It is easy to show that number of symmetric relations on a set $S$, with $~|S|=n~$ is $2^{\frac{n(n+1)}{2}}$ and (little bit tougher) that, number of anti-symmetric relations is $2^n3^{\binom{n}{2}}$( here it was asked). But, how many relations are their which are both anti-symmetric and symmetric?
I have tried it, but no idea how to proceed. It's little bit confusing. I have searched for it in this site, this problem was not asked before. Any help will be appriciated.
A relation on $S$ that is both symmetric and anti-symmetric is just a subset of $I_S = \{(s, s)\ :\ s \in S\}$. And any subset is a valid example. So the answer is $2^n$.