A Study on K Approximation spaces and their applications in decision making problems

Abstract

Rough set theory defined by Pawlak is one of the efficient mathematical tool widely used in data analytics, image processing, neural networks etc. Usually in defining rough sets, we consider an information system I = (U,A) where U is a non-empty finite set of objects and A is a finite set of fuzzy attributes. In general, I = (U ,R) is called as an approximation space, where U is a non-empty finite set of objects and R is an arbitrary equivalence relation on U. Using this, several works have been done on the algebraic study of rough sets. In this thesis, we extend the definition of approximation space to define k-approximation space in which the k-information system I = (U, R1, R2, . . . , Rk) where U is a non-empty finite set of objects and R1, R2, . . . , Rk are k-equivalence relations on U. For every subset X of U, we define k-rough set denoted by k RS (X) = (RSR1(X),RSR2(X),. . . ,RSR,(X)), where RSRi (X) is the rough set corresponding to X with respect to Ri. This concept is very powerful because of the following reasons. Firstly, for any given subset X of U using k-rough set the objects of X with respect to k distinct partitions can be compared easily. Secondly, this helps to study the influence of one partition over the other on the subsets of U. When k = 2, 2 RS(X) = (RSR, (X), RSR2 (X)) is called as the twin rough set defined on the twin approximation space I = (U, R1, R2). Soft set like rough set is also an efficient mathematical tool in capturing uncertainties. A Soft set on a non-empty finite set U is an ordered pair G = (F,A) where F : A -+ P(U), A is a set of attributes possessed by the objects in U. The idea of k-rough set is extended to covering based k-soft rough set, covering based k-hypergraph rough set and covering based k-minimal soft rough set. We also made an attempt to handle situations in which the system will have objects with various attributes and each attribute is determined in terms of many newline newlineiv newlineparameters. We consider an information system I = (U,A), where U be a non-empty finite