List all functions from $\displaystyle \{a,b\}$ to $\displaystyle \{1,2\}$.

For some reason I keep missing one!! I know that I should have $\displaystyle 2^2$ functions.

So I start by mapping a to 1 and 2, leaving b alone.

Then I map b to 1 and 2, leaving a alone.

Then I map a to 1, and b to 2.

But i'm missing one!

Is it a null function? One that doesn't map a and b to anything?