A single counterexample is enough because a general claim says that a statement is true in every permitted case. Finding even one permitted case where it is false directly contradicts the word “every”.
A universal claim can be written as
meaning “for every in the domain , the property is true.” Its logical negation is
meaning “there exists at least one permitted value for which the property is false.” Therefore, producing that one value disproves the original claim completely.
This distinction matters because testing many cases only provides evidence, not proof. A universal statement covers an entire domain, including cases that have not been checked. By contrast, a counterexample targets the claim’s weakest logical requirement: if every case must work, one failure is sufficient to make the statement false.
For example, consider the claim:
For every , the number is prime.
Choose , which belongs to :
Since is composite, the expression is not prime when . Thus, the universal claim is false.
The counterexample must satisfy all the conditions in the claim. For example, a negative number cannot disprove a statement restricted to positive integers because it lies outside the stated domain.
A common misconception is that several counterexamples are needed to make the disproof convincing. This is incorrect: one valid counterexample is logically conclusive, although many examples supporting a claim can never prove that it is universally true.
This is IB Mathematics: Analysis and Approaches HL, sub-topic AHL 1.15: Proof by induction, contradiction and counterexample, and may be tested analytically on Paper 1 or Paper 2. Do not merely state a value: substitute it, show that it satisfies the original conditions, and explain precisely why the resulting case contradicts the claim. The syllabus explicitly states that giving a counterexample without explaining why it works is insufficient.
