User:Jshflynn/Language Product Distributes over Union

Theorem
Let $\Sigma$ be an alphabet.

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

Then $V(W \cup Y) = (VW) \cup (VY)$

Proof
\begin{align*} \,V(W \cup Y) &= \{x \circ y: x \in V \land y \in W \cup Y\} \\ &\text{By the definition of set union:} \\ &= \{x \circ y: x \in V \land (y \in W \lor y \in Y)\} \\ &\text{Conjunction is distributive over disjunction:} \\ &= \{x \circ y: (x \in V \land y \in W) \lor (x \in V \land y \in Y)\} \\ &\text{By the definition of language product and set union:} \\ &= (VW) \cup (VY) \end{align*} Hence the result. $\blacksquare$