## 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 , 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.**

At first thought, it might seem that all prime numbers are odd. This is because it seems that all even numbers are not prime as 2 is a factor. However, by definition, 2 is a prime number but it is not odd and so we have found an example of when the statement is not true. This disproves the statement by counterexample.

### Example 3

** Find any values for p and q for which the following statement isn’t true: **

The tendency in this situation is to try various positive values of p and q until the statement isn’t true. However, the statement IS true for all positive values and so negative values of p and q must be tried. Bear in mind that you can only square root a positive number (without complex numbers) and so both p and q must both be negative. Try, for example, p=-2 and q=-2. The left hand side is square root of pq which is 2 whereas the right hand side is -2. Since 2 is not less than or equal to -2, we have found a counterexample.