Propetries of ideals and zero sets

Question

Why we obtain $I(X_1 \cup X_2) = I(X_1) \cap I(X_2)$ and $Z(J_1 \cup J_2)= Z(J_1) \cap Z(J_2)$?

• for a subset $X⊆\mathbb{A}^n,$ $I(X)$ is the ideal of $f∈k[x_1,⋯,x_n]$ with $f|X=0$ and
• for a subset $J⊆k[x_1,⋯,x_n],$ $Z(J)$ is the closed subset of $\mathbb{A}^n$ defined by $∩_{f∈J}Z(f).$

I would like to have example of it.

For example $X_1 = \{(1:0:0)\}$ is point. $X_2 =\{(x-y)\}$ is line.

Is it correct $X_1 \cup X_2 = \{(x-y)\}$ or no? How can I write: $I(X_1 \cup X_2)$ and $I(X_1) \cap I(X_2)$? And how we can get example $Z(J_1 \cup J_2)= Z(J_1) \cap Z(J_2)$?

Answer 1

$$f|(X_1\cup X_2)=0\iff f|X_1=0\text{ and }f|X_2=0.$$$$\cap_{f\in J_1\cup J_2}Z(f)=\bigcap_{f\in J_1}Z(f)\cap\bigcap_{f\in J_2}Z(f).$$