actually these are families of subsets, so my answer is probably not correct.
1. How many unlabeled families of subsets of a -set are there?
It is just ? Because you can have a subset with or elements?
2. Prove that the number of unlabeled families of subsets of an -set satisfies .
So . Thus we want to show that . Now means that there is a function such that for some constant . Maybe we can rewrite the LHS as ?