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."

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.