Negating x^2 + y^2 > 0 for All x,y

[Caldus]

How do I write the negation of:

For all x > 0, x^2 + y^2 > 0 for all y.

I thought it might be this:

There exists x < or = to 0 such that x^2 + y^2 < or = to 0 for one y value.

Thanks.

 

[roch]

Originally posted by Caldus
How do I write the negation of:

For all x > 0, x^2 + y^2 > 0 for all y.

I thought it might be this:

There exists x < or = to 0 such that x^2 + y^2 < or = to 0 for one y value.

Thanks.

I think that was close but not exact, it is:

There exist one x > 0 such that x^2+y^2 < or = 0 for one y.

The thing is that there is no statement about x < 0. So that there must be no statement for x <0 in the negation.

******************

Maybe an better formulation (and equivalent) of the problem is:

How do I write the negation of:

For all x>0 and for all y, x^2 + y^2 > 0.

Result:

There exist one x>0 and there one y such that x^2 + y^2 <= 0.

*********************

I hope it did help...

 

[Caldus]

Whoops, I did that wrong. The actual statements are (for the problem, not the solution):

For every x >0, x^2 + y^2 > 0 for all y.

Close enough I guess?

 

[NateTG]

$$\forall x>0 \exists y>0 \] s.t. \[ x^2+y^2 \leq 0$$

 

[Caldus]

Originally posted by NateTG
$$\forall x>0 \exists y>0 \] s.t. \[ x^2+y^2 \leq 0$$

What is that in English? Thanks.

 

[HallsofIvy]

What is that in English? Thanks.

"For all x greater than 0, there exist a y> 0

such that $$x^2+ y^2\leq0$$"

(It is, by the way, false.)

 

[NateTG]

Right, but it is the negation of the statement he made.