I really cannot seem to get my head around the definition of cardinal exponentiation with regards to finite sets:
$|X|^{|Y|}=|X^Y|$
How would one even begin to prove this? Isn't $X^Y$ the set of all functions with the domain $Y$ and the codomain $X$?
First, when either $X$ or $Y$ is infinite, this is just a definition of the left side, and there is nothing to be proven. The only thing to be proven is that when $X$ and $Y$ are both finite, then $|X|^{|Y|}$ (which is $m^n$ for some nonnegative integers $m,n$) is still equal to $|X^Y|$. This actually follows from a simple combinatorial argument. To assemble $f \in X^Y$, you independently choose one of the $|X|$ members of $X$ to be the value of $f(y)$ for each $y \in Y$. In other words you have to choose a member of $X$ a total of $|Y|$ times, and the order of the choices matters. Since $Y$ has $|Y|$ elements, the number of ways to do this is $|X|^{|Y|}$. (Note that this still works when $Y=\emptyset$, since $A^\emptyset=\{ \emptyset \}$ for every $A$. This means that when $X=Y=\emptyset$ you are forced into the convention $0^0=1$.)
