Is there a y that does not exist for all x such that y^2 = x?

[1MileCrash]

Homework Statement

Give a useful negation for:

\forall x > 0, \exists y > 0 s.t. y^2 = x

Homework Equations

The Attempt at a Solution

I'm not sure how to do this, I have

\exists y > 0 s.t. \forall x > 0, y^2 = x

Which says "there exists a y that for all x, y^2 = x"...which is obviously incorrect, as there is no y that when squared equals EVERYTHING.

 

[ArcanaNoir]

How about, "there exists an x greater than zero such that for all y greater than zero, y squared is not equal to x"

 

[SammyS]

If you were looking for an example to counter this (a counter example), what would need to be true for that example ? ... and, wouldn't it take just one example (at least as regards x)?

 

[1MileCrash]

If you were looking for an example to counter this (a counter example), what would need to be true for that example ? ... and, wouldn't it take just one example (at least as regards x)?

What do you mean counter it? I can't counter the first statement, it's true.

 

[ArcanaNoir]

If you were looking for an example to counter this (a counter example), what would need to be true for that example ? ... and, wouldn't it take just one example (at least as regards x)?

This supports my suggestion.

 

[ArcanaNoir]

A negation of "for all x, this is true" is "there exists an x such that this is not true." You say "there exists an x such that for all (for any arbitrary) y this is not true" because it has to be not true for all y. if it's only not true for some y, then there exists a y such that it is true, and you've lost it.

 

[SammyS]

What do you mean counter it? I can't counter the first statement, it's true.

Well yes, it is true for x & y being real numbers. That doesn't mean that you can't coming up with criteria that would need to hold for a counter-example, if such existed.

 

[1MileCrash]

Okay, I get it! So what you're saying, is that if there WERE a counter example, it would satisfy Arcana's negation statement, correct?

 

[SammyS]

That's what I'm saying if indeed, Arcana's negation statement is correct.