User:Jshflynn/Language Product Distributes over Union

Theorem
Let $\Sigma$ be an alphabet.

Let $V, W$ and $Y$ be formal languages over $\Sigma$.

Let $\circ$ denote concatenation and $\circ_L$ denote language product.

Then $\circ_L$ is distributive over $\circ$. That is,


 * $V \circ_L \left({W \cup Y}\right) = \left({V \circ_L W} \right) \cup \left({V \circ_L Y}\right)$

And


 * $\left({W \cup Y}\right) \circ_L V = \left({W \circ_L V}\right) \cup \left({Y \circ_L V}\right)$

Proof
Hence language product is left distributive over union.

The proof that language product is right distributive follows similarly.