# Counterexample

A counterexample is a specific case which shows that a general statement is false.

Example 1:

Provide a counterexample to show that the statement

"Every quadrilateral has at least two congruent sides"

is not always true.

Any scalene quadrilateral will serve as a counterexample.

For a conditional (if-then) statement, a counterexample must be an instance which satisfies the hypothesis , but not the conclusion .

Example 2:

Provide a counterexample to show that the statement

If $pq=x$ , then $p=\frac{x}{q}$

is not true for all real numbers $p$ , $q$ , and $x$ .

Let $p=1$ , $q=0$ and $x=0$ .

Then $\left(1\right)\left(0\right)=0$ but $1\ne \frac{0}{0}$ , since division by $0$ is undefined.