Proof, logic and mathematical reasoning
The Proof, logic and mathematical reasoning domain tests whether you can build a valid deductive argument rather than merely compute an answer. Questions ask you to prove statements about integers, inequalities and algebraic identities, to disprove false claims with a single counterexample, and to identify the logical structure - hypothesis, conclusion and converse - of a given statement.
Logical structure of a statement
In "if P then Q", P is the hypothesis and Q the conclusion. A direct proof assumes P and derives Q. The converse "if Q then P" is a different statement and needs its own proof. The contrapositive "if not Q then not P" is logically equivalent to the original and is often the easier route. "P is necessary for Q" means Q ⇒ P; "P is sufficient for Q" means P ⇒ Q.
Standard methods
- Direct proof: algebraic manipulation from the hypothesis to the conclusion.
- Proof by exhaustion: split into finitely many cases and check each one.
- Proof by contradiction: assume the negation of the statement and derive a falsehood.
- Disproof by counterexample: exhibit one case satisfying the hypothesis but failing the conclusion.
Worked example
Prove that n² + n is even for every integer n.
Factorise: n² + n = n(n + 1), the product of two consecutive integers. Exactly one of them is even, so the product is even. For a formal write-up, use exhaustion on parity. If n = 2k, then n² + n = 4k² + 2k = 2(2k² + k). If n = 2k + 1, then n² + n = (2k + 1)(2k + 2) = 2(2k + 1)(k + 1). Every integer is even or odd, so the result holds in both cases, as required.
Traps that cost marks
- Arguing from one example to a general claim; an example never proves a universal statement.
- Proving the converse and presenting it as the original result.
- In contradiction proofs, assuming the conclusion instead of its negation.
- Offering a counterexample that does not actually satisfy the hypothesis, so it disproves nothing.
- Ending with "as required" before the required statement has been reached.