Do all the solutions have to be in an affine variety?

Question

An affine variety $V(X)$ is the zero-locus of a set of polynomials. So if the variety is generated by the polynomial $y-x=0$ in $\mathbb{R}^2$, then do all the solutions (i.e., every point satisfying $y=x$) have to be in $V(X)$, or could certain points, such as $(0,0)$ or $(1,1)$ be omitted?

Answer 1

Yes, by definition, $V(X)$ means all the solutions. It's perfectly ok to talk about open subsets of this set, e.g. $V(X)$ minus the origin --- the fancy name for such things is quasi-affine varieties --- but you can't give those the same name!