  Although some information-theoretic measures of uncertainty or granularity
have been proposed in rough set theory, these measures are only dependent on
the underlying partition and the cardinality of the universe, independent of
the lower and upper approximations. It seems somewhat unreasonable since the
basic idea of rough set theory aims at describing vague concepts by the lower
and upper approximations. In this paper, we thus define new
information-theoretic entropy and co-entropy functions associated to the
partition and the approximations to measure the uncertainty and granularity of
an approximation space. After introducing the novel notions of entropy and
co-entropy, we then examine their properties. In particular, we discuss the
relationship of co-entropies between different universes. The theoretical
development is accompanied by illustrative numerical examples.
