Study of definable subsets in covering approximation spaces of rough sets

2011 ◽  
Author(s):  
Nianbai Fan ◽  
Gongzhu Hu ◽  
Hui Liu
Keyword(s):  
Rough Sets ◽  
Approximation Spaces

scholarly journals Two Types of Intuitionistic Fuzzy Covering Rough Sets and an Application to Multiple Criteria Group Decision Making

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 462 ◽  
Author(s):  
Jingqian Wang ◽  
Xiaohong Zhang
Keyword(s):  
Decision Making ◽  
Rough Set ◽  
Group Decision Making ◽  
Rough Sets ◽  
Group Decision ◽  
Multiple Criteria ◽  
Approximation Spaces ◽  
Intuitionistic Fuzzy ◽  
Definition Of ◽  
Covering Rough Set

Intuitionistic fuzzy rough sets are constructed by combining intuitionistic fuzzy sets with rough sets. Recently, Huang et al. proposed the definition of an intuitionistic fuzzy (IF) β -covering and an IF covering rough set model. In this paper, some properties of IF β -covering approximation spaces and the IF covering rough set model are investigated further. Moreover, we present a novel methodology to the problem of multiple criteria group decision making. Firstly, some new notions and properties of IF β -covering approximation spaces are proposed. Secondly, we study the characterizations of Huang et al.’s IF covering rough set model and present a new IF covering rough set model for crisp sets in an IF environment. The relationships between these two IF covering rough set models and some other rough set models are investigated. Finally, based on the IF covering rough set model, Huang et al. also defined an optimistic multi-granulation IF rough set model. We present a novel method to multiple criteria group decision making problems under the optimistic multi-granulation IF rough set model.

On the union and intersection operations of rough sets based on various approximation spaces

2015 ◽  
Vol 292 ◽  
pp. 214-229 ◽  
Author(s):  
Xiaohong Zhang ◽  
Jianhua Dai ◽  
Yucai Yu
Keyword(s):  
Rough Sets ◽  
Approximation Spaces

scholarly journals Topological Interpretation of Rough Sets

2014 ◽  
Vol 22 (1) ◽  
pp. 89-97 ◽  
Author(s):  
Adam Grabowski
Keyword(s):  
Rough Sets ◽  
Upper Bounds ◽  
Topological Spaces ◽  
Approximation Spaces ◽  
Extremally Disconnected ◽  
Topological Interpretation ◽  
Topological Closure ◽  
Mizar Mathematical Library ◽  
Abstract Spaces

Summary Rough sets, developed by Pawlak, are an important model of incomplete or partially known information. In this article, which is essentially a continuation of [11], we characterize rough sets in terms of topological closure and interior, as the approximations have the properties of the Kuratowski operators. We decided to merge topological spaces with tolerance approximation spaces. As a testbed for our developed approach, we restated the results of Isomichi [13] (formalized in Mizar in [14]) and about fourteen sets of Kuratowski [17] (encoded with the help of Mizar adjectives and clusters’ registrations in [1]) in terms of rough approximations. The upper bounds which were 14 and 7 in the original paper of Kuratowski, in our case are six and three, respectively. It turns out that within the classification given by Isomichi, 1st class subsets are precisely crisp sets, 2nd class subsets are proper rough sets, and there are no 3rd class subsets in topological spaces generated by approximations. Also the important results about these spaces is that they are extremally disconnected [15], hence lattices of their domains are Boolean. Furthermore, we develop the theory of abstract spaces equipped with maps possessing characteristic properties of rough approximations which enables us to freely use the notions from the theory of rough sets and topological spaces formalized in the Mizar Mathematical Library [10].

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