  In this paper, we exhibit the equivalence between different notions of unique
range sets, namely, unique range sets, weighted unique range sets and
weak-weighted unique range sets under certain conditions.\par Also, we present
some uniqueness theorems which show how two meromorphic functions are uniquely
determined by their two finite shared sets. Moreover, in the last section, we
make some observations that help us to construct other new classes of unique
range sets.
