set notation format: quick question

[renegade05]

Just a brain fart.... I am working on a proof and need some formatting help.

I have the following: \(\displaystyle \frac{-y}{x^2+y^2}\) Now it's initially given that \(\displaystyle y≠0\)

How can I write that \(\displaystyle x^2+y^2 > 0\) for all x and y, where again \(\displaystyle y≠0\) in set notation?

Something like \(\displaystyle \{x,y: x^2+y^2 > 0, x,y \epsilon \mathbb{R}, y≠0\}\) ??

I don't know... it doesn't look right. how can i make it correct?

Also, how can I write formally for this math proof that since \(\displaystyle x^2+y^2 > 0\) for all x and y, where again \(\displaystyle y≠0\) Then lets say the variable \(\displaystyle z=\frac{-y}{x^2+y^2}\) only depends on the value of y on whether z is positive or negative. Or essentially \(\displaystyle z=-y\) when all we are considered about is the sign. I hope this makes sense. I can I state this elegantly?

 

[daon2]

if \(\displaystyle y\neq 0\) then it is already true that \(\displaystyle x^2+y^2 >0\) for all \(\displaystyle x\). Why would you need a set to describe a true statement? The set that you have written down is equal to the set of all real numbers, or if you meant all points \(\displaystyle (x,y)\) such that \(\displaystyle x^2+y^2>0\) with \(\displaystyle y\neq 0\), then it is all of \(\displaystyle \mathbb{R}^2\) minus the x-axis..

For your second question, I'm not too sure what you mean.

 

[renegade05]

I guess I mean is there a formal way of writing z only depends on y when it comes to z being either positive or negative. Or is this self evident ?

If I leave the proof at: \(\displaystyle z=\frac{-y}{x^2+y^2}\) is it complete, if I am trying to proof that z only depends on y whether it is positive and negative. And the term \(\displaystyle x^2+y^2\) doesnt affect the sign? or do i need to write something further?