What is a contrapositive in math?

Answer 1

Contrapositives are an idea in logic which is very useful in math.

We say that A implies B if whenever Statement A is true then we know that statement B is also true.

So, Say that A implies B, written:

A -> B

The contrapositive of this statement is:

Not-B -> Not-A

Remember "A implies B" means that B must be true if A is true, so if we know that B is falce, we can deduce that A couldn't be true, so it must be falce.

With truth tables it can easily be shown that

"A -> B" IF AND ONLY IF "Not-B -> Not-A"

So when using the contrapositive, no information is lost.

In math, this is often used in proofs when, while trying to demonstrate that A implies B, it is easier to show that Not-B implies Not-A and hence that A implies B.