# Meaning of $A^B$ where $A$ and $B$ are sets [duplicate]

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?

• You're exactly right. It's the set of maps from $B$ to $A$. To get why this is good notation, convince yourself that $A^n$ is naturally the set of maps from $\{1,2,...n\}$ to $A$. – WoolierThanThou Dec 18 '19 at 10:09
Yes, $$A^B=\{\text{functions from B to A}\}$$. With this notation,$$\left\lvert A^B\right\rvert=\lvert A\rvert^{\lvert B\rvert}.$$
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 $$C$$.