Let \(F\) be a family of nonempty subsets of \([n]=\{1,\dots,n\}\) such that no two disjoint subsets of \(F\) have the same union. In other words, for \(F =\{ A_1,A_2,\dots, A_k\},\) there exists no two sets \(I, J\subseteq [k]\) with \(I\cap J =\emptyset\) and \(\bigcup_{i\in I}A_i = \bigcup_{j\in J} A_j\). Determine the maximum possible size of \(F\).

For the new version of POW 2021-21, the best solution was submitted by 이재욱 (전기및전자공학부 2018학번, +4). Congratulations!

Here is the best solution of problem 2021-21.