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 in combinatorics 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?