1. ## unlabeled subsets

1. How many unlabeled families of subsets of a $\displaystyle 3$-set are there?

It is just $\displaystyle 4$? Because you can have a subset with $\displaystyle 0,1,2$ or $\displaystyle 3$ elements?

2. Prove that the number $\displaystyle F(n)$ of unlabeled families of subsets of an $\displaystyle n$-set satisfies $\displaystyle \log_{2} F(n) = 2^n +O(n \log n)$.

So $\displaystyle F(n) = n+1$. Thus we want to show that $\displaystyle 2^{2^{n}+O(n \log n)} = n+1$. Now $\displaystyle O(n \log n)$ means that there is a function $\displaystyle g(n)$ such that $\displaystyle |g(n)| < c(n \log n)$ for some constant $\displaystyle c$. Maybe we can rewrite the LHS as $\displaystyle 2^{2^n} \cdot 2^{O(n \log n)}$?

2. actually these are families of subsets, so my answer is probably not correct.

3. I know that there are $\displaystyle 2^{2^3}$ labeled subsets. The power set $\displaystyle |\mathcal{P}(S)| = 2^n$ for example. Isn't this what is meant by families of subsets?