Definition:Counterexample

Definition
Let $X$ be the statement:
 * $\forall x \in S: P \left({x}\right)$

That is:
 * For all the elements $x$ of a given set $S$, the property $P$ holds.

Such a statement may or may not be true.

Let $Y$ be the statement:
 * $\exists y \in S: \neg P \left({y}\right)$

That is:
 * There exists at least one element $y$ of the set $S$ such that the property $P$ does not hold.

It follows immediately by De Morgan's laws that if $Y$ is true, then $X$ must be false.

Such a statement $Y$ is referred to as a counterexample to $X$.

Also see

 * Proof by Counterexample