Showing that the set difference of two affine varieties need not be an affine variety

Question

I was wondering what would be an example of two affine varieties $A, B$ over the field of complex numbers $\mathbb{C}$ such that $A\setminus B$ is not an affine variety? I was initially thinking about $A = V(x^2 + y^2 - 1)$ and $B = V((x - 1)(y - 1))$, so that $A\setminus B = A\cap B^c = \{(z, w) \in \mathbb{C}^2\mid z^2 + w^2 = 1\land (z - 1)(w - 1) \neq 0\}$ with the reasoning that $(z, w) = (1, 0)$ and $(z, w) = (0, 1)$ would still be vanishing points for the circle. But then I realized that we are probably adding some polynomial to $x^2 + y^2 - 1$ such that it does not vanish at $(1, 0), (0, 1)$.