TITLE: A strengthening of Frankl's union-closed conjecture?
QUESTION [12 upvotes]: Frankl's union-closed conjecture states that if $F$ is a finite union-closed family of sets (i.e. a family that is closed under taking unions), then there must be an element that belongs to at least half the sets.
1) Does anyone know an example of a finite union-closed family $F$ such that the set $\mathcal{A}(F)$ of elements that belong to at least half the sets of $F$ is not a member of $F$? [edit: this is answered below by Thomas Bloom]
2) Does anyone know an example where no member of $F$ is a subset of $\mathcal{A}(F)$?

REPLY [8 votes]: As for question 2), I just found the following example:
Let $\mathcal{P}$ be the family of all non-empty subsets of $\{1,2,3,4\}$, and $\mathcal{F}$ be the family whose elements are all the sets $P\cup\{5,6,7,8,9\}$ with $P\in\mathcal{P}$ and 
the 4 sets
$$\{1,5,6,7,8\},\{2,5,6,7,9\},\{3,5,6,8,9\},\{4,5,7,8,9\}.$$
This union-closed family has 19 members. 1,2,3 and 4 belong to 9 sets, 6,7,8,9 to 18 sets and 5 belongs to 19 sets. Thus $\mathcal{A}(F)=\{5,6,7,8,9\}$ and no member of the family is a subset of $\mathcal{A}(F)$.