How to prove/disprove distribution of division over union
Ok so i figured it out. Let $L_1$, $L_2$ be relation sets where $L_2$ is a subset of $L_1$. We know $R$ and $S$ are subsets of $L_1$ and $X$ is a subset of $L_2$. Since we do not know whether $X$ is a subset of $R$ and $S$, we can give the following counter example:

Suppose 
$$\begin{align*}
  L_1 &amp;=(1,2,3,4,5,6,7,8), \\
  R &amp;= (1,2,3,4), \\
  S &amp;= (5,6,7)\; \text{and}\\ X &amp;= (1,3).
\end{align*}$$
Then, $(R \cup S) / X = (2,4,5,6,7)$ whereas $R / X$ and $S / X$ are illegal.