If Forall x Reverse Element yPx y is true does it necessaril

If Forall x Reverse Element yP(x, y) is true, does it necessarily follow that Reverse Element x Forall yP(x, y) is true? If it is, prove it. Otherwise, give a counterexample that justifies your answer.

Solution

False.

Let, P(x,y) be :x^2+y=0

So for all real x there exist some y so that P(x,y)=0

and y=-x^2

But if y>0

x^2+y cannot be 0 for any x because x^2>=0 for all real x

Hence, second statement does not follow