Union of definable sets

Question

I tried to prove this question but without a success: Let $K_1$ and $K_2$ be definable sets, prove that $K_1\cup K_2$ is definable.

What I tried to do is to assume: $K_1=Ass(X)=\{ v|v \vDash X \}$ and $K_2=Ass(Y)=\{ v|v \vDash Y \}$ and to say that $K_1\cup K_2=Ass(X \cap Y)$. but I couldn't prove that: $K_1\cup K_2 \supseteq Ass(X \cap Y)$ .

I would like to get help with this proof

Answer 1

HINT: Remember that $K$ is definable if there is some $\varphi_K(x)$ such that $m\in K\iff M\models\varphi_K(m)$. Now also remember that $x\in A\cup B$ if and only if $x\in A\lor x\in B$.

Answer 2

For sets of formulas $X, Y$, define $$X \vee Y=\{\varphi \vee \psi \;| \; \varphi \in X, \psi \in Y\}$$

Now, if $K_1=\mathrm{Ass}(X)$ and $K_2=\mathrm{Ass}(Y),$ it should be straightforward that $K_1 \cup K_2 = \mathrm{Ass}(X \vee Y)$ (try to convince yourself of this).