# Complexity of generating power sets

Suppose I have two sets $$A$$ and $$B$$ containing integers. Let $$B'$$ be the power set of $$B$$. Then suppose I have an algorithm that enumerates all possible pairings of elements in $$A$$ and $$B'$$ to apply a function $$f$$ to them and check something.

I understand that generating a power set is $$O(2^n)$$, but what complexity class does this belong to, and how do I prove it?

Complexity time $$O(2^n)$$ is called $$\text{DTIME}(O(2^n))$$ and is a sub-complexity class of E which is in turn a sub-complexity class of EXPTIME.
To prove you can do it within $$O(2^n)$$ time, you have to provide an algorithm for that and show the algorithm runs in no more than $$O(2^n)$$.