\(\sc{SET}\ \sc{COVERING}\ \) Given a finite set \(U\), a finite family \(\mathcal{S}\) of subsets of \(U\) and a natural number \(k\), does there exist a subfamily \(\mathcal{S}' \subseteq \mathcal{S}\) of cardinality \(\leq k\) such that \(\bigcup \mathcal{S}' = U\)?