# On Some Generalized Relations

VIT University

^{1}14 Nov 2014-pp 714-718

TL;DR: Several generalised relations are defined where the rough set definition considered is that of Iwinski and many properties of these relations can be used in different fields of computer science like, databases, artificial intelligence, granular computing and soft computing.

Abstract: Crisp set relations have been generalised to fuzzy relations, intuitionistic fuzzy relations, rough relations and also some hybrid relations related to these notions. Rough set is a fruitful model to capture imprecision in data. There are two approaches to the rough set notion. The approach by Pawlak is topological by nature and the approach of Iwinski is algebraic by nature. In this paper we define several generalised relations where the rough set definition considered is that of Iwinski and prove many properties of these relations. These relations can be used in different fields of computer science like, databases, artificial intelligence, granular computing and soft computing. We illustrate by example the structures of most of the concepts introduced in the paper.

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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.

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TL;DR: Various properties are proved, which are connected to the operations and relations over sets, and with modal and topological operators, defined over the set of IFS's.

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TL;DR: It is argued that both notions of a rough set and a fuzzy set aim to different purposes, and it is more natural to try to combine the two models of uncertainty (vagueness and coarseness) rather than to have them compete on the same problems.

Abstract: The notion of a rough set introduced by Pawlak has often been compared to that of a fuzzy set, sometimes with a view to prove that one is more general, or, more useful than the other. In this paper we argue that both notions aim to different purposes. Seen this way, it is more natural to try to combine the two models of uncertainty (vagueness and coarseness) rather than to have them compete on the same problems. First, one may think of deriving the upper and lower approximations of a fuzzy set, when a reference scale is coarsened by means of an equivalence relation. We then come close to Caianiello's C-calculus. Shafer's concept of coarsened belief functions also belongs to the same line of thought. Another idea is to turn the equivalence relation into a fuzzy similarity relation, for the modeling of coarseness, as already proposed by Farinas del Cerro and Prade. Instead of using a similarity relation, we can start with fuzzy granules which make a fuzzy partition of the reference scale. The main contribut...

2,452 citations

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TL;DR: This paper introduces and discusses the concept of fuzzy rough sets, a type of rough set that is similar to a ULTIMATE model but with some properties of a Turing-complete system.

318 citations