Distance between relations

Question

Assume I have two binary relations $R_{1}$ and $R_{2}$ on a set $S$, i.e., two subsets of $S \times S$. I want to compare these two relations in terms of similarity. Is there any distance metric that I can apply? The two relations are both complete, reflexive, antisymmetric and transitive. Alternatively, we could assume that both relations are irreflexive, asymmetric, transitive, and $a \neq b \Longrightarrow aRb \lor bRa$.