If xyP x y is true does it necessarily follow that xyP x y i

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

Solution

It is not necessarily true.

P(x,y) is \'xy= 1\' , the universe of discourse for x is the set of positive integers, and the universe of discourse for y is the set of real numbers.

xyP (x, y) means:

For every positive integer x there is a real number y such that xy= 1. This proposition is true.

For x = 5(+ve integer) , y = 1/5 (real number)

xyP (x, y) means:

There exists a real number y such that, for every positive integer x ,xy= 1. This proposition is false.

For y = 5(real no.) , x = 1/5 (which is not a +ve integer)

Hence, If xyP (x, y) is true, it is not necessary that xyP (x, y) is true.