  We introduce a variant of the vertex-distinguishing edge coloring problem,
where each edge is assigned a subset of colors. The label of a vertex is the
union of the sets of colors on edges incident to it. In this paper we
investigate the problem of finding a coloring with the minimum number of colors
where every vertex receives a distinct label. Finding such a coloring
generalizes several other well-known problems of vertex-distinguishing
colorings in graphs.We show that for any graph (without connected component
reduced to an edge or a single vertex), the minimum number of colors for which
such a coloring exists can only take 3possible values depending on the order of
the graph. Moreover, we provide the exact value for paths, cycles and complete
binary trees.
