Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. a) ∀x∃y(x2 = y)

The statement is true. If x and y are both real numbers, the statement is:

"for every x, there exist y such that "

This is true, because you can pick any real number, square it, and obtain another real number, y. The relation is not surjective, i.e. we will not use all possible values for y, but it doesn't matter. The statement is only asking to find a value for x^2, which we can always do.