Quantifiers in Set Builder Statement Can the following sets be different because of change of quantifiers?
$$ 
\begin{aligned}
A &= \{ x : \exists{y}\ni P(x, y) \} \\
B &= \{ x : \forall{y}\,P(x, y) \}
\end{aligned}
$$
For example, the definition of the domain of a graph $ G $ is $ \{ x: \exists{y}\ni (x, y) \in G \} $. What's the difference if we say $ \{ x: \forall{y}\ni (x, y) \in G \} $?

$ \{ x \: | \: \exists{y} \: (x, y) \in G \} $ means all $x$ such that there is some $y$ such that $(x,y) \in G$ ... so that is indeed the domain of the graph.
$ \{ x\: | \: \forall {y} \: (x, y) \in G \} $ would be all $x$ such that for all $y$ it is true that $(x,y) \in G$ ... so that would be the points $x$ that are connected to all points $y$ ... which is different.
Example:
Suppose $ G$ is defined on $\{1,2,3\}$, and let's say $G = \{(1,1),(1,2),(1,3),(2,1)\}$.  Then $ \{ x \: | \: \exists {y} \: (x, y) \in G \} $ would be $\{1,2\}$, since both $ 1$ and $2$ map to some point (but $3$ does not), but $ \{ x\:|\: \forall {y} \: (x, y) \in G \} = \{1\}$, since only $1$ is connected to all points.
