How Do You Use a Counterexample to Disprove a Statement?

A counterexample disproves a universal statement by providing one valid case for which the statement is false. One counterexample is sufficient, but you must explain clearly why it contradicts the original claim.

This is part of IB Mathematics: Analysis and Approaches HL, AHL 1.15: proof by induction, contradiction and counterexamples, and may be tested on Paper 1 or Paper 2.

The Reasoning

A universal statement claims that a property holds for every permitted value. Symbolically, it has the form

$$\text{For all } x \in S,\ P(x) \text{ is true}.$$

To disprove it, select one value $x=a$ that satisfies the stated domain but makes $P(a)$ false. Testing several examples that work does not prove the statement, while one example that fails disproves it completely.

Consider the claim:

For every $n \in \mathbb{N}$, $n^2+41n+41$ is prime.

Choose $n=41$, which is permitted because $41 \in \mathbb{N}$. Then

$$41^2+41(41)+41=41(41+41+1)=41 \times 83.$$

This number is composite because it has factors other than $1$ and itself. Therefore, $n=41$ is a counterexample, so the original statement is false.

A common misconception is that simply stating a value is enough. In AHL 1.15, the IB syllabus explicitly requires an explanation of why the example is a counterexample.

Exam Technique

For a command such as disprove, write the chosen value, verify that it belongs to the domain, show the calculation, and finish with an explicit conclusion. Do not attempt to prove the opposite statement generallyone fully justified counterexample earns the required reasoning credit.