User:Jshflynn/Intersection of Linguistic Structures
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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.
$\blacksquare$