Let $G$ be a simple graph. For a general edge coloring of a graph $G$ (i.e., not necessarily a proper edge coloring) and a vertex $v$ of $G$, denote by $S(v)$ the set (not a multiset) of colors used to color the edges incident to $v$. For a general edge coloring $f$ of a graph $G$, if $S(u)\neq S(v)$ for any two different vertices $u$ and $v$ of $G$, then we say that $f$ is a point-distinguishing general edge coloring of $G$. The minimum number of colors required for a point-distinguishing general edge coloring of $G$, denoted by $\chi _{0}(G)$, is called the point-distinguishing chromatic index of $G$. In this paper, we determine the point-distinguishing chromatic index of the union of paths and propose a conjecture.