Pagh's problem

We are given as inputs ${\displaystyle k}$ subsets ${\displaystyle X_{1},X_{2},\dots ,X_{k}}$ over a universe ${\displaystyle U=\{1,\dots ,k\}}$.

We must accept updates of the following kind: Given a pointer to two subsets ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$, create a new subset ${\displaystyle X_{1}\cap X_{2}}$.

After each update, we must output whether the new subset is empty or not.