Axiomatic approaches of the approximation operation based on rough sets and fuzzy rough sets

Rough set theory is an important tool for approximate reasoning about data. Axiomatic systems of rough sets are significant for using rough set theory in logical reasoning systems. In this paper, we propose a unified lower approximation axiomatic system for arbitrary binary relation based generalized rough sets. As the dual of axiomatic systems for lower approximation, a unified upper approximation axiomatic characterization of rough sets is also given. A binary relation can generate a lower approximation operation and an upper approximation operation. We prove that such a binary relation is unique. Furthermore, we can use the same expression to characterize the lower and upper approximations of fuzzy rough sets.