# Some Properties of Rough Sets on Fuzzy Approximation Spaces and Applications

VIT University

^{1}01 Jan 2017-pp 179-188

TL;DR: This paper shows through examples that some of the results established in this direction by De et al. have been found to be faulty and establish some more properties of these rough sets.

Abstract: The notion of Rough sets introduced by Pawlak has been extended in many directions to enhance its modelling power. One such approach is to reduce the restriction of the base relation being an equivalence relation. Adding the flavour of fuzzy sets to it a fuzzy proximity relation was used to generate a fuzzy approximation space by De et al. in 1999 and hence the rough sets on fuzzy approximation spaces could be generated. These are much more general than the basic rough sets and also the rough sets defined on proximity relations. However, some of the results established in this direction by De et al. have been found to be faulty. In this paper we show through examples that the results are actually faulty and provide their correct versions. Also, we establish some more properties of these rough sets. A real life application is provided to show the application of the results.

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TL;DR: Two types of multigranular rough sets on fuzzy approximation spaces (optimistic and pessimistic) are introduced, several of their properties are studied and how this notion can be used for prediction of rainfall are illustrated.

Abstract: Basic rough set model introduced by Pawlak in 1982 has been extended in many directions to enhance their modeling power. One such attempt is the notion of rough sets on fuzzy approximation spaces by De et al in 1999. This basic model uses equivalence relation for its definition, which decompose the universal set into disjoint equivalence classes. These equivalence classes are called granules of knowledge. From the granular computing point of view the basic rough set model is unigranular in character. So, in order to handle more than one granular structure simultaneously, two types of multigranular rough sets, called the optimistic and pessimistic multigranular rough sets were introduced by Qian et al in 2006 and 2010 respectively. In this paper, we introduce two types of multigranular rough sets on fuzzy approximation spaces (optimistic and pessimistic), study several of their properties and illustrate how this notion can be used for prediction of rainfall. The introduced notions are explained through several examples.

9 citations

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01 Aug 1996

TL;DR: A separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.

Abstract: A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.

52,705 citations

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TL;DR: This approach seems to be of fundamental importance to artificial intelligence (AI) and cognitive sciences, especially in the areas of machine learning, knowledge acquisition, decision analysis, knowledge discovery from databases, expert systems, decision support systems, inductive reasoning, and pattern recognition.

Abstract: Rough set theory, introduced by Zdzislaw Pawlak in the early 1980s [11, 12], is a new mathematical tool to deal with vagueness and uncertainty. This approach seems to be of fundamental importance to artificial intelligence (AI) and cognitive sciences, especially in the areas of machine learning, knowledge acquisition, decision analysis, knowledge discovery from databases, expert systems, decision support systems, inductive reasoning, and pattern recognition.

7,185 citations

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TL;DR: New definitions of lower and upper approximations are proposed, which are basic concepts of the rough set theory and are shown to be more general, in the sense that they are the only ones which can be used for any type of indiscernibility or similarity relation.

Abstract: This paper proposes new definitions of lower and upper approximations, which are basic concepts of the rough set theory. These definitions follow naturally from the concept of ambiguity introduced in this paper. The new definitions are compared to the classical definitions and are shown to be more general, in the sense that they are the only ones which can be used for any type of indiscernibility or similarity relation.

963 citations

### "Some Properties of Rough Sets on Fu..." refers background in this paper

...Attempts have been made to define generalized approximation spaces instead of knowledge bases in [4] and [5]....

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TL;DR: In tolerance approximation spaces the lower and upper set approximations are defined and the tolerance relation defined by the so called uncertainty function or the positive region of a given partition of objects have been chosen as invariants in the attribute reduction process.

Abstract: We generalize the notion of an approximation space introduced in [8] In tolerance approximation spaces we define the lower and upper set approximations We investigate some attribute reduction problems for tolerance approximation spaces determined by tolerance information systems The tolerance relation defined by the so called uncertainty function or the positive region of a given partition of objects have been chosen as invariants in the attribute reduction process We obtain the solutions of the reduction problems by applying boolean reasoning [1] The solutions are represented by tolerance reducts and relative tolerance reducts

955 citations

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TL;DR: The rough sets theory was originally founded on the idea of approximating a given set by means of indiscernibility binary relation, which was assumed to be an equivalence relation (reflexive, symmetric and transitive), but now the assumption of symmetry and transitivity is relaxed.

Abstract: The rough sets theory proposed by Pawlak [8,9] was originally founded on the idea of approximating a given set by means of indiscernibility binary relation, which was assumed to be an equivalence relation (reflexive, symmetric and transitive). With respect to this basic idea, two main theoretical developments have been proposed: some extensions to a fuzzy context (e.g. Dubois and Prade, [1,2], Slowinski and Stefanowski, [13,14,15], Yao, [19]) and some extensions of the indiscernibility relation by means of more general binary relations (e.g. Nieminen, [7], Lin, [5], Marcus, [6], Polkowski, Skowron and Zytkow, [10], Skowron and Stepaniuk, [11], Slowinski, [12], Slowinski and Vanderpooten, [16,17,18], Yao and Wong, [20]). In the latter extensions, we wish to point out the proposal of Slowinski and Vanderpooten( [16,17,18]) who introduced and characterized a general definition of rough approximations using a similarity relation which is a reflexive binary relation, relaxing the assumption of symmetry and transitivity.

253 citations