Are $\forall x\exists yP(x,y)$ and $\forall x\exists yP(y,x)$ equivalent?

Question

1 . For all x, there exists y such that P(x,y) where P is predicate.

2. For all x, there exists y such that P(y,x)

Are both these statements equivalent?

Answer 1

Let us take the domain of both $x$ and $y$ to be the set of all persons.

Let $P(x, y)$ denote "x loves y". So $$\forall x \exists y(P(x, y))$$ means "Everyone loves someone."

However, $P(y, x)$ denotes "y loves x" or, "x is loved by y": $$\forall x \exists y(P(y, x))$$ means that "everyone is loved by someone."

Answer 2

Consider $$P(x,y) \ \ :\Longleftrightarrow \ \ \ x>y$$ in the universe $\mathbf N=\{1,2,3, \dotsc \}$.