Covering-Based Rough Approximation Operators And Rough Fuzzy Normal Subgroups

Rough set is a soft computing method for dealing with indiscernibility and inaccurate data.The rough set was proposed by Pawlak,the Poland mathematician.William Zhu investigated the covering rough set as an extension of famous Pawlak rough set theory,which are transformed from equivalence relation to general covering.Though several scholars have researched on covering rough set theory in different aspects and depths,it is still not perfect and needs to be further explored.This article based on several classical covering rough approximation operators,the innovation and main consequences are as following:1.William Zhu only discussed the relationships between union reducible and approximation operators by the original covering,but it did not involve the intersection reducible of covering.To address this issue,in this paper,we mainly discuss the correlations between the intersection reducible and approximation operators in the original covering.2.The concept of induced covering was proposed by Yiyu Yao et al.(2012).In this work,we have further explored of induced covering and the relationships between the covering approximation operators defined in different induced covering.Moreover,a new finding of this paper is,the approximate operators defined by induced covering,are consistent with the classical approximation operators defined by William Zhu.3.In terms of the congruent relation of groups,we replace the equivalence relation of rough set by congruent relation,thus making the elements of rough set blurred.In such a way,we can explore the fuzzy properties of rough set.Also,we give the definitions of(7)?,?(8)-rough fuzzy normal subgroup,(7)?,?(8)-rough fuzzy quotient group and explore the relationships between them.