Determine the truth value and state why.

For all x, there exists a y: x=y²

hmmm?

If you have worked with me, then you know what I'm going to say. Let's rewrite this to help our visual memory. Since most of us are accustomed to seeing the "y" on the left-hand side of the equations, let's write this:

from: For all x, there exists a y: x=y²

to: For all x, there exists a y: y²=x

Solution:

y²=x

sqrt(y²) = sqrt(x)

y = sqrt(x).

For this to be true, then x must be only non-negative integers. Therefore, our original statement is false. You can also let x=2 and clearly show that the statement is false.