Say I have two sets
$$A = \big\{ x\in \mathbb{R}^n : f(x)=0, g(x)=0 \big\} \, , \\ B = \big\{ x\in \mathbb{R}^n : h(x)=0, j(x)=0 \big\} \, , $$
described by some implicit equations. How can I describe the union $A \cup B$ in terms of implicit equations? I guess, just like the intersection $A \cap B$ is described by the union of the equations, the union must be described by the $intersection$ of the equations. But how do you actually compute this?
$A=V(f)\cap V(g)$ and $B=V(h)\cap V(j)$ where $V(f)=\{x:f(x)=0\}$ etc. Therefore \begin{align} A\cup B&=(V(f)\cap V(g))\cup(V(h)\cap V(j))\\ &=(V(f)\cup V(h))\cap (V(f)\cup V(j))\cap (V(g)\cup V(h))\cap (V(g)\cup V(j)). \end{align} But $V(f)\cup V(h)=V(fh)$ etc., so $$A\cup V(fh)\cap V(fj)\cap V(gh)\cap V(gj),$$ the locus of vanishing of the four functions $fh$, $fj$, $gh$, $gj$.