Let P (x, y) be a propositional function if ꓯyꓱxP (x, y) is true does it necessarily follow that ꓱxꓯyP (x, y) is true? Justify your answer or give a counter-example

It is NOT TRUE

Step-by-step explanation:

ꓯyꓱxP (x, y)

means that for each value of y there exist x such that P (x, y) is true

whereas

ꓱxꓯyP (x, y)

means that there exists x such that for each value y, P (x, y) is true.

In the second case, the same x must work for every element y.

Counter-example

Consider

P (x, y) the following proposition

x-y = 0, for x, y integers.

Given an integer y, there is another integer x (namely, x=-y) such that

x-y = 0

so, ꓯyꓱxP (x, y) is TRUE

If ꓱxꓯyP (x, y) were TRUE, then would exist a single unique value of x such that P (x, y) is TRUE for every integer y.

Then P (x, 1) and P (x, 2) would be both TRUE and

x-1 = 0

x-2 = 0

and we conclude 1=2, which is a contradiction.

So ꓱxꓯyP (x, y) is NOT TRUE (FALSE)