\begin{definition}[Definition:Counterexample]
Let $X$ be the universal statement:
:$\forall x \in S: \map P x$
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 existential statement:
:$\exists y \in S: \neg \map P y$
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 $X$'''.
\end{definition}