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