I am new to predicate logic and am just learning about it; I encountered this problem on a quiz and was disconcerted by how I was supposed to answer it.
My answer for the quiz was: For every x, there exists a y where x points to y. I still don't know if I was correct.
I had a few confusions regarding this logic:
If x points to itself, would that be considered satisfactory to this logic? Can x point to multiple y values? Do all x's need to point to the same y value. Can a value for x also be a value for y, where it points to itself?
There was a question after this: $$\exists_y\forall_xA(x,y)$$
In this case, I answered: There is some y, where all x point to y. Which in my mind meant that x would only point to y.
So again, I was confused by my overthinking. Does y need to point to itself, or can y point to some other x?
$$\forall x \exists y A(x,y)$$ colloquially says that “each $x$ has a $y$ to point at.” That $y$ could be $x$. It also says nothing about each $x$ pointing to the same $y$; they could be different.
Conversely, $$\exists y \forall x A(x,y)$$ colloquially says that “some $y$ is pointed at by each $x$”. Now $y$ is included in the statement “by each $x$”… so yes! $y$ is pointing at itself in this case.
If we wanted each $x$ to only point at this $y$ we would have to quantify as follows: $$\exists y \forall x (A(x,y) \land \forall z (A(x,z) \rightarrow z=y))$$ can you see why?