Disproof by Counterexample

Disproof is the opposite of proof – instead of showing that something is true, we must show that it is false. Any statement that makes inferences about a set of numbers can be disproved by finding just one example for which it does not work.

Example 1

Disprove by counterexample that for any $a,b\in{\mathbb Z}$, if $a^2=b^2$, then $a=b$.

Note that ${\mathbb Z}$ is the set of all positive or negative integers. If an $a$ and $b$ such that $a\ne b$ and $a^2=b^2$, then the statement is disproved. Choosing any integer for $a$ and then choosing $b=-a$ will accomplish this. For example, let $a=4$ and $b=-4$. In this case $a^2=16$ and $b^2=16$ and so we have found an example where $a^2=b^2$ but $a\ne b$ and thus disproving the statement.

Example 2

Prove or disprove the statement that all prime numbers are odd.
2 is a prime number but it is not odd and so we have found an example of when the statement is not true – disproof by counterexample.