Find a logical predicate for some given conditions

Question

I am trying to give an example of a predicate $P(x, y)$ such that $\exists x$ $\forall y$ $P(x, y)$ and $\forall y$ $\exists x$ $P(x, y)$ have different truth values. I am struggling to think of such a predicate. Could I have a hint as to how to approach this problem?

Answer 1

How about $x=y$?

About your comment: If $\exists x\forall y(x=y)$, then your universe contains just one element. If your universe contains just one element, then every $P(x,y)$ is either true or false for all $x$ and $y$, and $\forall y\exists x \,P(x,y)\iff \exists x \forall y\, P(x,y)$.

Answer 2

Hint. The formula $\exists x \,\forall y. P(x,y)$ wants a universal $x$, which works for all $y$, regardless of $y$'s nature. Where is in $\forall y\, \exists x. P(x,y)$ the $y$ may depend on the particular $x$ given. Now think of some problem, which depends on $y$, where the solution $x$ exists for every $y$, but differs.