Question 93
Let $S$ be a set with $n$ elements. Take a positive integer $k$. Let $A_1, A_2, \dots, A_k$ be any distinct subsets of $S$. For each $i$ take $B_i = A_i$ or $B_i = S - A_i$. Find the smallest $k$ such that we can always choose $B_i$ so that $\bigcup_{1 \le i \le k} B_i = S$.