Are there twice as many generalized partitions as partitions?

Question

A partition of a set $S$ is a subset of the powerset of $S$, which covers $S$, is pairwise disjoint, and does not contain the empty set. If we drop the last condition, we get what I call a generalized partition of the set $S$. It is basically just a partition which may possibly contain the empty set. I want to prove that for every finite set $F$, the cardinality of the set of generalized partitions of $F$ is twice the cardinality of the set of partitions of $F$. Is that statement even true?

Answer 1

Of course. For every true partition $P$ you have two generalized partitions: $P$ and $P\cup\{\emptyset\}$.