In particular, if R and S are sets of polynomials over a field, then the sets of points where polynomials of R and S are simultaneously zero are Z(R) and Z(S), respectively.
Then:
Z(R) $\cap$ Z(S) = Z(R $\cup$ S)
And
Z(I) $\cup$ Z(J) = Z(IJ) (where I and J are ideals generated by R and S, resp.)
It seems almost obvious that the intersection or union of two affine algebraic sets would still be an affine algebraic set, however I'm not sure how to go about proving the specific equality. I'm sure it's something basic and I'm just overthinking it.