Showcasing $V(I \cap J)=V(I) \cup V(J)$ in Polynomials

[evinda]

Hello! (Wave)

I want to show that if $I,J$ ideals of $K[x_1, x_2, \dots , x_n]$, then $V(I \cap J)=V(I) \cup V(J)$.

Do I have to show that $ V(IJ)=V(I\cap J)$ and then $V(IJ)=V(I)\cup V(J)$? (Thinking)

If so, that's what I have tried:

$x \in V(IJ) \leftrightarrow (f_i \cdot g_j)(x)=0$, where $f_i \in I$ and $g_j \in J$

$\leftrightarrow f_i(x) \cdot g_j(x)=0 \leftrightarrow f_i(x)=0 \text{ OR } g_j(x)=0 \leftrightarrow x \in V(I) \text{ OR } x \in V(J)$ $\leftrightarrow x \in V(I)\cup V(J)$

So, $V(IJ)= V(I)\cup V(J)$.

Is it right so far? Or have I done something wrong? (Thinking)

How could I continue?

 

[evinda]

Is there also an other way, to show that $V(I \cap J)=V(I) \cup V(J)$? (Thinking)