Finding an example to disprove a quantified statement Find an example of $P(x,y)$ where $\forall x \exists y P(x,y)$ is false ($x \in \mathbb{R})$.
My interpretation of this statement is that for all $x$ in $\mathbb{R}$, there must be a $y$ where $P(x,y)$ is true. One example of $P(x,y)$ where the quantified statement would be false is:
$$
P(x,y) \equiv  \frac{y}{x} = 1
$$
This would make the above statement false because if $x$ was $0$, then $\frac{y}{0}$ would be undefined, right? At this point, I don't see a value of $y$ that would make $\frac{y}{x}$ evaluate to $1$.

Your $P$ is not defined for every possible pair $(x, y)$, therefore it is not a statement.
However you could use
$P(x, y) :\Leftrightarrow xy = 1$,
in which case $\forall x \in \mathbb{R} : \exists y \in \mathbb{R} : P(x,y)$ is false, since $x = 0$ is a counterexample.
