Proof by Counterexample

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

(For all the elements $x$ of a given set $S$, the property $P$ holds.)

Such a statement may or may not be true.

An example of such a statement which is definitely not true is:
 * All Englishmen are cowards.

For example, it is generally accepted that Lord Horatio Nelson (whatever other character flaws he may or may not have had) was most definitely not a coward.

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

(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 $X$.

Proving, or disproving, a statement in the form of $X$ by establishing the truth or falsehood of a statement in the form of $Y$ is known as the technique of proof by counterexample.