I know that set cover problem is NP-complete. We are given a universe $U$ (a set) of $n$ elements and a collection $S$ of $m$ sets whose union equals the universe and an integer $k$. The set cover problem is to identify a sub-collection of $S$ of size $k$ or less whose union equals the universe?
Now if I remove or less, and ask instead that is there a sub-collection of $S$ of size $k$ whose union equals the universe? Is this still NP-complete?
Here, of course, $k<m$.
My answer is that the problem is still NP-complete. From the initial problem we simply keep the same instance and we find a set cover of size $s\leqslant k$ then the modified problem has a set cover of size $k$ if $s=k$.