How wrong is the following in predicate logic

Question

If $P(x,y)$ means that $x$ is perpendicular to $y$. And $Q(x,y)$ means that $x$ is parallel to $y$, how wrong is the following written in predicate logic:

Every $x$ is perpendicular to $y$. In predicate: "$\forall x, \forall y \in P(x,y)$"

There exists $y$ that is parallel to $x$. In predicate: "For $x$ there $\exists y\in Q(x,y)$"

I know that the upper aren't totally correct. My question is how wrong are they?

Answer 1

Every $x$ is perpendicular to $y.$
In predicate: "$\forall x, \forall y \in P(x,y)$"

Literally: $∀x\:P(x,y).$ Here, the variable $y$ appears to be free.

(On the other hand, “Every $x$ is perpendicular to every $y”$ translates as $∀x∀y\:P(x,y).\,)$

Your suggestion just reads as “For each $(x,y)$ couple for which $P$ is true...”.

Both $P(a,b)$ and $(a,b){\in}P$ mean that $(a,b)$ satisfies $P(x,y).$

There exists $y$ that is parallel to $x.$
In predicate: "For $x$ there $\exists y\in Q(x,y)$"

Literally: $∃y\:Q(y,x).$

Answer 2

It's VERY wrong. It doesn't parse:

$$P, Q \subseteq S^2$$ which means $P$ and $Q$ are sets of ordered pair, therefore, their elements are of the form: $(a,b)$ and not $a$.

You can do this instead:

$$\forall x, \exists y P(x,y)$$ which says every $x$ is perpendicular to some $y$ without using elementhood.