# How to prove/disprove distribution of division over union

How can I prove distribution of division over union in relational algebra, i.e. the following:

$$(R \cup S) / X \overset{?}{=} (R / X) \cup (S / X)$$

• Maybe a proof my induction would do? – Tarik Feb 27 '18 at 2:41
• What have you tried? Where did you get stuck? We do not want to just hand you the solution; we want you to gain understanding. However, as it is we do not know what your underlying problem is, so we can not begin to help. See here for tips on asking questions about exercise problems. – Discrete lizard Feb 27 '18 at 7:37

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