorisation demonstrate closure; for example, the product of two odd numbers (2k + 1)(2m + 1) = 2(2km + k + m) + 1 is necessarily odd. Proof by exhaustion tests every finite possibility within a bounded domain, such as verifying a property for each digit or case. Disproof by counterexample disproves a universal conjecture "for all x, P(x)" by exhibiting a single specific value x₀ for which P(x₀) is false, such as demonstrating that n² + n + 41 fails to generate a prime when n = 41.
Your focus
- Construct algebraic proofs using deduction, exhaustion, and counterexample.
- Formulate general algebraic expressions for even, odd, and consecutive integers.
- Identify and state clear counterexamples to disprove false universal conjectures.
Proof (AS Unit 1: Pure Mathematics A) exam tips
Marking Points
- stating correct algebraic definitions of terms (e.g. 2k for even, 2k + 1 for odd)
- rigorous algebraic manipulation, expansion, and factorisation to display the required property
- a clear concluding statement linking the algebraic result back to the original proposition
- identifying and evaluating a specific counterexample that disproves a universal claim
Examiner Tips
- 💡Define all variables explicitly at the start, such as stating that k is an integer.
- 💡End a deductive proof with a concluding sentence, such as 'which is a multiple of 2, hence even'.
Common Mistakes
- using specific numerical examples instead of general algebraic variables in a deductive proof
- assuming the truth of the statement being proved within the intermediate algebraic working
- testing only a few cases and falsely asserting that proof by exhaustion is complete