Meaning of $A^B$ where $A$ and $B$ are sets

Question

I've tried to google this with a variety of search terms, but most results refer to the power set, which is $2^S$, not $A^B$.

Does it mean the set of all maps (functions) from the set $B$ to the set $A$? What is the definition of this notation?

Thank you for reading!

Answer 1

Yes, $A^B=\{\text{functions from $B$ to $A$}\}$. With this notation,$$\left\lvert A^B\right\rvert=\lvert A\rvert^{\lvert B\rvert}.$$

Answer 2

As others have said, yes, $A^B$ means the set of functions from $B$ to $A$. I would add that, with this notation, we also have the "exponential law" $$ (A^B)^C \cong A^{B \times C} $$ for any sets $A, B, C$, where $B \times C$ is the Cartesian product of $B$ and $C$.