 Solution

If the basis has $r\le k$ members, it has $2^r$ subsets, each specifying a possible union. Different subsets can produce the same union, so the number of distinct generated sets is at most $2^r$. Consequently,Here $\mathcal{F}$ is a set family of distinct sets. If a representation lists duplicate input sets, discard those duplicates before applying this bound. The empty union is included.