The First Distributive Law

There are two distributive laws of sets. The first law shows how the union operation distributes over the intersection operation. The second law shows us how the intersection operation distributes over the union operation. For the arbitrary sets$A$, $B$, and $C$, we can write these laws as follows:

Let’s analyze the first distributive law and see the reason why it holds.

Distribution of the union over an intersection

We can take an arbitrary element belonging to the set on the left-hand side and show its membership in the set on the right-hand side and vice versa. First, we show that $A\cup (B\cap C)\subseteq (A\cup B)\cap$ ...