Given an integer $N$ and a set of integers in $[1; N]$. Find a minimal set of integer arithmetical progressions such as given set can be covered using operations $A \cap B$, $A \cup B$ and $\overline A$ and the coverage has no excess points in $(-\infty; N]$.
Am I right assuming that this problem can be reduced to the set cover problem? I couldn't come up with a strict proof for some reason yet.