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 , if , then .
Note that is the set of all positive or negative integers. If an and such that and , then the statement is disproved. Choosing any integer for and then choosing will accomplish this. For example, let and . In this case and and so we have found an example where but 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.