$x,y \in \mathbb{N}$
$\exists x \exists y (x + y = 0) \lor (x * y = 0)$
In this propositional function, is it true that the $x$ and $y$ in $(x * y = 0)$ are free variables?
If so, am I allowed to assign a random value to the $x$ and $y$ so that I can get the truth value of the propositional function?
edit: additional information
I've been taught that $\exists x \exists y [P(x,y) \lor Q(x,y)]$ is different from $\exists x \exists y P(x,y) \lor Q(x,y)$ and that the existential quantifier is distributable in $\exists x \exists y [P(x,y) \lor Q(x,y)]$
We can define the function $FV$, the set of free variables in a formula, on the set of formulae in our language recursively (any text should have a precise definition). If $\theta$ is an atomic formula, then $FV(\theta)$ is defined as the variables in the formula (as there are no quantifiers). $(x*y=0)$ is an atomic formula, so $FV((x*y=0))=\{x,y\}$
The truth of a formula which contains free variables depends on the value of the variables.