There are various ways to prove mathematical statements. One of the possible methods which might come in handy in certain situations is called proof by contradiction. To prove a statement we first assume that the statement is false and then deduce something that contradicts either the condition, or the assumption itself, or just common sense. Due to the contradiction, we have to conclude that the first assumption must have been wrong, so the statement is actually true.
A closely related method is called contrapositive proof. An example should make the idea quite clear. Consider the statement “if the joke is funny, then I will be laughing". Another completely equivalent way of saying it would be “if I am not laughing, then the joke is not funny". The second statement is known as the contrapositive of the first statement.
We can often prove a statement by proving its contrapositive. Many statements are proven by deriving a contradiction. However, one can often rewrite them as either a direct proof or a contrapositive proof.
Let’s take a look at both of these techniques.