If and only if

The phrase if and only if, often abbreviated as iff, is common in the language of mathematics.

Normally, when we say something like “$\bc{B}$ if $\rc{A}$” or “if we know that $\rc{A}$ is true then $\bc{B}$ is also true”, we are only allowed to make a deduction in one direction. For instance, if somebody tells us the following:

If somebody is a professional basketball player then they are always tall.

And we hear about somebody who is a professional basketball player, then we may conclude that they must be tall.

However, if we hear about somebody who is tall, it would be a mistake to conclude that they are a professional basketball player. There are plenty of tall people who don’t even play basketball at all.

This kind of one-way rule is called an implication. As in, playing professional basketball implies that you’re tall, but being tall does not imply that you play professional basketball.

For this reason implications are written symbolically as a one-way arrow:

We use if and only if when we have implications in both directions: A implies B and B implies A. For instance

A number is even if and only if if is divisible by two.

In this case we can make implications in both directions:

• If 8 is even, it must be divisible by two.
• If 13 is not divisible by two, it must not be even.

A relation of this type is called an equivalence, and it’s written symbolically with a double arrow:

We call it an equivalence, because if this holds, then the set of things for which $\rc{A}$ is true must be exactly the same as the set of things for which $\bc{B}$ is true. That is, the two statements are equivalent.