I have a system of inequalities $$|z-a_k|\le R_k$$ where $z=x+iy$ (complex number) and $a_k$ and $R_k$ are real numbers for $k=1, \dots, n$. Basically the inequality above shows circle with center $a_k$ and radius $R_k$. The question here is, if I write $n$ inequalities as a system of inequalities and then solve this system, the solution will be the intersection of $n$ inequalities. But I want to find the union of $n$ inequalities. Is there any way to do that?
The union of a bunch of sets is the complement of the intersection of the complements of the sets. So $|z-a_k|\gt R_k$ gives the complement of the disk, the system of such inequalities gives the intersection of the complements, then you want the complement of that.