Prove or Disprove .

Question

$\vDash \forall x (\alpha \lor \beta) \implies ( \forall x \alpha \lor \forall x \beta) $.

I am not able to start this question, any help would be appreciated.

Thanks in advance!

Answer 1

If $\alpha(x)$ is true of some $x$ and false of others in some model, then taking $\beta(x) = \neg\alpha(x)$ gives a counterexample: a model in which the sentence is false.

For example, let $\alpha(x)$ be $x=1$ and $\beta(x) = \neg\alpha(x) = x\ne 1$. Of course it's true that $$ \forall x\,(x=1 \lor x\ne 1). $$ (The sentence is valid). But it's not valid that $$ \forall x\, x=1 \lor \forall x\, x\ne 1, $$ because in, say, the integers, or in any model with more than one element, not every $x$ is equal to $1$, and not every $x$ is unequal to $1$.

Answer 2

HINT: The statement Every integer is even or odd is an instance of $\forall x(\alpha\lor\beta)$.