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Top1. Introduction
The rough set theory introduced by Pawlak (1982, 1991) is an excellent mathematical implement for the analysis of uncertain, imprecise, inconsistently erratic and nebulous description of objects. The rough set philosophy is based on the posit that every object of the universe is associated with a certain amount of information. This information is expressed in the form of some attributes utilized for object description. With veneration to the available information about the objects characterized with kindred description are verbalized to be indiscernible. The indiscernibility relation thus generated constitutes a mathematical basis of the rough set theory. It induces equivalence classes which can be used to acquire knowledge about the objects of the information system. The rough set is predicated on the rudimentary conception as approximation of sets. The approximation of set is a pair of lower and upper approximation, predicated on the equivalence relation to induce knowledge about the universe. But, authentic life requisite is pretty different, and rough set technique cannot be applied due to the restrictive condition of equivalence relation. For a consummate description of authentic life system, frequently one would require by far more detailed data than a human being could apperceive and understand simultaneously for ever. It is also seen that in real life data, two different objects may have attribute values that are almost identical, if not exactly identical. It indicates that attribute values could be identical up to certain extent which cannot be applied by using rough set model. To this end, rough set is generalized to fuzzy environment as fuzzy rough set and rough fuzzy set (Dubois & Prade, 1990). Further the concept of equivalence relation is generalized to binary relations (Pawlak & Skowron, 2007a; Kondo, 2006; Zhu, 2007), neighborhood systems (Lin, 1989), Boolean algebras (Liu, 2005; Pawlak & Skowron, 2007b), fuzzy lattices (Liu, 2008), completely distributive lattices (Chen, Zhang, Yeung, & Tsang, 2006) etc. to overcome the limitation of equivalence relation.
Alternatively, the equivalence relation was generalized to fuzzy proximity relation and the concept of rough set on fuzzy approximation space was introduced (De, 1999). Further, Tripathy and Acharjya (2008) made paramount attempt in the study of rough set on fuzzy approximation space. The notation of fuzzy approximation space that depends on the concept of fuzzy proximity relation generalizes indiscernibility to almost indiscernibility and rough set on fuzzy approximation space reduces to rough set, on certain conditions. Rough set on fuzzy approximation space is further generalized to rough set on intuitionistic fuzzy approximation space (Tripathy & Acharjya, 2009). But, all these techniques extract knowledge from an information system defined over a single universe. In recent years, it is seen that, an information system establishes relation between two universes. Keeping this in mind, rough set has been extended to rough set on two universal sets (Liu, 2010). Many of its properties have been researched by Tripathy & Acharjya (2012, 2013). Further, rough set on two universal sets was extended to fuzzy rough set over two universes (Weihua, Wenxin & Yufeng, 2013). Knowledge reductions in generalized approximation space over two universes are also discussed by Weihua, Wentao & Shuqun (2015). Additionally, data reduction has become one of the most important of recent research. It is generally considered as a preprocessing technique for data mining and machine learning. The reduction in data can be carried out in two ways such as horizontal reduction or feature reduction and vertical reduction or sample reduction.