Often in maths we want to prove statements of the form “If A, then B.” For example: “If a number is divisible by \(4\), then it’s even.”
Usually, we prove such statements using something called a direct proof. In the example above, a direct proof would start by imagining we have some number — we don’t know which one — but we know it has the property “divisible by \(4\)”, and then using this information to work out that the number must be even.
However, this kind of direct reasoning can sometimes be tricky. Luckily, there’s another way! The idea is that a statement of the form “If A, then B” means exactly the same thing as “If not \(B\), then not \(A\).”
This second way of writing it is called the contrapositive, and we call “not \(B\)” the negation of \(B\). Here’s an everyday example: “If it rains, then I take my umbrella." is exactly the same as saying “If I don’t take my umbrella, then it’s not raining."
When we use this method in maths, we often say we’re proving by contrapositive: instead of proving “If \(A\) then \(B\)”, we prove “If not \(B\) then not \(A\).” Using this idea, to prove the example above, it would be the same as to prove the statement: ”If a number is not even, then it’s not divisible by \(4\)".
We sometimes write “If \(A\) then \(B\)” as \(A \implies B\), which is pronounced “\(A\) implies \(B\)”. Its contrapositive is: \(\text{not }B \implies \text{not }A.\) This way of thinking often makes a proof much simpler. Let’s see some examples to learn how to use this method.