FOL formula and check valididty of this?

Question

I think the following is logically valid, but my TA says it's not logically valid.

$ \forall x (A(x) \to B(x)) \to ( \exists x A(x) \vee \exists x B(x)) $

Who Can Clarify me about this Formula ?

Answer 1

Your formula is false in the structure whose carrier set is $\{42\}$ and the interpretation of $A$ and $B$ are both $\varnothing$.

Since there is a structure where the formula is false, it is not logically valid.

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If you looks at the possible models of your formula, with interpretaions $\hat A$, $\hat B$ of the predicates $A$ and $B$, the part $\forall x(A(x)\to B(x))$ says that $\hat A\subseteq \hat B$. And $\exists x A(x)\lor \exists x B(x)$ says that eat least one of $\hat A$ and $\hat B$ is non-empty.

However just because $\hat A$ is a subset of $\hat B$ doesn't mean the sets can't be empty.