Proof of a function composition on a power set

Question

$X$ is power set of $A$ minus $\emptyset$ : $X=P(A)\setminus \emptyset$ (and $A$ isn't empty).

There are two relations from the $X$: $R = \{(U,V)\vert U ⊂ V\}$ and $S = \{(U,V) \vert U \cap V \ne \emptyset\}$.

I need to prove that $S = R \circ R^{-1}$.

I don't even know how to begin.

Answer 1

Here's a start for you:

To see that $S = R \circ R^{-1}$, we first need to know $R^{-1}$. By definition, $$R^{-1} = \{(U,V) ~|~ U \supset V\}.$$ Now that we know this, consider $R \circ R^{-1}$. This is the following relation, $$R \circ R^{-1} = \{(U,V) ~|~ \exists W \text{ such that } (U,W) \in R^{-1}, \text{ and } (W,V) \in R\}.$$ But $(U,W) \in R^{-1}$ means that $W \subset U$, and $(W,V) \in R$ means that $W \subset V$. So what can you conclude from this?

Answer 2

If a set $V$ (which must be non empty by definition of $X$) is subset of two sets $U$ and $W$, then $U\cap W$ is not empty, because $V\subset U\cap W$.