Solution
= Solution
If the basis has $r\leq k$ 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,
$$
\boxed{|\mathcal F|\leq2^k.}
$$
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.