# Fulton example section $2.4$, help needed in clarification

This is an example in Fulton, Algebraic Curves.

Let $$V= V(xw-yz) \subset \mathbb{A^{4}(k)}$$, and $$\Gamma(V)$$, the co-ordinate ring of $$V$$ is $$k[x,y,z,w]/ (xw-yz)$$.

and, $$\bar{x}, \bar{y}, \bar{z}, \bar{w}$$ denote residues of $$x,y,z,w$$ in $$\Gamma(V)$$.

Then $$\frac{\bar{x}}{\bar{y}}= \frac{\bar{z}}{\bar{w}} =f \in k(V)$$, (the field of rational functions on $$V$$) is defined at $$P=(x,y,z,w) \in V$$ if $$y,w, \neq 0$$

My question- I think points $$P$$ where f is defined should be where $$\bar{z}, \bar{y} \neq \bar{0}$$, the points $$(x,y,z,w)$$ where $$y,z$$ are not in $$I(V)= (xw-yz)$$. How come the book says otherwise? I'm confused, can someone help?

Thanks.