On various approximation spaces that is the closeness of union and intersection operations of rough approximation pairs, i.e.
A natural question is as follows:
Whether the above union and intersection operations are binary operation on RSðUÞ, that is, whether the union and section operations of rough approximation pairs are closed in RSðUÞ. http://dx.doi.org/10.1016/j.ins.2014.09.007 0020-0255/ 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author at: Department of Mathematics, College of Arts and Sciences, Shanghai Maritime University, Shanghai 201306, China.
E-mail addresses: email@example.com, firstname.lastname@example.org (X. Zhang).
Information Sciences 292 (2015) 214–229
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Information SciencesðRðXÞ;RðXÞÞ d ðRðYÞ;RðYÞÞ ¼ ðRðXÞ [ RðYÞ;RðXÞ [ RðYÞÞ; ðRðXÞ;RðXÞÞ e ðRðYÞ;RðYÞÞ ¼ ðRðXÞ \ RðYÞ;RðXÞ \ RðYÞÞ:inter-Rough set theory was proposed by Pawlak [22,23]. Rough set is a mathematical tool for dealing with uncertainty and it is widely applied to pattern recognition, image processing, feature selection, rule extraction, decision supporting, granular computing, data mining and knowledge discovery from large data sets (see [24,42]).
In the basic theoretical researches on classical rough sets and various generalized rough sets, algebra approach is widely applied (see [1,3–8,10,12,16–19,21,25,27,28,30,32,34,35,38–40,43–45]). For Pawlak’s approximate space ðU;RÞ, where U is the universe and R is an equivalence relation on U, let X # U, we consider the approximation pairs ðRðXÞ;RðXÞÞ, which is also called rough sets. Denote RSðUÞ ¼ fðRðXÞ;RðXÞÞjX # Ug. The union and intersection operations of rough approximation pairs can be defined by:Received in revised form 31 August 2014
Accepted 7 September 2014
Available online 18 September 2014
Rough approximation pair
Fuzzy rough set
Rough fuzzy set 1. Introduction(lower approximation, upper approximation). We present that the union and intersection operations of rough approximation pairs are closed for classical rough sets and two kinds of covering based rough sets, but not for twenty kinds of covering based rough sets and the generalized rough sets based on fuzzy approximation space. Moreover, we proved that the union and intersection operations of rough fuzzy approximation pairs are closed and a bounded distributive lattice can be constructed. 2014 Elsevier Inc. All rights reserved.Xiaohong Zhang a,b,⇑, Jianhua Dai c, Yucai Yu d aDepartment of Mathematics, College of Arts and Sciences, Shanghai Maritime University, Shanghai 201306, China b School of Computer Science, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China cCollege of Computer Science, Zhejiang University, Hangzhou 310027, China d Shaoxing High School, Shaoxing, Zhejiang 312000, China a r t i c l e i n f o
Received 6 April 2013 a b s t r a c t
Algebraic structures and lattice structures of rough sets are basic and important topics in rough sets theory. In this paper we pointed out that a basic problem had been overlooked,the union and intersection operations of rough sets based onjournal homepage: www.elsevier .com/locate / ins tient s gener appro
Theorem 2.1 . Let ðU;RÞ be a Pawlak’s approximation space. For any X; Y # U, there exists Z # U such that
Theor ð2U ;\ we can give the formula of the above Z in Theorem 2.1 and W in Theorem 2.2: 0 0 0 0 0 where appro
Let and s
X. Zhang et al. / Information Sciences 292 (2015) 214–229 215sðXÞ ¼ fx 2 UjsðxÞ # Xg; sðXÞ ¼ fx 2 UjsðxÞ \ X – ;g:
The sets sðXÞ and sðXÞ are called s-lower and s-upper approximation of X respectively.
By the Example 164(a) and Fig. 25 in  we know that, for tolerance approximation spaces, the union and intersection operations of rough approximation pairs ðsðXÞ; sðXÞÞ are not closed.
Based on tolerance approximation spaces, Wasilewski and Slezak introduced the notions of T-lower and T-upper approximations in . By this generalized rough sets, a double Heyting algebra is established in , but we will show that their main results are questionable, because the union and intersection operations of T-rough approximation pairs are not proven to be closed.ximation space. ðU; sÞ is a tolerance approximation space, denote sðxÞ ¼ fy 2 Ujðx; yÞ 2 sg, where x 2 U. Define operations s : 2U ! 2U : 2U ! 2U such that, for any X # U:3. On generalized rough sets based on tolerance approximation spaces and determined by preorder relation
Tolerance approximation spaces were introduced by Skowron and Stepaniuk in . Let U be a universe and s # U U. s is called tolerance relation on U if it is reflexive and symmetric. If s is tolerance relation on U, then ðU; sÞ is called toleranceR1ðX0;Y 0Þ ¼ [f½sRjs 2 U; ½sR # ðX \ YÞ0; ½sR X 0; ½sR Y 0g:W ¼ ð[fT # UjT # Y ; T \ R1ðX ;Y Þ ¼ ;gÞ \ X ;
R1ðX;YÞ ¼ [f½sRjs 2 U; ½sR # X [ Y ; ½sR X; ½sR Yg;Z ¼ ð[fT # UjT # X; T \ R1ðX; YÞ ¼ ;gÞ [ Y; where;[; 0;R; ;;UÞ is a strong topological rough algebra (where 0 denote the complement, see Theorem 1 in ). Moreover,The above theorems are generalized to strong topological rough algebras (see Theorem 6 and Theorem 7 in ). In fact, for Pawlak’s approximation space ðU;RÞ, denote the power set of U by 2U , then ð2U ;\;[;0; ;;UÞ is a Boolean algebra andem 2.2 . Let ðU;RÞ be a Pawlak’s approximation space. For any X; Y # U, there exists W # U such that
RðWÞ ¼ RðXÞ \ RðYÞ; RðWÞ ¼ RðXÞ \ RðYÞ:RðZÞ ¼ RðXÞ [ RðYÞ; RðZÞ ¼ RðXÞ [ RðYÞ:RðAÞ ¼ fx 2 Uj½xR # Ag;
RðAÞ ¼ fx 2 Uj½xR \ A – ;g:
The pair ðRðAÞ;RðAÞÞ is referred to as rough set approximation of A.
The union and intersection operations of Pawlak’s classical rough sets (approximation pairs) are closed, which were first obtained by Pomykala and Pomykala in  and further discussed in [1,3,9,14].et consisting of equivalence classes of R, and ½xR the equivalence class containing x. Given an arbitrary set A # U, in al it may not be possible to describe A precisely in ðU;RÞ. One may characterize A by a pair of lower and upper ximations:In this paper, we will solve the above problem for various rough set models. The results of this paper show that the union and intersection operations of rough approximation pairs are not closed for some kinds of generalized rough sets, and the closeness problem is often overlooked in some literatures. We will also point out an open problem. 2. On Pawlak’s classical rough sets