# Help negate this statement

1. Aug 31, 2011

### 1MileCrash

1. The problem statement, all variables and given/known data

Give a useful negation for:

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

2. Relevant equations

3. 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.

2. Aug 31, 2011

### 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"

3. Aug 31, 2011

### SammyS

Staff Emeritus
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)?

4. Aug 31, 2011

### 1MileCrash

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

5. Aug 31, 2011

### ArcanaNoir

This supports my suggestion.

6. Aug 31, 2011

### 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.

7. Aug 31, 2011

### SammyS

Staff Emeritus
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.

8. Aug 31, 2011

### 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?

9. Aug 31, 2011

### SammyS

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