  A pair $(\mathcal{A},\mathcal{B})$ of families of subsets of an $n$-element
set is called cancellative if whenever $A,A'\in\mathcal{A}$ and
$B\in\mathcal{B}$ satisfy $A\cup B=A'\cup B$, then $A=A'$, and whenever
$A\in\mathcal{A}$ and $B,B'\in\mathcal{B}$ satisfy $A\cup B=A\cup B'$, then
$B=B'$. It is known that there exist cancellative pairs with
$|\mathcal{A}||\mathcal{B}|$ about $2.25^n$, whereas the best known upper bound
on this quantity is $2.3264^n$. In this paper we improve this upper bound to
$2.2682^n$. Our result also improves the best known upper bound for Simonyi's
sandglass conjecture for set systems.
