The law of the excluded middle asserts that a statement is true or it is false, it cannot be anything in between. We can use this as another method of proof. We assume that the statement is false and proceed logically to show that this gives a statement that we definitely know is false such as 1 = 0 or the Moon is made of cheese. Thus our assumption must be wrong, the statement can't be false – it leads to something ridiculous – so the statement is true.

This method is called proof by contradiction. The name comes from the fact that assuming that the statement is false is later contradicted by some other fact. It is also known by the name reductio ad absurdum which when translated means reduction to the absurd.

Simple examples of proof by contradiction

The first example is just to show you the idea of proof by contradiction. The statement is easier to prove by a direct method as we have seen in Theorem 20.1.

Example 23.1

Suppose that n is an odd integer. Then n2 is an odd integer.

Proof. Assume the contrary. That is, we suppose that n is an odd integer but that the conclusion is false, i.e. n2 is an even integer.