I don't quite get the definition yet. How do I show that a relation $R : A \mapsto B$ is a partial function iff $R \circ R^{-1} \subseteq id_B$?
I'm getting multiple definitions of partial definitions. one says that it's only a partial relation if for all $a \in A$ and $b_1, b_2 \in B$, we have $aRb_1 \land aRb_2 \implies b1=b2$. But I can't see how this definition will be useful here.
I think it's easier to work with the contrapositives here.
If $R$ is not a partial function, then there is some $a \in A$ and some $b_1 \ne b_2$ in $B$ such that $a\,R\,b_1$ and $a\,R\,b_2$. Use the definition of $R \circ R^{-1}$ to deduce that $b_1\ (R \circ R^{-1})\ b_2$, and hence $R \circ R^{-1} \not\subseteq \mathrm{id}_B$.
Conversely, if $R \circ R^{-1} \nsubseteq \mathrm{id}_B$ then there will be some $b_1 \ne b_2$ in $B$ such that $b_1\, (R \circ R^{-1})\, b_2$. Unwind the definition of $R \circ R^{-1}$ to deduce that there is some $a \in A$ such that $a\, R\, b_1$ and $a\, R\, b_2$, meaning that $R$ is not a partial function.