How do quantifiers apply to mark variables as free and bound? Given something like this $\forall xP(x,y) \implies (\exists y Q(y) \land Q(x))$ Wouldn't both x and y be both free and bound? My friend is telling me that x would be bound and y would be free but I am not really understanding his logic.
What am I missing?
 A: The terms "bound" and "free" apply to each occurrence of a variable, not globally across the entire formula. In your example, on the left side of the $\implies$, the $x$ is bound and the $y$ is free, while on the right side of the $\implies$, the $x$ is free and the $y$ is bound.
This is assuming the formula is read as $(\forall xP(x,y)) \implies (\exists y Q(y) \land Q(x))$, as the spacing seems to suggest.  If the $\forall x$ is meant to apply to the entire rest of the formula, then of course all occurrences of all $x$ are bound.
A: *

*$$\forall xP(x,y) \implies (\exists y Q(y) \land Q(x))$$ can be
equivalently written as $$\forall w P(w,y) \implies (\exists z Q(z)
\land Q(x)).$$ Now it's clearer that $x$ and $y$ are free variables
(can't be renamed), and that there are also two bound variables
(each of which can be renamed). Adopting a slightly different
notation makes the sentence syntactically even sweeter: $$\forall w
Pwy \implies (\exists z Qz \land Qx).$$

*Similarly, $$\forall x Px\land \forall x Qx\;\equiv\;\forall x
Px\land \forall y Qy.$$
