# Math Help - unlabeled subsets

1. ## unlabeled subsets

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

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

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

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

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

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