Suppose I have a set A and I want to consider all the functions $f:x \rightarrow A$ for $x \in A$. How do I define the cardinality of the set of such functions?
I can't get my head around the cardinal arithmetic on this; even a hint would be greatly appreciated.
For a given $x \in A$, the number of such functions is $|A|^{|x|}$. If you want the number of such functions for any $x$, simply sum $|A|^{|x|}$ over $x \in A$. Without knowing more about $A$ we cannot simplify further.