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Rough Set Theory Based on Two Universal Sets and Its Applications Knowledge-Based Systems 23 (2010) 110–115 Contents lists available at ScienceDirect Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys Rough set theory based on two universal sets and its applications Guilong Liu * School of Information Science, Beijing Language and Culture University, Beijing 100083, China article info abstract Article history: For two universal sets U and V, we define the concept of solitary set for any binary relation from U to V. Received 22 February 2008 Through the solitary sets, we study the further properties that are interesting and valuable in the theory Accepted 4 June 2009 of rough sets. As an application of crisp rough set models in two universal sets, we find solutions of the Available online 13 June 2009 simultaneous Boolean equations by means of rough set methods. We also study the connection between rough set theory and Dempster–Shafer theory of evidence. In particular, we extend some results to arbi- Keywords: trary binary relations on two universal sets, not just serial binary relations. We consider the similar prob- Rough set lems in fuzzy environment and give an example of application of fuzzy rough sets in multiple criteria Lower and upper approximations decision making in the case of clothes. Fuzzy set Boolean matrix Ó 2009 Elsevier B.V. All rights reserved. Dempster–Shafer theory of evidence 1. Introduction This paper is organized as follows: In Section 3, for two univer- sal sets U and V, we define the concept of solitary set for any binary Rough set theory was developed by Pawlak as a formal tool for relation from U to V. Through the solitary sets, we list the further representing and processing information in database. In Pawlak properties that are interesting and valuable in the theory of rough rough set theory [19,20], the lower and upper approximation opera- sets. In Section 4, as an application of crisp rough set models in two tors are based on equivalence relation. However, the requirement of universal sets, we have found an algorithm for the simultaneous an equivalence relation in Pawlak rough set models seem to be a very Boolean equations by means of rough set methods. In Section 5, restrictive condition that may limit the applications of the rough set we study the relationship between rough set theory and Demp- models. Thus one of the main directions of research in rough set ster–Shafer theory of evidence. In particular, we extend some re- theory is naturally the generalization of the Pawlak rough set sults to arbitrary binary relations on two universal sets, not just approximations. For instance, the notations of approximations serial binary relations. In Section 6, we study the basic properties are extended to general binary relations [1,11,12,22,25,32–34,37], of fuzzy rough sets in two universal sets. In Section 7, we give a neighborhood systems [6], coverings [36], completely distributive case for the multiple criteria decision making on choices of clothes lattices [2], fuzzy lattices [16] and Boolean algebras [14,23]. designs. Section 8 concludes the paper. On the other hand, the generalization of rough sets in fuzzy environment is another topic receiving much attention in recent 2. Preliminaries years [4–9,21,24]. Based on equivalence relation, the concept of fuzzy rough set was first proposed by Dubois and Prade [4] in Wong et al. [29] generalized the rough set models using two the Pawlak approximation space. Yao [33] gave a unified model distinct but related universal sets. Let U and V denote two univer- for both rough fuzzy sets and fuzzy rough sets based on the anal- sal sets of interest and R a binary relation which is a subset of the ysis of level sets of fuzzy sets. Pei [3,24] considered the approxima- Cartesian product U  V. We called the triplet ðU; V; RÞ an approx- tion problems of fuzzy sets in fuzzy information systems results in imation space. PðUÞ; PðVÞ denote the power sets of U; V, theory of fuzzy rough sets. Li and Zhang [5] analyzed crisp binary respectively. relations and rough fuzzy approximations. With respect to R, we define right neighborhood rðxÞ of an ele- Rough set models on two universal sets can be interpreted by ment x in U, the R-relative set of x in U, to be the set of y in V with both generalized approximation spaces and the notions of interval the property that x is R-related to y. Thus, in symbols, structures [29]. Much research [5,24,30] has been done for these rðxÞ¼fy 2 VjxRyg. Similarly, the left neighborhood lðyÞ of an ele- models. In this paper, we attempt to conduct a further study along ment y in V is lðyÞ¼fx 2 UjxRyg. this line. By the notation rðxÞ, elements in U may be viewed as equivalent if they have the same R-relative set. Thus, an equivalence relation between the elements can be formally defined. Let E denote the * Tel.: +86 10 82303656. U E-mail address: [email protected] equivalence relation on U. 0950-7051/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2009.06.011 G. Liu / Knowledge-Based Systems 23 (2010) 110–115 111 Definition 2.1 [15]. Two elements x and x0 in the universal set U Proof. (1) We only need to prove that ðRYÞðxÞ¼1 if and only if 0 0 0 are equivalent if rðxÞ¼rðx Þ, that is, xEUx if and only if rðxÞ¼rðx Þ. _y2rðxÞYðyÞ¼1. Indeed, if ðRYÞðxÞ¼1, then x 2 RY, hence there exist some z 2 V such that z 2 rðxÞ\Y; that is, z 2 rðxÞ; YðzÞ¼1, there- The equivalence relation EU partitions the set U into disjoint fore _ YðyÞ¼1, and vice versa. A similar argument works to subsets. We use ½x to denote an equivalence class in U containing y2rðxÞ E prove part (2). h an element x 2 U. Similarly, the equivalence relation EV on V can be 0 0 defined by: yEV y if and only if lðyÞ¼lðy Þ. Furthermore, if universal sets U and V are finite sets, then a bin- Suppose that U; V and W are universal sets, R is a relation ary relation from U to V can be represented by a Boolean matrix from U to V, and S is a relation from V to W. We can then define and a subset of V can be represented by a column Boolean vector. a relation, the composition of R and S, written as S R. The rela- Note that Part (1) of Proposition 3.1 can be restated as the follow- tion S R is a relation from U to W and is defined as follows. If a ing equivalent form: is in U and c is in W, then aðS RÞc if and only if for some b in V, we have aRb and bSc. Binary relations R; EU ; EV have the following ðRYÞðxÞ¼_y2V ðRðx; yÞ^YðyÞÞ: properties. We have the following proposition. Proposition 2.1 [15]. Let ðU; V; RÞ be the approximation space, Proposition 3.2. Let U ¼fx1; x2; ...; xmg and V ¼fy ; y ; ...; y g be EU; EV as above, then EV R ¼ R ¼ R EU. 1 2 n With the approximation space ðU; V; RÞ, we define the lower and two finite universal sets, R a binary relation form U to V, MR the m  n upper approximation operators R; R : PðVÞ!PðUÞ by [34] matrix representing R, and Y a subset of V. Then RY ¼ M Y; RY ¼fx 2 UjrðxÞ # Yg; and RY ¼fx 2 UjrðxÞ\Y–;g; R where MR Y is the Boolean product of m  n matrix MR and column respectively. The pair RY ¼ðRY; RYÞ is referred to as the rough set of Boolean vector Y. Y 2 PðVÞ. Note that the lower and upper approximation operators R; R can Definition 3.1 [17]. Let U; V be two universal sets and R be a also be represented by equivalence class x as follows: ½ E binary relation from U to V,ifx 2 U and rðxÞ¼;, we call x a solitary element with respect to R. The set of all solitary elements with RY ¼[rðxÞ # Y ½xE and RY ¼[rðxÞ\Y–;½xE: respect to R is called the solitary set and denoted as S, i.e., Example 2.1. Let U ¼fx ; x ; x ; x g; V ¼fy ; y ; y ; y ; y g, relation 1 2 3 4 1 2 3 4 5 S ¼fxjx 2 U; rðxÞ¼;g: R is given by its corresponding matrix: 0 1 11001 Recall that a binary relation on U is called a serial relation if B C every element x 2 U, has at least one element y 2 U such that B 11100C B C xRy. In other words, S . @ A: ¼; 00010 Through the solitary set, we list the algebraic properties that are 11100 interesting and valuable in the theory of rough sets as follows: Then U=EU ¼ffx1g; fx2; x4g; fx3gg; V=EV ¼ffy1; y2g; fy3g; fy4g; fy5gg. Proposition 3.3. Let R be an arbitrary binary relation from U to V Let Y ¼fy1; y2; y5g, then RY ¼fx1; x2; x4g; RY ¼fx1g. and S be the solitary set respect to R. Then the lower and upper approximation operators satisfy the following properties: for subsets X; YinV, 3. Algebraic properties of rough sets (1) RY ¼[y2Y lðyÞ; This section studies the algebraic structure of the lower and C C (2) R;¼S, R;¼;, RV ¼ U and RV ¼ S , where S denotes the upper approximation operators of rough sets in two universal sets.
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