Algebraic Geometry Question - on ideals of algebraic sets

[slevvio]

Hello everyone, I was wondering if I could get some advice for the following problem:

I have two algebraic sets X, X', i.e. X = V(J), Y = V(J'), and let I(X),I(Y) be the ideals of these sets, i.e. I(X) ={x \in X | f(x) = 0 for all x \in X}. I am trying to show that I(X \cap Y) is not always equal to I(X) + I(Y), so I have tried many examples of ideals of \mathbb{C}[X] but I am not getting anywhere.

Any help would be appreciated!

Thanks

 

[mathwonk]

look in C[X,Y], and try a couple of sets that meet tangentially.