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:

\[\rc{A} \Rightarrow \bc{B}\p\]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*.