  The union vertex-distinguishing chromatic index $\chi'_\cup(G)$ of a graph
$G$ is the smallest natural number $k$ such that the edges of $G$ can be
assigned nonempty subsets of $[k]$ so that the union of the subsets assigned to
the edges incident to each vertex is different. We prove that $\chi'_\cup(G)
\in \left\{ \left\lceil \log_2\left(n +1\right) \right\rceil, \left\lceil
\log_2\left(n +1\right) \right\rceil+1 \right\}$ for a graph $G$ on $n$
vertices without a component of order at most two. This answers a question
posed by Bousquet, Dailly, Duch\^{e}ne, Kheddouci and Parreau, and
independently by Chartrand, Hallas and Zhang.
