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=6 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.