A universal claim dies to a single witness: exhibit one case where every hypothesis holds and the conclusion fails. The bank's kills are worth memorising — 2 against 'every prime is odd'; x = 2 against '|x| > x'; x = 1/2 against 'x² > x'; (1, 1) against '(x + y)² = x² + y²' with its missing 2xy; 2 and −2 against 'a² = b² implies a = b'; 6 dividing 4 × 3 against the divisibility implication; supplementary angles against equal sines implying equal angles; n = 40 giving 41² and n = 41 giving 41 × 43 against n² + n + 41 always prime.
Hypothesis-false examples prove nothing: 3 × 5 says nothing about a claim whose premise needs an even product, and confirming instances (x = −5 satisfies |x| > x) can never disprove. The counterexample must satisfy every premise and break the conclusion — check both halves before committing. Hunt where claims are weakest: boundary values 0 and 1, the interval (0, 1] where squaring shrinks, sign changes, and the n = 40/41-style self-referential inputs examiners love.
Jumbled proofs yield to the same instinct in reverse: read the lines in order, label each given, definition or deduction, and stop at the first line that needs more than the lines above it. That overreaching universal step is exactly where a counterexample hides — test it with the boundary values above. Remember the quantifier flip (disproof of 'for all' is 'there exists … not'), distrust verifying instances dressed as argument, and never accept a 'counterexample' nobody evaluated: 3 × 4 = 12 is even, so n = 3 refutes nothing. A disproof never needs more than one line of working after the witness — exhibit it, verify both halves, stop; anything longer is a proof wearing a disguise.
Study method: attempt five timed questions, log every miss with its topic code, redo misses from scratch within 48 hours, then re-attempt the topic a week later. Pair each hub with its drill link below and a fortnightly full mock.
Place this topic in the wider test with the syllabus checklist and Paper 1 vs Paper 2, learn the timing system in how to prepare, and check what scores mean in scores explained.