User:Jshflynn/Intersection of Linguistic Structures

Theorem
Let $(V, \circ)$ and $(W, \circ)$ be linguistic structures.

Then $(V \cap W, \circ)$ is a linguistic structure.

Proof
Let $x, y \in V \cap W$. As $x, y \in V$ we have $x \circ y \in V$.

Also $x, y \in W$ so $x \circ y \in W$.

By definition of union:


 * $x \circ y \in V \land x \circ y \in W \implies x \circ y \in V \cap W$

So:


 * $\forall x, y \in V \cap W: x \circ y \in V \cap W$

Hence $(V \cap W, \circ)$ is a linguistic structure.