It can take some work to show that something is not an affine variety.
For example, consider the set
[tex: X = \{(x,x) : x \in \mathbb{R}, x \neq 1} \
It can take some work to show that something is not an affine variety.
For example, consider the set
[tex: X = \{(x,x) : x \in \mathbb{R}, x \neq 1} \