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 $\mathcal{A}(F)$?



A strengthening of Frankl's union-closed conjecture?
The simplest example I found is the family


$$ \emptyset, \{1\}, \{1,2,3\}, \{1,2,4\}, \{1,2,3,4\} $$


This is a union-closed family with 5 elements. 1 appears in 4, 2 appears in 3, 3 and 4 both appear in 2, but $\{1,2\}$ is not in the family.


