@FMI c Kyung Moon Sa Co.

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1 Annals of Fuzzy Matheatics and Inforatics Volue 1, No. 1, (January 2011), pp ISSN c Kyung Moon Sa Co. Multi-granulation rough set: fro crisp to fuzzy case Xibei Yang, Xiaoning Song, Huili Dou, Jingyu Yang Received 16 Noveber 2010; Accepted 29 Noveber 2010 Abstract. Multi granulation is an iproveent of the classical rough set theory since it uses a faily of binary relations instead of of a single indiscernibility relation for the constructing of approxiation. In this paper, the ulti granulation rough set approach is further generalized into fuzzy environent. A faily of fuzzy T siilarity relations are used to define the optiistic and pessiistic fuzzy rough sets respectively. The basic properties about these fuzzy rough sets are then discussed. Finally, the relationships aong single relation based fuzzy rough set, optiistic and pessiistic ulti granulation fuzzy rough sets are addressed AMS Classification: 94D05, 68T37 Keywords: Fuzzy rough set, Fuzzy T -siilarity relation, Multi-granulation rough set, Multi-granulation fuzzy rough set Corresponding Author: Xibei Yang (yangxibei@hotail.co) 1. Introduction Rough set theory [12, 13, 14, 15], proposed by awlak, is a powerful tool which can be used to deal with the inconsistency probles by separation of certain and doubtful knowledge extracted fro the exeplary decisions. resently, rough set theory has been deonstrated to be useful in the fields such as knowledge discovery, decision analysis, data ining, pattern recognition and so on. However, it is well known that the indiscernibility relation is too restrictive for classification analysis in practical applications, especially in the fuzzy environent. Therefore, how to generalize the rough set odel to fuzzy case plays an iportant role in the developent of rough set theory. For exaple, in Ref. [6], the odel of rough fuzzy set is proposed by using the indiscernibility relation to approxiate a

2 fuzzy concept. Alternatively, the fuzzy rough odel is the approxiation of a crisp set or a fuzzy set in a fuzzy approxiation space. In Ref. [19], Sun et al. presented the interval valued fuzzy rough set by cobining the interval valued fuzzy set and rough set. By eploying an approxiation space constituted fro an intuitionistic fuzzy triangular nor, an intuitionistic fuzzy iplicator, and an intuitionistic fuzzy T equivalence relation, Cornelis et al. [4] defined the concept of intuitionistic fuzzy rough sets in which the lower and upper approxiations are both intuitionistic fuzzy sets in the universe. Zhou et al. proposed the intuitionistic fuzzy rough approxiation by constructive and axioatic approaches in Ref. [24]. Bhatt [1] presented the fuzzy rough sets on copact coputational doain. Zhao et al. [23] investigated the fuzzy variable precision rough sets by cobining fuzzy rough set and variable precision rough set with the goal of aking fuzzy rough set a special case. Hu et al. [7] proposed the Gaussian Kernel based fuzzy rough set, it uses the Gaussian Kernel to copute fuzzy T equivalence relation for objective approxiation. The sae authors proposed the fuzzy preference based rough sets in Ref. [8]. uyang et al. [11] presented the fuzzy rough odel which is based on the fuzzy tolerance relation. More details about recent advanceents of fuzzy rough set can be found in the literatures [2, 3, 5, 9, 10, 20, 21, 22]. bviously, it is not difficult to observe that the above fuzzy rough set approaches are proposed on the basis of one and only one fuzzy binary relation. However, it should be noticed that in Ref. [16, 17, 18], the authors argued that we often need to describe concurrently a target concept through ulti binary relations (e.g. equivalence relation, tolerance relation, reflexive relation and neighborhood relation) on the universe according to a user s requireents or targets of proble solving. Therefore, they proposed the concept of Multi Granulation Rough Set (MGRS) [16, 17, 18] odel which is based on a faily of indiscernibility relations. In their papers, Qian et al. said that the MGRS are useful in the following cases [18]: (1) we cannot perfor the intersection operations between two different attributes sets; (2) decision akers ay be independent for the sae project; (3) extract decision rules fro distributive inforation systes and groups of intelligent agents through using rough set approaches. The purpose of this paper is to further generalize the MGRS to fuzzy environent. In our approach, a faily of fuzzy T siilarity relations is used for constructing ulti granulation fuzzy rough set. To facilitate our discussion, we first present awlak s rough set, Qian s MGRS in Section 2. In Section 3, the optiistic and pessiistic ulti granulation fuzzy rough approxiations are defined as the generalization of classical fuzzy rough set and Qian s MGRS. Not only the basic properties about the ulti granulation fuzzy rough set are discussed, but also the relationships aong several fuzzy rough sets are presented. Results are suarized in Section reliinary knowledge on rough sets In this section, we review soe basic concepts such as inforation syste, awlak s rough set, ulti granulation rough set and doinance based rough set. 56

3 2.1. awlak s rough set. Forally, let U be the universe of discourse, s a faily of equivalence relations on U, then the pair K (U, R) is referred to as a knowledge base. R R, A (U, R) is referred to as a approxiate space, we use U/R to represent the faily of all equivalence classes in ters of R (or classification of U). Such faily of equivalence classes is referred to as categories or concepts of R. Therefore, x U, [x] s used to denotes a category (equivalence class) in ters of R which contains x. Suppose that A R, then A (intersection of all equivalence relations in A) is also an equivalence relation, and will be denoted by IND(A), it is referred to as an indiscernibility relation over A. U/IN D(A) is the faily of all equivalence classes in ters of the set of equivalence relations A, each eleent in U/IND(A) is referred to as an A basic knowledge, [x] A {y U : (x, y) IND(A)} is the equivalence class of x in ters of the set of equivalence relations A. By the indiscernibility relation IN D(A), one can derive the lower and upper approxiations of an arbitrary subset X of U. They are defined as A(X) {x U : [x] A X} and A(X) {x U : [x] A X } respectively. The pair [A(X), A(X)] is referred to as the awlak s rough set of X with respect to the set of attributes A Multi granulation rough set. The ulti granulation rough set is different fro awlak s rough set odel because the forer is constructed on the basis of a faily of binary relations instead of a single indiscernibility relation. The concept of ulti granulation rough set theory was firstly proposed by Qian et al. In their appraoch, two different odels have been defined. The first one is the optiistic ulti granulation rough set [16, 18], the second one is the pessiistic ulti granulation rough set [17] ptiistic MGRS. Definition 2.1. [18] Let K be a knowledge base in which A {R 1, R 2,, R } R, then X U, the optiistic ulti granulation lower and upper approxiations are denoted by (X) and (X) respectively where (2.1) (2.2) (X) {x U : [x] A1 X [x] A2 X [x] A X}; (X) ( X); [x] Ri (1 i ) is the equivalence class of x in ters of the equivalence relation, X is the copleent of set X. 57

4 By the lower and upper approxiations (X), the optiistic ulti granulation boundary region of X is (2.3) BN (X) (X) (X) and (X). Theore 2.2. Let K be a knowledge base in which A {R 1, R 2,, R } R, then X U, we have (2.4) (X) {x U : [x] R1 X [x] R2 X [x] R X }. roof. By Definition 2.1, we have x (X) x / ( X) [x] R1 ( X) [x] E2 ( X) [x] R ( X) [x] R1 X [x] R2 X [x] R X By Theore 2.2, we can see that though the optiistic upper approxiation is defined by the copleent of the lower approxiation, it can also be considered as a subset in which each object has a non epty intersection with the target in ters of each subset of attributes essiistic MGRS. Definition 2.3. [17] Let K be a knowledge base in which A {R 1, R 2,, R } R, then X U, the pessiistic ulti granulation lower and upper approxiations are denoted by (X) and (X) respectively where (2.5) (2.6) (X) {x U : [x] R1 X [x] R2 X [x] R X}; (X) ( X). By the lower and upper approxiations (X), the pessiistic ulti granulation boundary region of X is (2.7) BN (X) 58 (X) (X) and (X).

5 Theore 2.4. Let K be a knowledge base in which A {R 1, R 2,, R } R, then X U, we have (2.8) (X) {x U : [x] R1 X [x] R2 X [x] R X }. roof. By Definition 2.3, we have x (X) x U ( X) [x] R1 ( X) [x] R2 ( X) [x] R ( X) [x] R1 X [x] R2 X [x] R X Siilar to the upper approxiation of optiistic MGRS, the upper approxiation of pessiistic MGRS can also be represented as a subset in which each object has a non epty intersection with the target in ters of one of the subsets of attributes. More details about MGRS can be found in Ref. [18] Single granulation fuzzy rough set. The notion of fuzzy sets provides a convenient tool for representing vague concepts by allowing partial eberships. resently, the concept of fuzzy rough sets was proposed and studied by any authors. We briefly review soe of the basic concepts of the fuzzy rough set. Firstly, we need the following notions. A triangular nor, or shortly t nor, is an increasing, associative and coutative apping T : [0, 1] [0, 1] [0, 1] that satisfies the boundary condition ( x [0, 1], T (x, 1) x). The ost popular continuous t nors are: (1) in operator T M (x, y) in{x, y}; (2) algebraic product: T (x, y) x y; (3) bold intersection(also called the Lukasiewicz t nor): T L (x, y) ax{0, x+ y 1}. A triangular conor, or shortly t conor, is an increasing, associative and coutative apping S : [0, 1] [0, 1] [0, 1]that satisfies the boundary condition ( x [0, 1], S(x, 0) x). The well known continuous t nors are: (1) ax operator S M (x, y) ax{x, y}; (2) probabilistic su: S (x, y) x + y x y; (3) bounded su: S L (x, y) in{1, x + y}. A negator N is a decreasing apping [0, 1] [0, 1] satisfying N(0) 1 and N(1) 0. The negator N S (x) 1 x is referred to as the standard negator. A negator N is involutive of N(N(x)) x for each x [0, 1], an involutive negator is continuous and strictly decreasing. If U is the universe of discourse, then the faily of all fuzzy sets on U is denoted by F (U). Moreover, a fuzzy relation on U is a fuzzy subset R F (U U) where R : U U [0, 1]. 59

6 The fuzzy relation s referred to as serial iff x U, there is a y U such that R(x, y) 1; the fuzzy relation s referred to as reflexive iff R(x, x) 1 holds for each x U; the fuzzy relation is referred to as syetric iff R(x, y) R(y, x) ( x, y U); the fuzzy relation is referred to as T transitive iff T (R(x, y), R(y, z)) R(x, z). If a fuzzy relation s reflexive, syetric and T transitive, then it is referred to as a fuzzy T siilarity relation. Moreover, it should be noticed that there is a canonical one-to-one correspondence between fuzzy T siilarity relation and fuzzy T partition. Based on the definition of T siilarity relation, we can further generalize awlak s knowledge based to fuzzy environent. Let U be the universe of discourse, s a faily of fuzzy T siilarity relations on U, then the pair K (U, R) is referred to as a fuzzy T knowledge base. R R, F A (U, R) is referred to as a fuzzy T approxiate space. Given a fuzzy T knowledge base K, Suppose that A {R 1, R 2,, R } R is a faily of fuzzy T siilarity relations, then R A { : i 1, 2,, } (intersection of all fuzzy T siilarity relations in A ) is also a fuzzy T siilarity relation. Therefore, x, y U, R A (x, y) { (x, y) : i 1, 2,, }. The fuzzy rough approxiations in ters of fuzzy T siilarity relation in ters of R A is defined as follows. Definition 2.5. [6] Let U be the universe of discourse, R A is a fuzzy T siilarity relation on U, then F F (U), then fuzzy rough lower approxiation and fuzzy rough upper approxiation of F are denoted by R(F ) and R(F ) respectively, whose eberships for each x U are: (2.9) R A (F )(x) S ( 1 R A (x, y), F (y) ), (2.10) y U R A (F )(x) T ( R A (x, y), F (y) ). y U Theore 2.6. [6] Let U be the universe of discourse, R A is a fuzzy T siilarity relation on U, then F, F F (U), the fuzzy rough approxiations have the following properties: (1) R A (U) R A (U) U; (2) R A ( ) R A ( ) ; (3) R A (F ) F R A (F ); (4) R A (F ) R A ( F ); (5) R A (F ) R A ( F ); (6) R A (F F ) R A (F ) R A (F ); (7) R A (F F ) R A (F ) R A (F ); (8) R A (F F ) R A (F ) R A (F ); (9) R A (F F ) R A (F ) R A (F ); (10) R A (R A (F )) R A (F ); (11) R A (R A (F )) R A (F ); 60

7 (12) F F R A (F ) R A (F ); (13) F F R A (F ) R A (F ). 3. Constructive approach to ulti granulation fuzzy rough set Following Qian s work, it is natural to introduce the ulti granulation concept into fuzzy environent. This is what will be discussed in this section. In the following, we will use a faily of fuzzy T siilarity relations instead of a single fuzzy T siilarity relation to define the concept of ulti granulation fuzzy rough set. Siilar to Qian s approach, the optiistic and pessiistic ulti granulation fuzzy rough sets will be presented respectively ptiistic ulti granulation fuzzy rough set. Definition 3.1. Let U be the universe of discourse, {R 1, R 2,, R } s a faily of fuzzy T siilarity relations on U, then F F (U), the optiistic ulti granulation fuzzy rough lower approxiation upper approxiation of F are denoted by (F ) and (F ) respectively, whose eberships for each x U are: (3.1) (3.2) If (F ) (F )(x) (F )(x) y U y U S ( 1 (x, y), F (y) ), T ( (x, y), F (y) ). (F ), then F is referred to as optiistic definable under the ulti granulation fuzzy T environent; otherwise, F is referred to as optiistic [ undefinable. (F ), (F ) ] is referred to as a pair of optiistic rough approxiation of F under the ulti granulation fuzzy T environent. In the following, let us exaine soe properties of such fuzzy rough approxiation. Theore 3.2. Let U be the universe of discourse, {R 1, R 2,, R } F R(U) is a faily of fuzzy relations on U, then F, F F (U), the optiistic ulti granulation fuzzy rough set has the following properties: (1) (F ) F (F ); (2) (3) ( ) (F ) (F ), ( ), 61 (F ) (U) (F ); (U) U;

8 (4) (5) ( ( F ) (X) ) (X), (F ), ( (6) F F roof. (1) x U, (F ) (F ), ( F ) (X) ) (F ) (F ); (F ). (X); (F )(x) (F )(x) S ( 1 (x, y), F (y) ) y U S ( 1 (x, x), F (x) ) S ( 0, F (x) ) F (x) T ( (x, y), F (y) ) y U T ( (x, x), F (x) ) T ( 1, F (x) ) F (x) (2) x U, ( )(x) S ( 1 (x, y), 0 ) y U ( 1 Ri (x, y) ) ( 1 Ri (x, x) ) 0 y U Siilarly, it is not difficult to obtain that x U, ( ). (U)(x) T ( (x, y), U(y) ) y U T ( (x, y), 1 ) (x, y) 1 y U y U Siilarly, it is not difficult to obtain that 62 (U) U.

9 (3) x U, (F )(x) S ( 1 (x, y), F (y) ) (F )(x) y U Siilarly, it is not difficult to obtain that (4) By the proof of 1, since ( ( By 3, we have (F )) (F )) (F ) (F ). (F ) F, then by the result of 4, we have (F ). Therefore, it ust be proved that (F ). (F ) (F ). Therefore, ( (F )) ( (F )) ( (F )) ( (F )) (F ) (F ) Siilarly, it is not difficult to obtain that (5) x U, ( F )(x) ( (X) ) S ( 1 (x, y), 1 F (y) ) y U y U 1 1 ( ( 1 T Ri (x, y), F (y) )) y U T ( (x, y), F (y) ) Siilarly, it is not difficult to obtain that 63 (F )(x) ( F ) (F ). (X).

10 (6) x U, since F F, then (F )(x) S ( 1 (x, y), F (y) ) y U S ( 1 (x, y), F (y) ) y U (F )(x) Siilarly, it is not difficult to obtain that (F ) (F ). In the above theore, (1) says that the optiistic ulti granulation lower and upper approxiations satisfy the contraction and extension respectively; (2) says that the optiistic ulti granulation fuzzy rough set satisfies the norality and conorality; (3) expresses the relationships between ulti granulation and single granulation fuzzy rough sets; (4) and (5) shows the idepotency and copleent of ulti granulation fuzzy rough set respectively; (6) shows the onotone of optiistic ulti granulation fuzzy rough approxiation w.r.t. the variety of fuzzy target. These results are all ulti granulation generalization of the properties we showed in Theore 2.6. Theore 3.3. Let U be the universe of discourse, {R 1, R 2,, R } F R(U) is a faily of fuzzy relations on U, F 1, F 2,, F n F (U), the optiistic ulti granulation fuzzy rough set has the following properties: (1) (2) (3) ( n ( n F j ) (F j ) ), n ( F j ) ( n ( (F j ) ), n ( n F j ) (F j ) ) ; n ( F j ) n ( (F j ) ), n ( F j ) n ( 64 n ( F j ) n ( (F j ) ) ; (F j ) ).

11 roof. (1) x U, n ( F j )(x) S ( n 1 (x, y), ( F j )(y) ) y U y U S ( n 1 (x, y), F j (y) ) y U n y U n S ( 1 (x, y), F j (y) ) S ( 1 (x, y), F j (y) ) n (( n (F j )(x) (F j ) ) (x) ) ( ( n (F j ) )) (x) Siilarly, it is not difficult to prove (2) x U, n ( n ( F j ) (F j ) ). n ( F j )(x) S ( n 1 (x, y), ( F j )(y) ) y U y U S ( n 1 (x, y), F j (y) ) y U n y U n ( n S ( 1 (x, y), F j (y) ) S ( 1 (x, y), F j (y) ) (F j )(x) ) ( n ( (F j ) )) (x) Siilarly, it is not difficult to prove 65 n ( F j ) n ( (F j ) ).

12 (3) By 3 of Theore 3.2, since (F ) (F ), then n ( F j ) n ( F j ). Moreover, by (8) of Theore 2.6, it is not difficult to observe that n ( F j ) follows that n (F j ) n ( F j ) n ( Siilarly, it is not difficult to obtain n (F j ) (F j ) ). n ( n ( F j ) n ( (F j ) ), it (F j ) ). In the above theore, not only ulti fuzzy T relations, but also a faily of fuzzy targets are approxiated. These forulas express the relationship between the optiistic fuzzy rough approxiations of a single set and the optiistic fuzzy rough approxiations of ulti sets under the ulti granulation fuzzy T environent essiistic ulti granulation fuzzy rough set. Definition 3.4. Let U be the universe of discourse, {R 1, R 2,, R } s a faily of fuzzy T siilarity relations on U, then F F (U), the pessiistic ulti granulation fuzzy rough lower approxiation upper approxiation of F are denoted by (F ) and (F ) respectively, whose eberships for each x U are: (3.3) (3.4) If (F ) (F )(x) (F )(x) y U y U S ( 1 (x, y), F (y) ), T ( (x, y), F (y) ). (F ), then F is referred to as pessiistic definable under the ulti granulation fuzzy T environent; otherwise, F is referred to as pessiistic undefinable. [ (F ), (F ) ] is referred to as a pair of pessiistic rough approxiation of F under the ulti granulation fuzzy T environent. The following are properties of pessiistic ulti granulation fuzzy rough approxiation. Theore 3.5. Let U be the universe of discourse, {R 1, R 2,, R } s a faily of fuzzy T siilarity relations on U, then F F (U), the pessiistic ulti granulation fuzzy rough set has the following properties: 66

13 (1) (2) (3) (4) (5) (F ) F ( ) (X) ( (X), ( X) (X) ) (F ); ( ), (X), (X) (X), (U) (X); ( ( X) (U) U; (X) ) (X); roof. The proof of Theore 3.5 is siilar to the proof of Theore 3.2. (X); Siilar to the optiistic case, in Theore 3.5, (1) says that the pessiistic ulti granulation lower and upper approxiations satisfy the contraction and extension respectively; (2) says that the pessiistic ulti granulation fuzzy rough set satisfies the norality and conorality; (3) expresses the relationships between pessiistic ulti granulation and single granulation fuzzy rough sets; (4) and (5) shows the idepotency and copleent of pessiistic ulti granulation fuzzy rough set respectively; (6) shows the onotone of pessiistic ulti granulation fuzzy rough approxiation w.r.t. the variety of fuzzy target. These results are also ulti granulation generalization of the properties we showed in Theore 2.6. Theore 3.6. Let U be the universe of discourse, {R 1, R 2,, R } F R(U) is a faily of fuzzy relations on U, F 1, F 2,, F n F (U), the pessiistic ulti granulation fuzzy rough set has the following properties: n ( n (1) ( X j ) (X j ) ) n ( n, ( X j ) (X j ) ) ; (2) (3) n ( X j ) n ( X j ) n ( n ( (X j ) ), (X j ) ), n ( X j ) n ( X j ) n ( n ( roof. The proof of Theore 3.6 is siilar to the proof of Theore 3.3. (X j ) ) ; (X j ) ). Theore 3.6 expresses the relationship between the pessiistic fuzzy rough approxiations of a single set and the pessiistic fuzzy rough approxiations of ulti sets under the ulti granulation fuzzy T environent Relationships aong several odels. 67

14 Theore 3.7. Let U be the universe of discourse, {R 1, R 2,, R } F R(U) is a faily of fuzzy relations on U, F F (U), we have (3.5) (3.6) roof. x U, we have (F )(x) (F ) (F ) Siilarly, it is not difficult to prove y U y U (F ), (F ). S ( 1 (x, y), F (y) ) S ( 1 (x, y), F (y) ) (F )(x). (F ) (F ). The above theore express the relationship between optiistic and pessiistic ulti granulation fuzzy rough approxiation. Forula (3.5) tells us that the pessiistic ulti granulation fuzzy lower approxiation is included into the optiistic ulti granulation fuzzy lower approxiation, forula (3.6) tells us that the pessiistic ulti granulation fuzzy upper approxiation includes optiistic ulti granulation fuzzy upper approxiation. Theore 3.8. Let U be the universe of discourse, {R 1, R 2,, R } F R(U) is a faily of fuzzy relations on U, F F (U), i 1, 2,,, we have (3.7) (3.8) (3.9) (3.10) (F ) (F ) (F ), (F ) (F ), (F ), (F ) (F ). roof. It can be derived directly fro Definition 2.5, Definition 3.1 and Definition

15 The above theore expresses the relationships between single granulation fuzzy rough approxiation and optiistic(pessiistic) ulti granulation fuzzy rough approxiation. Forula (3.7) says that the single granulation fuzzy lower approxiation is included into the optiistic ulti granulation fuzzy lower approxiation; forula (3.8) says that the single granulation fuzzy upper approxiation includes the optiistic ulti granulation fuzzy upper approxiation; forula (3.9) says that the single granulation fuzzy lower approxiation includes the pessiistic ulti granulation fuzzy lower approxiation; forula (3.10) says that the single granulation fuzzy upper approxiation is included into the pessiistic ulti granulation fuzzy upper approxiation. 4. Conclusions In this paper, the constructive approach is used to generalize the ulti granulation rough set to fuzzy environent. The odels of optiistic and pessiistic fuzzy rough approxiations are then defined respectively. These ulti granulation fuzzy rough set are constructed on the basis of a faily of fuzzy T siilarity relations instead of a single fuzzy relation. In our further research, the approach to attribute reductions w.r.t. the proposed fuzzy rough odel is an iportant issue to be addressed. Acknowledgeents. This work is supported by the Natural Science Foundation of China (No ). References [1] R. B. Bhatt, M. Gopal, n the copact coputational doain of fuzzy rough sets, attern Recogn. Lett. 26 (2005) [2] D.G. Chen, W.X. Yang, F.C. Li, Measures of general fuzzy rough sets on a probabilistic space, Infor. Sci. 178 (2008) [3] D.G. Chen, S.Y. Zhao, Local reduction of decision syste with fuzzy rough sets, Fuzzy Sets and Systes 161 (2010) [4] C. Cornelis, M.D. Cock, E.E. Kerre, Intuitionistic fuzzy rough sets: at the crossroads of iperfect knowledge, Expert Syst. 20 (2003) [5] T. Q. Deng, Y. M. Chen, W. L. Xu, Q. H. Dai, Anovel approach to fuzzy rough sets based on a fuzzy covering, Infor. Sci. 177 (2007) [6] D. Dubios, H. rade, Rough fuzzy sets and fuzzy rough sets, Int. J. Gen. Syst. 17 (1990) [7] Q.H. Hu, L. Zhang, D.G. Chen, W. edrycz, D.R. Yu, Gaussian kernel based fuzzy rough sets: Model, uncertainty easures and applications, Int. J. Approx. Reason. 51 (2010) [8] Q.H. Hu, D.R. Yu, M.Z. Guo, Fuzzy preference based rough sets, Infor. Sci. 180 (2010) [9] G.L. Liu, Generalized rough sets over fuzzy lattices, Infor. Sci. 178 (2008) [10] J.S. Mi, Y. Leung, H.Y. Zhao, T. Feng, Generalized fuzzy rough sets deterined by a triangular nor, Infor. Sci. 178 (2008) [11] Y. uyang, Z.D. Wang, H.. Zhang, n fuzzy rough sets based on tolerance relations, Infor. Sci. 180 (2010) [12] Z. awlak, Rough sets theoretical aspects of reasoning about data, Kluwer Acadeic ublishers (1992). [13] Z. awlak, A. Skowron, Rudients of rough sets, Infor. Sci. 177 (2007) [14] Z. awlak, A. Skowron, Rough sets: Soe extensions, Infor. Sci. 177 (2007) [15] Z. awlak, A. Skowron, Rough sets and boolean reasoning, Infor. Sci. 177 (2007)

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