Let $\mathscr{A} $ be a set of sets. Let's denote $\{A \setminus B : A,B \in \mathscr{A}\}$ by $\mathscr{A} \setminus \mathscr{A} $.

The *Marica-Schönheim* theorem says that $|\mathscr{A} \setminus \mathscr{A}| \geq |\mathscr{A}|$ for every **finite** $\mathscr{A}$.

This immediately implies the result for countably-infinite $\mathscr{A}$, since if we had $|\mathscr{A} \setminus \mathscr{A}|=n $ is finite, then taking a subset of size $n+1$ out of $\mathscr{A}$ gives a contradiction.

There seems to be no natural one-to-one mapping $\mathscr{A} \to \mathscr{A} \setminus \mathscr{A} $, so this raises the question:

Do we have $|\mathscr{A} \setminus \mathscr{A}| \geq |\mathscr{A}|$ for families $\mathscr{A}$ of arbitrary cardinality?

I asked it on math.SE a while ago.