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.

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 $F$?

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 $\mathcal{A}(F)$?

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.

Does anyone know an example of a finite union-closed family $F$ such that the set of elements that belong to at least half the sets of $F$ is not a member of $F$?

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.

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 $F$?