Counterexample

Summary

A counterexample to a universal statement $\forall x \in X, \, P(x)$ is an element $a \in X$ such that $P(a)$ is false. One counterexample shows that the universal statement is false.

Remarks

The number $2$ is a counterexample to "each prime number is odd". This works because of the negation of a universal statement: $\neg \forall x \, P(x)$ is equivalent to $\exists x \, \neg P(x)$.

Explanations

L1 human

To show that a general claim is wrong, one bad case is enough. "Each continuous function is differentiable" is false: $f(x) = |x|$ is continuous but not differentiable at $0$.