A single failure kills a 'for all' claim: A claim that something is true for all values can be destroyed by a single value where it fails — a counterexample.
You don't prove anything in general; you just show one case that breaks it.
IB-style question — disprove a claim
Disprove the statement: 'Every prime number is odd.'
Step by step
- Find one prime that is not odd.
- 2 is prime, but it is even.
Final answer
The number 2 is an even prime — a counterexample — so the statement is false.
IB-style question — a famous near-miss
Disprove: 'n² − n + 41 is prime for every positive integer n.'
Step by step
- It works for n = 1, 2, 3, … but try n = 41.
- Substitute.
- 1681 = 41 × 41 is not prime.
Final answer
n = 41 gives 41², which is composite — a counterexample.
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Try the awkward values first: Good places to look for a counterexample: 0, 1, negatives, fractions, and edge cases. These often break a claim that 'feels' true for ordinary positive whole numbers.
IB-style question — squares
Disprove: 'If a² = b² then a = b.'
Step by step
- Try a positive and its negative.
- Check the condition.
Final answer
a = 2, b = −2 satisfy a² = b² yet a ≠ b — a counterexample.
IB-style question — a fraction breaks it
Disprove: 'For every real number x, x² ≥ x.'
Step by step
- Whole numbers obey it, so try a fraction between 0 and 1.
- Compare.
Final answer
x = ½ gives x² < x, so the claim is false.