Proof by Counter-example

Consider a statement of the form

x M, if P(x) then Q(x).

Suppose that we wish to prove that this statement is false. In order to disprove this statement, we have to find a value of x in M for which P(x) is true and Q(x) is false. Such an x is called a counterexample.

Furthermore, proving that this statement is false is equivalent to showing that its negation is true. The negation of the above statement is

x in M such that P(x) and not Q(x).

x  M | P(x) ~Q(x).

Finding an x that makes the above statement true will disprove the original statement.