Can someone please explain to me these two concepts and how they differ from each other? I'm taking a class entitled Math & Computers which emphasizes algebraic geometry in a symbolic-computational setting. That being said, the class is not very oriented towards explaining and understanding these ideas as it is assumed that I've already had experience in dealing with them. Most specifically, I'm talking about affine varieties and vanishing ideals in the realm of polynomial rings. Is the vanishing ideal a subset of the affine variety?