Union-closed family

A set family $\mathcal{A}$ is union-closed when $A\cup B\in \mathcal{A}$ for every $A,B\in \mathcal{A}$.

The union-closed sets conjecture states that every finite nontrivial union-closed family has an element contained in at least half of its members.

For $x,y\in [0,1]$, the binary entropy and the golden ratio $\phi$ satisfy

Every finite union-closed family other than $\{\varnothing \}$ has an element belonging to at leastof its members. The proof applies the binary entropy product inequality coordinate by coordinate to the union of two independent uniform members of the family.