A note on equal unions in families of sets

A family of sets has the equal union property if and only if there exist two nonempty disjoint subfamilies having the same union. We prove that any n nonempty subsets of an n-element set have the equal union property if the sum of their cardinalities exceeds n(n+1)/2. This bound is tight. Among families in which the sum of the cardinalities equals n(n+1)/2, we characterize those having the equal union property.