Question

How to get a counterexample to this quantified statement where the universe for all variables consists of real numbers?

Answer 1

∀x∃y(y^2=x)

Answer 2

Is it saying that for all x there exist a y that makes the statement true?

Answer 3

wouldnt the counterexample be like if x is negative there can't be a y to equal x since it is squared?

Answer 4

Yes. This statement is not true. For example if x = -1, then there is no real number y such that y^2 = -1.

Answer 5

That would be it.

Answer 6

Okay thank you! new at this stuff so I get confused a lot

Answer 7

For the record, the logical negation of your statement is this:
\[NOT \ (\forall x \ \exists y\ (y^2=x)) \equiv \exists x \ \forall y \ (\ NOT(y^2 =x)) \equiv \exists x \ \forall y \ (y^2 \neq x)\]
The general rule is to turn the "for all" into "there exists" and vice versa, and the negate any other statements. That's what we've done here.

Answer 8

"...and TO negate any other statements"