Showing that an ideal is not generated by two elements

Question

Let $X=V(x_1, x_2)$ and $Y=V(x_3, x_4)$ be affine varieties on $\Bbb C^4$ where $\Bbb C$ is the complex number. Then, I have to show that the ideal $I(X ∪ Y)$ cannot be generated by two elements. I think I need to find some contradiction... Could anyone show me how to give a proof?

Answer 1

HINT: Monomial ideals, leading terms ( Groebner bases etc)

If $P_k$ generate a monomial ideal $I$ then the leading terms of $P_k$ will also generate $I$. But the minimal number of monomials generating $I$ is $4$.