Determining which formula descibes an English sentence in FOL

Question

Let the natural language sentence be: "For every person, there is a food that the person likes".

Define P(x) = "$\underbrace{\_\_\_}_{x}$ is a person". F(x) = "$\underbrace{\_\_\_}_{x}$ is a food". L(x, y) = "$\underbrace{\_\_\_}_{x}$ likes $\underbrace{\_\_\_}_{y}$".

Which one of these formulas represent the English sentence in FOL?

1.) $\forall x. \exists y. P(x) \implies (F(y) \land L(x, y))$

2.) $\forall x. \exists y. (P(x) \land F(y)) \implies L(x,y)$

Answer 1

The first one.

Following natural language, we have to write: $\forall x \ (Px \to \exists y \ (Fy \land L(x,y))$.

Using Prenex transformation, due to the fact that $y$ is not free in $Px$, we get the equivalent:

$\forall x \ \exists y \ (Px \to (Fy \land L(x,y))$.