How many inverse relations

Question

How many inverse relations are there for an n-element set?

I know that $R \circ R^{-1}=R^{-1} \circ R$ where $R$ is an invertible relation, but that's as far as I can get.

Answer 1

Every relation has an inverse relation, so this question is really asking how many relations there are on some ground set $A$. A relation is a subset of $A\times A$, which has size $|A\times A|=|A|\times |A|=n^2$. There are $$2^{n^2}$$ subsets of $A\times A$, and therefore this quantity of inverse relations.