What does $A^{B}$ mean?

Question

Assume, that A and B are finite sets.

What notion $$A^{B}$$ does mean?

Have been looking for awhile now.

Answer 1

That symbol usually denotes the set of all functions $B\to A$.

I suspect the notation is suggested by the fact that the size $|A^B|$ of this set is $|A|^{|B|}$.

Addendum: To see this, consider some simple examples.

Suppose $A$ consists of a single point, say $A=\{a\}$. Then the only function $f:B\to \{a\}$ is the constant function with $f(b)\equiv a$ for each $b\in B$. In this case, $|A^B|=1$, which gibes with $|A|^{|B|}=1^{|B|}=1$ for any finite set $B$.

If instead we suppose that $B$ consists of a single point, say $B=\{b\}$, then again the only functions $f:\{b\}\to A$ are constant functions; there is one such function for each choice of the image point $f(b)\equiv a\in A$ (this is almost "cheating" since every function whose domain is a single point is a constant function, but it is still true), and each distinct choice yields a distinct constant function. So in this case, we have $|A^B|=|A|$, which gibes with $|A|^{|B|}=|A|^1 = |A|$ for any finite set $A$.

I invite you to consider the next simplest case with $|A|=2$ or $|B|=2$.

The even simpler cases with $|A|=0$ or $|B|=0$ work as well, although you will have to think hard about what the set of functions $B\to A$ is if either $A=\varnothing$ or $B=\varnothing$.

Answer 2

For any sets, finite or not, $$A^B = \{ f : B \to A \}$$ the set of functions from $B$ to $A$. For example, in that sense $\mathbb R^2$ is the set of functions $f : 2 = \{ 0, 1\} \to \mathbb R$.