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A comparison of two kinds of definitions of rough approximations based on a similarity relation
Information Sciences 181 (2011) 2587–2596
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A comparison of two kinds of definitions of rough approximations based on a similarity relation E.A. Abo-Tabl Mathematics Department, Faculty of Science, Assiut University, Assiut, Egypt Department of Mathematics, Faculty of Science and Arts, Al-mithnab, Qassim University, P.O. Box 931, Buridah 51931, Al-mithnab, Kingdom of Saudi Arabia
a r t i c l e
i n f o
Article history: Received 23 March 2009 Received in revised form 22 December 2010 Accepted 2 January 2011 Available online 21 January 2011 Keywords: Rough sets Lower and upper approximations Similarity relation Accuracy measure
a b s t r a c t In the present paper, we investigate the three types of Yao’s lower and upper approximations of any set with respect to any similarity relation. These types based on a right neighborhood. Also, we define and investigate other three types of approximations for any similarity relations. These new types based on the intersection of the right neighborhoods. Moreover, we give a comparison between these types. Lastly, the relationship between these definitions is introduced. Ó 2011 Published by Elsevier Inc.
1. Introduction The theory of rough sets, proposed by Pawlak [13–17], is an extension of the set theory for the study of intelligent systems characterized by insufficient and incomplete information. The lower and the upper approximation operators are constructed by using a binary relation on the universe. Using the concepts of the lower and the upper approximations from the rough set theory, knowledge hidden in information systems may be unraveled and expressed in the form of decision rules [8,14,34]. There are at least two approaches for the development of the rough set theory, namely the constructive and the axiomatic approaches. In the constructive approach, binary relations on the universe, partitions of the universe, neighborhood systems, and Boolean algebras are all the primitive notions. The lower and the upper approximation operators are constructed by means of these notions [5,8,12–14,18,19,21,24–27,34]. The constructive approach is suitable for practical applications of the rough sets. On the other hand, the axiomatic approach, which is appropriate for studying the structures of rough set algebras, takes the lower and the upper approximation operators as primitive notions. In this approach, a set of axioms is used to characterize approximation operators that are the same as the ones produced using the constructive approach [9,25]. The foundations as well as a majority of the recent studies on the rough sets are focused on the constructive approaches. The classical rough approximations are based on equivalence relations, but this requirement is not satisfied in some situations. Thus the classical rough approximations has been extended to the similarity relation based rough sets [22], the tolerance relation based rough sets [20], the arbitrary binary relation based rough sets [11,25,26,30,31] and the covering-based rough sets [3,4,6,10,29,32,33]. Suppose we are given a finite nonempty set U of objects, called the universe. Indiscernibility reflects a total impossibility of distinguishing between objects considering the available information. This situation can be represented using a binary relation R defined on U, which is reflexive, symmetric, and transitive. Objects of U can be partitioned into indiscernibility (or equivalence) classes, which form the basic granules of the knowledge available through R. It is natural to extend E-mail address: [email protected] 0020-0255/$ - see front matter Ó 2011 Published by Elsevier Inc. doi:10.1016/j.ins.2011.01.007
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the indiscernibility concept to take account of situations where objects are not significantly distinct. This happens when the objects are described by imprecise data or when small differences are meaningless in the context of the study. This situation can be modeled using a binary relation R defined on U, which represents a certain form of similarity. The similarity or the tolerance relations have been studied extensively (see, e.g., [7], [28]). A basic difference between the indiscernibility and the similarity relations is that, in general, the similarity relations do not give rise to a partition of the set of objects. Information about similarity can be represented using the similarity classes for each object x 2 U. The reflexivity property cannot be relaxed since, as any object is trivially indiscernible with itself, it is, a fortiori, similar to itself. The most controversial property is symmetry. Most authors dealing with the similarity relations do impose this property. Notice, however, that the statement xRy, which means ‘‘y is similar to x’’, is directional; it has a subject y and a referent x and it is not equivalent, in general, to the statement ‘‘x is similar to y’’ as argued by Tversky [23]. For example, in the following statement: ‘‘a son resembles his father,’’ the son is the subject and the father is the referent. The inverse statement usually makes much less sense. We first investigate and study in Section 2 the three types of Yao’s lower and upper approximations of any set with respect to any similarity relation, which are based on the concept of right neighborhood. Moreover, we define and investigate other three types of approximations for any similarity relations, which are based on the intersection of the right neighborhoods. In addition, we give a comparison between all these definitions and show these comparisons in a table (Section 3). In Section 4, a simple example which show the differences between all definitions of rough approximation is then introduced. At last, some conclusion is presented in Section 5.
2. Yao’s approach of generalized rough sets Since reflexivity is a minimal requirement for any type of the similarity relation, we assume in the following that R is reflexive. Yao [25,26] extended the Pawlak’s definitions of a rough set to any similarity relation R as follows: Definition 2.1. For the pair (U, R), the set xR is defined as xR = {y 2 U: xRy} called the right neighborhood of an element x 2 U. Definition 2.2. Let R be any reflexive relation on a nonempty set U. For any subset A # U, the lower and the upper approximations of A according to R are then defined as
RðAÞ ¼ fx 2 U : xR # Ag; RðAÞ ¼ fx 2 U : xR \ A–/g: Obviously, if R is an equivalence relation, xR = [x]R and these definitions are equivalent to the original Pawlak’s definitions, where [x]R is called an equivalence class of x 2 U. We list the properties that are of interest in the theory of rough sets, let A, B # U: L1 : RðAÞ ¼ ½RðAc Þc , where Ac denotes the complement of A in U. L2. R(U) = U. L3. R(A \ B) = R(A) \ R(B). L4. R(A [ B) R(A) [ R(B). L5. A # B ) R(A) # R(B). L6. R(/) = /. L7. R(A) # A. L8 : A # RðRðAÞÞ. L9. R(A) # R(R(A)). L10 : RðAÞ # RðRðAÞÞ. U 1 : RðAÞ ¼ ½RðAc Þc . U 2 : Rð/Þ ¼ /. U 3 : RðA [ BÞ ¼ RðAÞ [ RðBÞ. U 4 : RðA \ BÞ # RðAÞ \ RðBÞ. U 5 : A # B ) RðAÞ # RðBÞ. U 6 : RðUÞ ¼ U. U 7 : A # RðAÞ. U 8 : RðRðAÞÞ # A. U 9 : RðRðAÞÞ # RðAÞ. U 10 : RðRðAÞÞ # RðAÞ. K. R(Ac [ B) # [R(A)]c [ R(B). LU: RðAÞ # RðAÞ.
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Properties L1 and U1 state that two approximations are dual to each other. Hence, properties with the same numbers may be regarded as dual properties. Properties L9, L10, U9 and U10 are expressed in terms of set inclusion. The standard version using set equality can be derived from L1–L10 and U1–U10. For example, it follows from L7 and L9 that R(A) = R(R(A)). It should also be noted that these properties are not independent. With respect to any subset A # U, the universe can be divided into three disjoint regions using the lower and the upper approximations:
BNDðAÞ ¼ RðAÞ RðAÞ; POSðAÞ ¼ RðAÞ; NEGðAÞ ¼ U RðAÞ: An element of the negative region NEG(A) definitely dose not belong to A, an element of the positive region POS(A) definitely belongs to A, and an element of the boundary region BND(A) possibly belongs to A. An accuracy measure of the set A # U according to any reflexive relation R is defined as:
lR ðAÞ ¼
jRðAÞj jRðAÞj
;
where j j denotes the cardinality of the set. As one can notice, 0 6 lR(A) 6 1. If A is definable in U then lR(A) = 1, if A is undefinable in U then lR(A) < 1. Proposition 2.1. For any reflexive relation R on a nonempty set U and for every A, B # U the properties L1–L7, U1–U7, K, and LU hold according to Definition 2.2. Proof. (L1–L5 and K) see [2] (L6) Since R is a reflexive relation on U, then x 2 xR for all x 2 U, also there does not exists x 2 U such that xR # /, hence R(/) = /. (L7) Assume that x 2 R(A), then xR # A. Since R is a reflexive relation on U, then x 2 xR for all x 2 U and there does not exists y 2 U A such that yR # A. Thus x 2 A and so R(A) # A. We can prove U1–U7 as the same as L1–L7. Also we can prove LU from L7 and U7. h The following example shows that the inverse in L4, L7, U4, U7 and K in Proposition 2.1 is not true in general. Example 2.1. Let R = {(a, a), (b, b), (c, c), (d, d), (a, b), (a, c), (b, c), (c, d), (d, b)} be a reflexive relation on a set U = {a, b, c, d}. Then aR = {a, b, c}, bR = {b, c}, cR = {c, d} and dR = {b, d}. If A = {b, c} and B = {c, d}, then RðAÞ ¼ fbg; RðBÞ ¼ fcg; RðA [ BÞ ¼ fb; c; dg; RðAÞ ¼ RðBÞ ¼ U; RðA \ BÞ ¼ fa; b; cg; RðAc [ BÞ ¼ fcg and [R(A)]c [ R(B) = {a, c, d}, ) R(A [ B) – R(A) [ R(B), RðA \ BÞ–RðAÞ\ RðBÞ; RðAc [ BÞ–½RðAÞc [ RðBÞ; RðAÞ–A and RðAÞ–A.
Remark. Let R be any reflexive relation, then for every A # U the properties L8–L10 and U8–U10 do not hold. Example 2.2 (Examples of the above properties do not hold). In Example 2.1, if A = {a, b}, then RðAÞ ¼ fa; b; dg and RðRðAÞÞ ¼ fdg, hence A RðRðAÞÞ and RðAÞ RðRðAÞÞ, ) L8 and L10 do not hold. If A = {b, c}, then R(A) = {b} and R(R(A)) = /, so R(A)6 # R(R(A)), ) L9 do not hold. If A = {a, b, c}, then R(A) = {b, c} and RðRðAÞÞ ¼ U, thus RðRðAÞÞ A and RðRðAÞÞ RðAÞ; ) U 8 ; U 10 do not hold. Finally, if A = {a, d}, then RðAÞ ¼ fa; c; dg and RðRðAÞÞ ¼ U, therefore RðRðAÞÞ RðAÞ; ) U 9 do not hold. Definition 2.3. Let R be any reflexive relation on a nonempty set U. For any subset A # U, the lower and the upper approximations of A according to R are then defined as
R0 ðAÞ ¼
[ fxR : xR # Ag;
R0 ðAÞ ¼ ½R0 ðAc Þc : And so the accuracy measure will be defined as:
l0R ðAÞ ¼
jR0 ðAÞj jR0 ðAÞj
:
Proposition 2.2. For any reflexive relation R on a nonempty set U and for every A, B # U the properties L1, L2, L4–L7, L9, U1, U2, U4– U7, U9, and LU hold according to Definition 2.3.
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Proof ((L1, L4–L7, and L9) see [2]). S (L2) Since for all x 2 U, we get x 2 xR # {xR: xR # U} = R0 (U), then U # R0 (U), but R0 (U) # U, therefore U = R0 (U). We can prove U1, U2, U4–U7 and U9 as the same as L1, L2, L4–L7 and L9. Also we can prove LU from L7 and U7. h The following example shows that the inverse in L4, L7, U4 and U7 in Proposition 2.2 is not true in general. Example 2.3. In Example 2.1, if A = {a, b} and B = {c, d}, then R0 ðAÞ ¼ /; R0 ðBÞ ¼ fc; dg; R0 ðA [ BÞ ¼ U; R0 ðAÞ ¼ fa; bg; R0 ðBÞ ¼ U and R0 ðA \ BÞ ¼ /; ) R0 (A [ B) – R0 (A) [ R0 (B) and R0 ðA \ BÞ–R0 ðAÞ \ R0 ðBÞ. If A = {a, b, d}, then R0 (A) = {b, d} and R0 ðAÞ ¼ U, hence R0 ðAÞ–A–R0 ðAÞ.
Remark. Let R be any reflexive relation, then for every A, B # U the properties L3, L8, L10, U3, U8, U10 and K do not hold. Example 2.4 (Examples of the above properties do not hold). In Example 2.1, if A = {b, c} and B = {c, d}, then R0 (A) = {b, c}, R0 (B) = {c, d} and R0 (A \ B) = /, so R0 (A \ B) – R0 (A) \ R0 (B), ) L3 do not hold.If A = {b} and B = {c}, then R0 ðAÞ ¼ fa; bg, R0 ðBÞ ¼ fa; cg and R0 ðA [ BÞ ¼ U, hence R0 ðA [ BÞ–R0 ðAÞ [ R0 ðBÞ; ) U3 do not hold.If A = {a, b}, then R0 ðAÞ ¼ fa; bg and R0 ðR0 ðAÞÞ ¼ /; ) A R0 ðR0 ðAÞÞ and R0 ðAÞ R0 ðR0 ðAÞÞ; ) L8 and L10 do not hold.If A = {b, c}, then R0 (A) = {b, c} and R0 ðR0 ðAÞÞ ¼ U; ) R0 ðR0 ðAÞÞ A and R0 ðR0 ðAÞÞ R0 ðAÞ; ) U 8 and U10 do not hold.If A = {a, b, d} and B = {a, b}, then R0 (A) = {b, d}, R0 (B) = / andR0 (Ac [ B) = {a, b, c}, hence [R0 (A)]c [ R0 (B) = {a, c}, thus, R0 (Ac [ B)6 # [R0 (A)]c [ R0 (B), ) K do not hold. Definition 2.4. Let R be any reflexive relation on a nonempty set U. For any subset A # U, the lower and the upper approximations of A according to R are then defined as
R00 ðAÞ ¼
[ fxR : xR \ A–/g;
R00 ðAÞ ¼ ½R00 ðAc Þc : And so the accuracy measure will be defined as:
l00R ðAÞ ¼
jR00 ðAÞj jR00 ðAÞj
:
Proposition 2.3. For any reflexive relation R on a nonempty set U and for every A; B # U the properties L1–L8, U1–U8, K and LU hold according to Definition 2.4. Proof. (U1–U5) see [2] (U6) Obviously, R00 ðUÞ # U. For all x 2 U, we have x 2 xR, then U # R00 ðUÞ and so R00 ðUÞ ¼ U. (U7) Since R is reflexive, then x 2 xR for all x 2 A, hence A # R00 ðAÞ. (U8) Assume that y 2 R00 ðR00 ðAÞÞ, then there exists x 2 U such that y 2 xR and xR \ R00 (A) – /, hence there exists z 2 U such that z 2 xR and z 2 R00 (A), so z R R00 ðAc Þ, thus xR \ Ac = /, then xR # A, hence y 2 A, therefore R00 ðR00 ðAÞÞ # A.. (K) Assume that y 2 R00 ðAc [ BÞ ¼ ½R00 ðAc [ BÞc c , then y R R00 ððAc [ BÞc , so for all x 2 U such that y 2 xR we get xR \ (Ac [ B)c = /, that implies xR # (Ac [ B), then xR \ Ac – / or xR \ B – /, hence y 2 R00 ðAc Þ or y R R00 ðBc Þ, thus y R R00 (A) or y 2 R00 (B) and so y 2 [R00 (A)]c [ R00 (B). Therefore R00 (Ac [ B) # [R00 (A)]c [ R00 (B). We can prove L1–L8 as the same as U1–U8. Also we can prove LU from L7 and U7. h The following example shows that the inverse in L4, L7, L8, U4, U7 and U8 in Proposition 2.3 is not true in general. Example 2.5. In Example 2.1, if A = {b, c} and B = {c, d}, then R00 (A) = /, R00 (B) = /, R00 (A [ B) = {d}, ) R00 (A [ B) – R00 (A) [ R00 (B). If A = {a, b} and B = {a, d}, then R00 ðAÞ ¼ R00 ðBÞ ¼ U and R00 ðA \ BÞ ¼ fa; b; cg; ) R00 ðA \ BÞ–R00 ðAÞ \ R00 ðBÞ. If A = {a, b}, then R00 ðAÞ ¼ /; R00 ðAÞ ¼ U; R00 ðR00 ðAÞÞ ¼ U and R00 ðR00 ðAÞÞ ¼ /; ) R00 ðAÞ–A–R00 ðAÞ; A–R00 ðR00 ðAÞÞ and A–R00 ðR00 ðAÞÞ.
Remark. Let R be any reflexive relation, then for every A # U the properties L9, L10, U9 and U10 do not hold. Example 2.6 (Examples of the above properties do not hold). In Example 2.1., If A = {a, b, c}, then R00 (A) = {a}, R00 (R00 (A)) = / and R00 ðR00 ðAÞÞ ¼ fa; b; cg; ) R00 ðAÞ R00 ðR00 ðAÞÞ and R00 ðR00 ðAÞÞ R00 ðAÞ; ) L9 and U10 do not hold. If A = {d}, then R00 ðAÞ ¼ fb; c; dg; R00 ðR00 ðAÞÞ ¼ U and R00 ðR00 ðAÞÞ ¼ fdg, so R00 ðR00 ðAÞÞ R00 ðAÞ and R00 ðAÞ R00 ðR00 ðAÞÞ; ) U 9 and L10 do not hold.
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3. Our approaches to generalized rough sets In this section, we introduce three types of generalized rough approximations with respect to any reflexive relation. All of those approximations are based on the intersection of the right neighborhoods. We can extend the Pawlak’s definitions of a rough set to any reflexive relation R as follows: Definition 3.1 [1]. For the pair (U, R), a set hpiR is the intersection of all right neighborhoods containing p, i.e.,
8 > < \p2xR ðxRÞ if 9x : p 2 xR; hpiR ¼ / otherwise: > :
Lemma 3.1 [1]. For any binary relation R on U if x 2 hyiR, then hxiR # hyiR. Proof. Let z 2 hxiR = \x2wR(wR). Then z is contained in every wR which contains x, and since also x is contained in every uR which contains y, then z is contained in every uR which contains y, i.e., z 2 hyiR. Then hxiR # hyiR. Definition 3.2. Let R be any reflexive relation on a nonempty set U. For any subset A # U, the lower and the upper approximations of A according to R are then defined as
aprðAÞ ¼ fx 2 U : hxiR # Ag; aprðAÞ ¼ fx 2 U : hxiR \ A–/g: Obviously, if R is an equivalence relation, hxi R = [x]R and these definitions are equivalent to the original Pawlak’s definitions. By using Definition 3.2 we can define the accuracy measure of any set A as the following:
lapr ðAÞ ¼
japrðAÞj : japrðAÞj
Proposition 3.1. For any reflexive relation R on a nonempty set U and for every A, B # U the properties L1–L7, L9, U1–U7, U9, K, and LU hold according to Definition 3.2. Proof c
c
c
c
c
(L1) ½aprðA Þ ¼ fx 2 U : hxiR \ A –/g ¼ fx 2 U : hxiR \ A ¼ /g ¼ fx 2 U : hxiR # Ag ¼ aprðAÞ: (L2) Since hxiR # U for all x 2 U, hence x 2 apr(U). Then U # apr(U). Also, since apr(U) # U. Thus, apr(U) = U. (L3) aprðA \ BÞ ¼ fx 2 U : hxiR # A \ Bg ¼ fx 2 U : hxiR # A ^ hxiR # Bg ¼ fx 2 U : hxiR # Ag \ fx 2 U : hxiR # Bg
¼ aprðAÞ \ aprðBÞ: (L4) Assume that x 2 apr(A) [ apr(B), then x 2 apr(A) or x 2 apr(B), hence hxiR # A or hxiR # B, so hxiR # A [ B, thus x 2 apr(A [ B), therefore apr(A [ B) apr(A) [ apr(B). (L5) Let A # B and x 2 apr(A), then hxiR # A and so hxi R # B, hence x 2 apr(B). Thus apr(A) # apr(B). (L6) Since R is a reflexive relation on U, then x 2 hxiR for all x 2 U, also there does not exists x 2 U such that hxiR # /, hence apr(/) = /. (L7) Assume that x 2 apr(A), then hxiR # A. Since R is a reflexive relation on U, then x 2 hxiR for all x 2 U and there does not exists y 2 U A such that hyiR # A. Thus x 2 A and so apr(A) # A. (L9) Let x 2 apr(A), we will show that hxiR # apr(A). Since x 2 apr(A), then hxiR # A. Let y 2 hxiR then hyiR # h xiR for all y 2 hxiR, hence h yiR # A for all y 2 hxiR. Thus y 2 apr(A) for all y 2 hxiR and so hxiR # apr(A), that implies x 2 apr(apr(A)). Hence, apr(A) # apr(apr(A)).(K) Assume that x R [apr(A)]c [ apr(B), then x R [apr(A)]c and x R apr(B), hence x 2 apr(A) and x R apr(B), that implies hxiR # A and hxi R6 # B, then hxiR6 # Ac and h xiR6 # B, hence hxiR6 # Ac [ B, thus x R apr(Ac [ B), therefore apr(Ac [ B) # [apr(A)]c [ apr(B). We can prove U1–U7 and U9 as the same as L1–L7 and L9. Also we can prove LU from L7 and U7. h The following example shows that the inverse in L4, L7, U4 and U7 in proposition 3.1 is not true in general. Example 3.1. In Example 2.1, we have haiR = {a, b, c}, h biR = {b}, hciR = {c} and hdi R = {d}. If A = {a, b} and B = {c}, then aprðAÞ ¼ fbg; aprðBÞ ¼ fcg; aprðAÞ ¼ fa; bg; aprðBÞ ¼ fa; cg; aprðA [ BÞ ¼ fa; b; cg and aprðA \ BÞ ¼ /; ) aprðA [ BÞ–aprðAÞ[ aprðBÞ and aprðA \ BÞ–aprðAÞ \ aprðBÞ. If A = {a}, then apr(A) = / and so A – apr(A). If A = {b}, then aprðAÞ ¼ fa; bg and so A–aprðAÞ.
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Remark. Let R be any reflexive relation, then for every A # U the properties L8, L10, U8, and U10 do not hold. Example 3.2. (Examples of the above properties do not hold). From Examples 2.1 and 3.1, if A = {a, b}, then aprðAÞ ¼ fa; bg and aprðaprðAÞÞ ¼ fbg, hence A aprðaprðAÞÞ and aprðAÞ aprðaprðAÞÞ; ) L8 and L10 do not hold. If A = {b}, then apr(A) = {b} and aprðaprðAÞÞ ¼ fa; bg, hence aprðaprðAÞÞ A and aprðaprðAÞÞ aprðAÞ, ) U8 and U10 do not hold. Definition 3.3. Let R be any reflexive relation on a nonempty set U. For any subset A # U, the lower and the upper approximations of A according to R are then defined as
[ fhxiR : hxiR # Ag;
apr0 ðAÞ ¼
apr0 ðAÞ ¼ ½apr0 ðAc Þc : By using Definition 3.3 we can define the accuracy measure of any set A as the following:
l0apr ðAÞ ¼
japr0 ðAÞj japr0 ðAÞj
:
Proposition 3.2. For any reflexive relation R on a nonempty set U and for every A, B U the properties L1–L7, L9, U1–U7, U9, K, and LU hold according to Definition 3.3. Proof (L1) ½apr0 ðAc Þc ¼ ½fapr 0 ðAc Þc gc c ¼ apr0 ðAÞ. S (L2) Since for all x 2 U, we get x 2 hxiR # {hxiR:hxi R # U} = apr0 (U), then U # apr0 (U), but apr0 (U) # U, therefore 0 U = apr (U). (L3) Assume that y 2 apr0 (A \ B), there exists x 2 U such that y 2 hxi R # (A \ B), so y 2 hxiR # A and y 2 hxiR # B, then y 2 apr0 (A) \ apr0 (B), therefore apr0 (A \ B) # apr0 (A) \ apr0 (B). Conversely, if y 2 apr0 (A) \ apr0 (B), then y 2 apr0 (A) and y 2 apr0 (B), there exist x1, x2 2 U such that y 2 hx1iR # A and y 2 h x2iR # B, hence y 2 hyi R # hx1iR # A and y 2 hyi R # hx2iR # B, so y 2 hyi R # (A \ B), thus y 2 apr0 (A \ B), therefore apr0 (A) \ apr0 (B) # apr0 (A \ B). (L4) Assume that y 2 apr0 (A) [ apr0 (B), then y 2 apr0 (A) or y 2 apr0 (B), hence there exist x1 or x2 2 U such that y 2 hx1iR # A or y 2 hx2iR # B, so y 2 hyi R # A [ B, thus y 2 apr0 (A [ B), therefore apr0 (A [ B) apr0 (A) [ apr0 (B). (L5) Let A # B and y 2 apr0 (A), then there exists x 2 U such that y 2 hxiR # A, so y 2 hxi R # B, hence y 2 apr0 (B), therefore apr0 (A) # apr0 (B). (L6) Since for all x 2 U we get hxi R – /, thus apr0 (/) = /. S (L7) Since apr0 (A) = {h xiR:hxiR # A}, then apr0 (A) # A. S (L9) Assume that y 2 apr0 (A), then there exists x 2 U such that y 2 hxiR # A, hence y 2 hxiR # {hxi R:hxiR # A} # A, so y 2 h xiR # apr0 (A) # A, then y 2 apr0 (apr0 (A)), therefore apr0 (A) # apr0 (apr0 (A)).(K) Assume that y 2 apr0 (Ac [ B), there exists x 2 U such that y 2 hxiR # (Ac [ B), so y 2 hyiR # (Ac) or y 2 h yiR # (B), then y 2 hyiR6 # (A) or y 2 apr0 (B), hence y R apr0 (A) or y 2 apr0 (B), that implies y 2 [apr0 (A)]c [ apr0 (B). Therefore, apr0 (Ac [ B) # [apr0 (A)]c [ apr0 (B ). We can prove U1–U7 and U9 as the same as L1–L7 and L9. Also we can prove LU from L7 and U7. h The following example shows that the inverse in L4, L7, U4 and U7 in Proposition 3.2 is not true in general. Example 3.3. From Examples 2.1 and 3.1, if A = {a, b} and B = {c, d}, then apr0 ðAÞ ¼ fbg; apr 0 ðBÞ ¼ fc; dg; apr0 ðA [ BÞ ¼ U; apr0 ðAÞ ¼ fa; bg; apr0 ðBÞ ¼ fa; c; dg and apr0 ðA \ BÞ ¼ /; ) apr 0 ðA [ BÞ–apr 0 ðAÞ [ apr0 ðBÞ and apr0 ðA \ BÞ–apr0 ðAÞ \ apr 0 ðBÞ. If A = {a, b}, then apr0 (A) = {b} and so apr0 (A) – A. If A = {c, d}, then apr0 ðAÞ ¼ fa; c; dg, hence A–apr 0 ðAÞ. Remark. Let R be any reflexive relation, then for every A, B U the properties L8, L10, U8, and U10 do not hold. Example 3.4. (Examples of the above properties do not hold). From Examples 2.1 and 3.1, if A = {a, b}, then apr 0 ðAÞ ¼ fa; bg and apr0 ðapr0 ðAÞÞ ¼ fbg, hence A apr 0 ðapr 0 ðAÞÞ and apr0 ðAÞ apr 0 ðapr0 ðAÞÞ; ) L8 and L10 do not hold. If A = {b}, then apr0 (A) = {b} and apr0 ðapr 0 ðAÞÞ ¼ fa; bg, thus apr0 ðapr 0 ðAÞÞ A and apr0 ðapr 0 ðAÞÞ apr 0 ðAÞ; ) U 8 and U10 do not hold. Definition 3.4. Let R be any reflexive relation on a nonempty set U. For any subset A # U, the lower and the upper approximations of A according to R are then defined as
apr00 ðAÞ ¼
[ fhxiR : hxiR \ A–/g;
apr00 ðAÞ ¼ ½apr00 ðAc Þc :
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By using Definition 3.4 we can define the accuracy measure of any set A as the following:
l00apr ðAÞ ¼
japr00 ðAÞj japr00 ðAÞj
:
Proposition 3.3. For any reflexive relation R on a nonempty set U and for every A, B # U the properties L1–L8, U1–U8, K, and LU hold according to Definition 3.4. Proof (U1) ½apr00 ðAc Þc ¼ ½fapr00 ðAc Þc gc c ¼ apr00 ðAÞ. (U2) Since for all x 2 U we get hxi R \ / = /, thus apr 00 ð/Þ ¼ /. (U3)
apr00 ðA [ BÞ ¼
[
fhxiR : hxiR \ ðA [ BÞ–/g ¼
[ [ [ fhxiR : ðhxiR \ AÞ [ ðhxiR \ BÞ–/g ¼ fhxiR : hxiR \ A–/g or fhxiR
: hxiR \ B–/g ¼ apr00 ðAÞ [ apr00 ðBÞ: (U4) Assume that y 2 apr 00 ðA \ BÞ, then there exists x 2 U such that y 2 hxiR and hxiR \ (A \ B) – /, so hxi R \ A – / and hxiR \ B – /, hence y 2 apr00 ðAÞ and y 2 apr 00 ðBÞ, thus y 2 apr00 ðAÞ \ apr00 ðBÞ. Therefore, apr00 ðA \ BÞ # apr 00 ðAÞ \ apr 00 ðBÞ. (U5) Assume that A # B and y 2 apr 00 ðAÞ, then there exists x 2 U such that y 2 hxiR and hxiR \ A – /, so hxiR \ B – /, hence y 2 apr00 ðBÞ, therefore apr 00 ðAÞ # apr00 ðBÞ. (U6) Obviously, apr 00 ðUÞ # U. Also, for all x 2 U, we have x 2 hxiR, then U # apr 00 ðUÞ and so apr00 ðUÞ ¼ U. (U7) For all x 2 A, we have x 2 hxi R, then A # apr00 ðAÞ. (U8) Assume that y 2 apr00 ðapr 00 ðAÞÞ, then there exists x 2 U such that y 2 hxiR and h xiR \ apr00 (A) – /, hence there exists z 2 U such that z 2 hxiR and z 2 apr00 (A), so z R apr00 ðAc Þ, thus hxiR \ Ac = /, then hxiR # A, hence y 2 A, therefore apr00 ðapr 00 ðAÞÞ # A.(K) Assume that y 2 apr 00 ðAc [ BÞ ¼ ½apr 00 ðAc [ BÞc c , then y R apr00 ðAc [ BÞc , so for all x 2 U such that y 2 hxiR we get hxiR \ (Ac [ B)c = /, so y 2 hyiR # h xiR # (Ac [ B), hence hyiR \ Ac – / or hyiR \ B – /, thus y 2 apr00 ðAc Þ or y R apr00 ðBc Þ, then y R apr00 (A) or y 2 apr00 (B), that implies y 2 [apr00 (A)]c [ apr00 (B). Therefore R00 (Ac [ B) # [R00 (A)]c [ R00 (B). We can prove L1–L8 as the same as U1–U8. Also we can prove LU from L7 and U7. The following example shows that the inverse in L4, L7, L8, U4, U7 and U8 in Proposition 3.3 is not true in general. Example 3.5. Let R = {(a, a), (b, b), (c, c), (d, d), (a, b), (b, a), (c, b)} be a reflexive relation on a set U = {a, b, c, d}. Then aR = {a, b}, bR = {a, b}, cR = {b, c} and dR = {d}, ) hai R = {a, b}, hbiR = {b}, hciR = {b, c} and hdiR = {d}. If A = {a, b} and B = {a, c}, then apr 00 ðAÞ ¼ fag; apr00 ðBÞ ¼ /; apr 00 ðA [ BÞ ¼ fa; b; cg; apr 00 ðAÞ ¼ fa; b; cg; apr00 ðBÞ ¼ fa; b; cg; apr00 ðA \ BÞ ¼ fa; bg; apr00 ðapr 00 ðAÞÞ ¼ fa; b; cg and apr 00 ðapr 00 ðBÞÞ ¼ /; ) apr 00 ðA [ BÞ–apr00 ðAÞ [ apr 00 ðBÞ; apr 00 ðA \ BÞ–apr 00 ðAÞ \ apr 00 ðBÞ; apr00 ðAÞ–A–apr 00 ðAÞ; A– 00 00 apr ðapr ðAÞÞ and B–apr 00 ðapr 00 ðBÞÞ.
Remark. Let R be any reflexive relation, then for every A # U the properties L9, L10, U9 and U10 do not hold. Example 3.6. (Examples of the above properties do not hold). In Example 3.5., If A = {a, d}, then apr 00 ðAÞ ¼ fa; b; dg; apr00 ðapr 00 ðAÞÞ ¼ U and apr 00 ðapr00 ðAÞÞ ¼ fa; dg; ) apr 00 ðapr00 ðAÞÞ apr00 ðAÞ and apr 00 ðapr00 ðAÞÞ apr00 ðAÞ; ) U 9 and L10 do not hold. If A = {a, b}, then apr00 (A) = {a}, apr00 (apr00 (A)) = / and apr 00 ðapr00 ðAÞÞ ¼ fa; bg, so apr00 (A)6 # apr00 (apr00 (A)) and apr 00 ðAÞ apr00 ðapr 00 ðAÞÞ; ) L9 and U10 do not hold. In Table 1 we summarize the properties of the above definitions of lower and upper approximation operators. Remark. For any reflexive relation R on a finite nonempty set U, we observe that:
L9 and U9 hold according to Definition 3.2 but it does not hold according to Definition 2.2. L3, U3 and K hold according to Definition 3.3 but it does not hold according to Definition 2.3. Definition 2.4 is the same as Definition 3.4. 4. The relationship between the two approaches For the sake of illustration, we study the relationship between Definitions 2.2 and 3.2. Below we present a simple example, using a nonsymmetric and nontransitive relation R, where each definition gives a different result. We can find the relationship between the lower (upper) approximation in Definitions 2.2 and 3.2 as follows:
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Table 1 The comparison between the properties of rough sets depending on Definitions 2.2, 2.3, 2.4, Definitions 3.2, 3.3 and 3.4 by using any reflexive relation R. A cross () indicates that the property is satisfied. Property
Def. 2.2
Def. 2.3
Def. 2.4
Def. 3.2
Def. 3.3
Def. 3.4
L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 K LU
Lemma 4.1. For any reflexive relation R on a nonempty set U and for every A # U, then R(A) # apr(A) and aprðAÞ # RðAÞ. Proof. Assume that a 2 R(A), then aR # A and so haiR # A, hence a 2 apr(A), therefore R(A) # apr(A). Also, suppose that a 2 aprðAÞ, then hai R \ A – /, hence aR \ A – /, and so a 2 RðAÞ, that implies aprðAÞ # RðAÞ.
h
Lemma 4.2. For any transitive relation R on a nonempty set U. If x 2 yR, then xR # yR. Proof. Assume that x 2 yR and z 2 xR, then yRx and xRz, hence by transitive yRz, and so z 2 yR, that implies xR # yR. h The following theorem gives us the relationship between the right neighborhood of any element x 2 U and the intersection of the right neighborhoods containing this element. Theorem 4.1. For any reflexive and transitive relation R on a nonempty set U. xR = hxiR for all x 2 U. Proof. The proof is immediately derived from Lemma 4.2.
h
In Table 2 we will use the relation R which comes from the Example 2.1 to compare the properties of the rough sets depending on Definitions 2.2 and 3.2. From Lemma 4.1 and Theorem 4.1 we can introduce the final theorem as follows. Theorem 4.2. For any reflexive relation R on a nonempty set U, then
POSR(A) # POSapr(A). NEGR(A) # NEGapr(A). BNDapr(A) # BNDR(A). lR(A) 6 lapr(A). If R is transitive, then our approach is equivalent to Yao’s approach.
One can study the relationship between Definition 2.3 (Definition 2.4) and Definition 3.3 (Definition 3.4). Also, one can illustrate the differences between them on the analogical example to the one which was showed above. It should also be noted that the family of neighborhoods produced by a similarity relation forms coverings of the universe. This property relates this study to many recent studies on the covering-based rough set model (see [29,31,32]). Abu-Donia [2] generalized the rough set approximations by using a family of binary relations. If this family consists of exactly single relation, then this generalization of approximations is equivalent to the Yao’s approaches. Additionally, our
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Table 2 The comparison between the boundary of the set and the accuracy measure of the set depending on Definitions 2.2 and 3.2 by using a reflexive relation R. A
R
R
apr
apr
BNDR
BNDapr
lR
lapr
{a} {b}
/ /
{a} {a, b, d}
/ {b}
{a} {a, b}
{a} {a, b, d}
{a} {a}
0 0
0
{c}
/
{a, b, c}
{c}
{a, c}
{a, b, c}
{a}
0
{d} {a, b}
/ /
{c, d} {a, b, d}
{d} {b}
{d} {a, b}
{c, d} {a, b, d}
/ {a}
0 0
{a, c}
/
{a, b, c}
{c}
{a, c}
{a, b, c}
{a}
0
{a, d}
/
{a, c, d}
{d}
{a, d}
{a, c, d}
{a}
0
{b, c}
{b}
U
{b, c}
{a, b, c}
{a, c, d}
{a}
{b, d}
{d}
U
{b, d}
{a, b, d}
{a, b, c}
{a}
{c, d}
{c}
U
{c, d}
{a, c, d}
{a, b, d}
{a}
{a, b, c}
{a, b}
U
{a, b, c}
{a, b, c}
{c, d}
/
{a, b, d}
{d}
U
{b, d}
{a, b, d}
{a, b, c}
{a}
{a, c, d}
{c}
U
{c, d}
{a, c, d}
{a, b, d}
{a}
{b, c, d}
{b, c, d}
U
{b, c, d}
U
{a}
{a}
1 4 1 4 1 4 1 2 1 4 1 4 3 4
1 2 1 2
1 1 2 1 2 1 2 2 3 2 3 2 3
1 2 3 2 3 3 4
approach and those generalizations are independent. Moreover, if the relations of this family are equivalence relations, then all generalizations of approximations are equivalent to the original Pawlak’s definitions. 5. Conclusion The theory of rough sets is typically studied based on the notion of an approximation space and the induced lower and upper approximations of subsets of a universe. In this paper, we introduce new definitions of lower and upper approximations, which are basic concepts of the rough set theory. These definitions follow naturally from the base of topological space {hxiR: x 2 U}. In the study, we use a reflexive relation in order to get hxiR – /. These new definitions, which differ from the classical definitions of approximations, are the only ones to properly characterize the set of lower approximation and the set of upper approximation when the similarity relation is reflexive, but not necessarily symmetric and transitive. This is why we suggest using our new definitions even when the relation is symmetric and transitive. The work presented in this paper utilized the approximations generated by similarity relation to get a specific type of the right neighborhoods and to apply them in order to get more accurate approximations. This approach is more general than that of Yao’s. Acknowledgements The author thanks the anonymous referees and the Editor-in-Chief, Professor Witold Pedrycz for their valuable suggestions in improving this paper. References [1] E.A. Abo-Tabl, Some Topological Concepts and their Applications from the Relations Point of View, PhD. Thesis-Assiut University, Egypt, 2008. [2] H.M. Abu-Donia, Comparison between different kinds of approximations by using a family of binary relations, Knowledge-Based Systems 21 (2008) 911–919. [3] W. Bartol, J. Miro, K. Pioro, F. Rossello, On the coverings by tolerance classes, Information Sciences 166 (14) (2004) 193–211. [4] Z. Bonikowski, E. Bryniarski, U. Wybraniec-Skardowska, Extensions and intentions in the rough set theory, Information Sciences 107 (1998) 149–167. [5] K. Chakrabarty, R. Biswas, S. Nanda, Fuzziness in rough sets, Fuzzy Sets and Systems 110 (2000) 247–251. [6] J. Dai, Rough 3-valued algebras, Information Sciences 178 (8) (2008) 1986–1996. [7] U. Hoehle, Quotients with respect to similarity relations, Fuzzy Sets and Systems 27 (1988) 31–44. [8] M. Kryszkiewicz, Rough set approach to incomplete information systems, Information Sciences 112 (1998) 39–49. [9] T.Y. Lin, Q. Liu, Rough approximate operators: axiomatic rough set theory, in: W. Ziarko (Ed.), Rough sets, Fuzzy Sets and Knowledge Discovery, Springer, Berlin, 1994, pp. 256–260. [10] G.L. Liu, Y. Sai, A comparison of two types of rough sets induced by coverings, International Journal of Approximate Reasoning 50 (2009) 521–528. [11] G.L. Liu, W. Zhu, The algebraic structures of generalized rough set theory, Information Sciences 178 (21) (2008) 4105–4113. [12] H.T. Nguyen, Some mathematical structures for computational information, Information Sciences 128 (2000) 67–89. [13] Z. Pawlak, Rough sets, International Journal of Computer and Information Science 11 (1982) 341–356. [14] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data, Kluwer Academic Publishers, Boston, 1991. [15] Z. Pawlak, A. Skowron, Rudiments of rough sets, Information Sciences 177 (1) (2007) 3–27. [16] Z. Pawlak, A. Skowron, Rough sets: some extensions, Information Sciences 177 (1) (2007) 28–40. [17] Z. Pawlak, A. Skowron, Rough sets and Boolean reasoning, Information Sciences 177 (1) (2007) 41–73. [18] J.A. Pomykala, Approximation operations in approximation space, Bulletin of the Polish Academy of sciences: Mathematics 35 (1987) 653–662. [19] M. Quafafou, A-RST: a generalization of rough set theory, Information Sciences 124 (2000) 301–316. [20] A. Skowron, J. Stepaniuk, Tolerance approximation spaces, Fundamenta Informaticae 27 (1996) 245–253. [21] R. Slowinski, D. Vanderpooten, Similarity relation as a basis for rough approximations, in: P.P. Wang (Ed.), Advances in machine intelligence and softcomputing, Department of Electrical Engineering, Duke University, Durham, NC. USA, 1997, pp. 17–33. [22] R. Slowinski, D. Vanderpooten, A generalized definition of rough approximations based on similarity, IEEE Transactions on Knowledge and Data Engineering 12 (2) (2000) 331–336.
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E.A. Abo-Tabl / Information Sciences 181 (2011) 2587–2596 A. Tversky, Features of Similarity, Psychological Review 84 (4) (1977) 327–352. U. Wybraniec-Skardowska, On a generalization of approximation space, Bulletin of the Polish Academy of Sciences: Mathematics 37 (1989) 51–61. Y.Y. Yao, Constructive and algebraic methods of theory of rough sets, Information Sciences 109 (1998) 21–47. Y.Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences 111 (1–4) (1998) 239– 259. Y.Y. Yao, T.Y. Lin, Generalization of rough sets using modal logic, Intelligent Automation and Soft Computing, An International Journal 2 (1996) 103– 120. L.A. Zadeh, Similarity relations and fuzzy orderings, Information Sciences 3 (4) (1971) 177–200. W. Zhu, Topological approaches to covering rough sets, Information Sciences 177 (2007) 1499–1508. W. Zhu, Generalized rough sets based on relations, Information Sciences 177 (22) (2007) 4997–5011. W. Zhu, Relationship between generalized rough sets based on binary relation and covering, Information Sciences 179 (2009) 210–225. W. Zhu, Relationship among basic concepts in covering-based rough sets, Information Sciences 179 (14) (2009) 2478–2486. W. Zhu, F.-Y. Wang, Reduction and axiomization of covering generalized rough sets, Information Sciences 152 (1) (2003) 217–230. W. Ziarko, Variable precision rough set model, Journal of Computer and System Sciences 46 (1993) 39–59.
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Information Sciences journal homepage: www.elsevier.com/locate/ins
A comparison of two kinds of definitions of rough approximations based on a similarity relation E.A. Abo-Tabl Mathematics Department, Faculty of Science, Assiut University, Assiut, Egypt Department of Mathematics, Faculty of Science and Arts, Al-mithnab, Qassim University, P.O. Box 931, Buridah 51931, Al-mithnab, Kingdom of Saudi Arabia
a r t i c l e
i n f o
Article history: Received 23 March 2009 Received in revised form 22 December 2010 Accepted 2 January 2011 Available online 21 January 2011 Keywords: Rough sets Lower and upper approximations Similarity relation Accuracy measure
a b s t r a c t In the present paper, we investigate the three types of Yao’s lower and upper approximations of any set with respect to any similarity relation. These types based on a right neighborhood. Also, we define and investigate other three types of approximations for any similarity relations. These new types based on the intersection of the right neighborhoods. Moreover, we give a comparison between these types. Lastly, the relationship between these definitions is introduced. Ó 2011 Published by Elsevier Inc.
1. Introduction The theory of rough sets, proposed by Pawlak [13–17], is an extension of the set theory for the study of intelligent systems characterized by insufficient and incomplete information. The lower and the upper approximation operators are constructed by using a binary relation on the universe. Using the concepts of the lower and the upper approximations from the rough set theory, knowledge hidden in information systems may be unraveled and expressed in the form of decision rules [8,14,34]. There are at least two approaches for the development of the rough set theory, namely the constructive and the axiomatic approaches. In the constructive approach, binary relations on the universe, partitions of the universe, neighborhood systems, and Boolean algebras are all the primitive notions. The lower and the upper approximation operators are constructed by means of these notions [5,8,12–14,18,19,21,24–27,34]. The constructive approach is suitable for practical applications of the rough sets. On the other hand, the axiomatic approach, which is appropriate for studying the structures of rough set algebras, takes the lower and the upper approximation operators as primitive notions. In this approach, a set of axioms is used to characterize approximation operators that are the same as the ones produced using the constructive approach [9,25]. The foundations as well as a majority of the recent studies on the rough sets are focused on the constructive approaches. The classical rough approximations are based on equivalence relations, but this requirement is not satisfied in some situations. Thus the classical rough approximations has been extended to the similarity relation based rough sets [22], the tolerance relation based rough sets [20], the arbitrary binary relation based rough sets [11,25,26,30,31] and the covering-based rough sets [3,4,6,10,29,32,33]. Suppose we are given a finite nonempty set U of objects, called the universe. Indiscernibility reflects a total impossibility of distinguishing between objects considering the available information. This situation can be represented using a binary relation R defined on U, which is reflexive, symmetric, and transitive. Objects of U can be partitioned into indiscernibility (or equivalence) classes, which form the basic granules of the knowledge available through R. It is natural to extend E-mail address: [email protected] 0020-0255/$ - see front matter Ó 2011 Published by Elsevier Inc. doi:10.1016/j.ins.2011.01.007
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the indiscernibility concept to take account of situations where objects are not significantly distinct. This happens when the objects are described by imprecise data or when small differences are meaningless in the context of the study. This situation can be modeled using a binary relation R defined on U, which represents a certain form of similarity. The similarity or the tolerance relations have been studied extensively (see, e.g., [7], [28]). A basic difference between the indiscernibility and the similarity relations is that, in general, the similarity relations do not give rise to a partition of the set of objects. Information about similarity can be represented using the similarity classes for each object x 2 U. The reflexivity property cannot be relaxed since, as any object is trivially indiscernible with itself, it is, a fortiori, similar to itself. The most controversial property is symmetry. Most authors dealing with the similarity relations do impose this property. Notice, however, that the statement xRy, which means ‘‘y is similar to x’’, is directional; it has a subject y and a referent x and it is not equivalent, in general, to the statement ‘‘x is similar to y’’ as argued by Tversky [23]. For example, in the following statement: ‘‘a son resembles his father,’’ the son is the subject and the father is the referent. The inverse statement usually makes much less sense. We first investigate and study in Section 2 the three types of Yao’s lower and upper approximations of any set with respect to any similarity relation, which are based on the concept of right neighborhood. Moreover, we define and investigate other three types of approximations for any similarity relations, which are based on the intersection of the right neighborhoods. In addition, we give a comparison between all these definitions and show these comparisons in a table (Section 3). In Section 4, a simple example which show the differences between all definitions of rough approximation is then introduced. At last, some conclusion is presented in Section 5.
2. Yao’s approach of generalized rough sets Since reflexivity is a minimal requirement for any type of the similarity relation, we assume in the following that R is reflexive. Yao [25,26] extended the Pawlak’s definitions of a rough set to any similarity relation R as follows: Definition 2.1. For the pair (U, R), the set xR is defined as xR = {y 2 U: xRy} called the right neighborhood of an element x 2 U. Definition 2.2. Let R be any reflexive relation on a nonempty set U. For any subset A # U, the lower and the upper approximations of A according to R are then defined as
RðAÞ ¼ fx 2 U : xR # Ag; RðAÞ ¼ fx 2 U : xR \ A–/g: Obviously, if R is an equivalence relation, xR = [x]R and these definitions are equivalent to the original Pawlak’s definitions, where [x]R is called an equivalence class of x 2 U. We list the properties that are of interest in the theory of rough sets, let A, B # U: L1 : RðAÞ ¼ ½RðAc Þc , where Ac denotes the complement of A in U. L2. R(U) = U. L3. R(A \ B) = R(A) \ R(B). L4. R(A [ B) R(A) [ R(B). L5. A # B ) R(A) # R(B). L6. R(/) = /. L7. R(A) # A. L8 : A # RðRðAÞÞ. L9. R(A) # R(R(A)). L10 : RðAÞ # RðRðAÞÞ. U 1 : RðAÞ ¼ ½RðAc Þc . U 2 : Rð/Þ ¼ /. U 3 : RðA [ BÞ ¼ RðAÞ [ RðBÞ. U 4 : RðA \ BÞ # RðAÞ \ RðBÞ. U 5 : A # B ) RðAÞ # RðBÞ. U 6 : RðUÞ ¼ U. U 7 : A # RðAÞ. U 8 : RðRðAÞÞ # A. U 9 : RðRðAÞÞ # RðAÞ. U 10 : RðRðAÞÞ # RðAÞ. K. R(Ac [ B) # [R(A)]c [ R(B). LU: RðAÞ # RðAÞ.
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Properties L1 and U1 state that two approximations are dual to each other. Hence, properties with the same numbers may be regarded as dual properties. Properties L9, L10, U9 and U10 are expressed in terms of set inclusion. The standard version using set equality can be derived from L1–L10 and U1–U10. For example, it follows from L7 and L9 that R(A) = R(R(A)). It should also be noted that these properties are not independent. With respect to any subset A # U, the universe can be divided into three disjoint regions using the lower and the upper approximations:
BNDðAÞ ¼ RðAÞ RðAÞ; POSðAÞ ¼ RðAÞ; NEGðAÞ ¼ U RðAÞ: An element of the negative region NEG(A) definitely dose not belong to A, an element of the positive region POS(A) definitely belongs to A, and an element of the boundary region BND(A) possibly belongs to A. An accuracy measure of the set A # U according to any reflexive relation R is defined as:
lR ðAÞ ¼
jRðAÞj jRðAÞj
;
where j j denotes the cardinality of the set. As one can notice, 0 6 lR(A) 6 1. If A is definable in U then lR(A) = 1, if A is undefinable in U then lR(A) < 1. Proposition 2.1. For any reflexive relation R on a nonempty set U and for every A, B # U the properties L1–L7, U1–U7, K, and LU hold according to Definition 2.2. Proof. (L1–L5 and K) see [2] (L6) Since R is a reflexive relation on U, then x 2 xR for all x 2 U, also there does not exists x 2 U such that xR # /, hence R(/) = /. (L7) Assume that x 2 R(A), then xR # A. Since R is a reflexive relation on U, then x 2 xR for all x 2 U and there does not exists y 2 U A such that yR # A. Thus x 2 A and so R(A) # A. We can prove U1–U7 as the same as L1–L7. Also we can prove LU from L7 and U7. h The following example shows that the inverse in L4, L7, U4, U7 and K in Proposition 2.1 is not true in general. Example 2.1. Let R = {(a, a), (b, b), (c, c), (d, d), (a, b), (a, c), (b, c), (c, d), (d, b)} be a reflexive relation on a set U = {a, b, c, d}. Then aR = {a, b, c}, bR = {b, c}, cR = {c, d} and dR = {b, d}. If A = {b, c} and B = {c, d}, then RðAÞ ¼ fbg; RðBÞ ¼ fcg; RðA [ BÞ ¼ fb; c; dg; RðAÞ ¼ RðBÞ ¼ U; RðA \ BÞ ¼ fa; b; cg; RðAc [ BÞ ¼ fcg and [R(A)]c [ R(B) = {a, c, d}, ) R(A [ B) – R(A) [ R(B), RðA \ BÞ–RðAÞ\ RðBÞ; RðAc [ BÞ–½RðAÞc [ RðBÞ; RðAÞ–A and RðAÞ–A.
Remark. Let R be any reflexive relation, then for every A # U the properties L8–L10 and U8–U10 do not hold. Example 2.2 (Examples of the above properties do not hold). In Example 2.1, if A = {a, b}, then RðAÞ ¼ fa; b; dg and RðRðAÞÞ ¼ fdg, hence A RðRðAÞÞ and RðAÞ RðRðAÞÞ, ) L8 and L10 do not hold. If A = {b, c}, then R(A) = {b} and R(R(A)) = /, so R(A)6 # R(R(A)), ) L9 do not hold. If A = {a, b, c}, then R(A) = {b, c} and RðRðAÞÞ ¼ U, thus RðRðAÞÞ A and RðRðAÞÞ RðAÞ; ) U 8 ; U 10 do not hold. Finally, if A = {a, d}, then RðAÞ ¼ fa; c; dg and RðRðAÞÞ ¼ U, therefore RðRðAÞÞ RðAÞ; ) U 9 do not hold. Definition 2.3. Let R be any reflexive relation on a nonempty set U. For any subset A # U, the lower and the upper approximations of A according to R are then defined as
R0 ðAÞ ¼
[ fxR : xR # Ag;
R0 ðAÞ ¼ ½R0 ðAc Þc : And so the accuracy measure will be defined as:
l0R ðAÞ ¼
jR0 ðAÞj jR0 ðAÞj
:
Proposition 2.2. For any reflexive relation R on a nonempty set U and for every A, B # U the properties L1, L2, L4–L7, L9, U1, U2, U4– U7, U9, and LU hold according to Definition 2.3.
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Proof ((L1, L4–L7, and L9) see [2]). S (L2) Since for all x 2 U, we get x 2 xR # {xR: xR # U} = R0 (U), then U # R0 (U), but R0 (U) # U, therefore U = R0 (U). We can prove U1, U2, U4–U7 and U9 as the same as L1, L2, L4–L7 and L9. Also we can prove LU from L7 and U7. h The following example shows that the inverse in L4, L7, U4 and U7 in Proposition 2.2 is not true in general. Example 2.3. In Example 2.1, if A = {a, b} and B = {c, d}, then R0 ðAÞ ¼ /; R0 ðBÞ ¼ fc; dg; R0 ðA [ BÞ ¼ U; R0 ðAÞ ¼ fa; bg; R0 ðBÞ ¼ U and R0 ðA \ BÞ ¼ /; ) R0 (A [ B) – R0 (A) [ R0 (B) and R0 ðA \ BÞ–R0 ðAÞ \ R0 ðBÞ. If A = {a, b, d}, then R0 (A) = {b, d} and R0 ðAÞ ¼ U, hence R0 ðAÞ–A–R0 ðAÞ.
Remark. Let R be any reflexive relation, then for every A, B # U the properties L3, L8, L10, U3, U8, U10 and K do not hold. Example 2.4 (Examples of the above properties do not hold). In Example 2.1, if A = {b, c} and B = {c, d}, then R0 (A) = {b, c}, R0 (B) = {c, d} and R0 (A \ B) = /, so R0 (A \ B) – R0 (A) \ R0 (B), ) L3 do not hold.If A = {b} and B = {c}, then R0 ðAÞ ¼ fa; bg, R0 ðBÞ ¼ fa; cg and R0 ðA [ BÞ ¼ U, hence R0 ðA [ BÞ–R0 ðAÞ [ R0 ðBÞ; ) U3 do not hold.If A = {a, b}, then R0 ðAÞ ¼ fa; bg and R0 ðR0 ðAÞÞ ¼ /; ) A R0 ðR0 ðAÞÞ and R0 ðAÞ R0 ðR0 ðAÞÞ; ) L8 and L10 do not hold.If A = {b, c}, then R0 (A) = {b, c} and R0 ðR0 ðAÞÞ ¼ U; ) R0 ðR0 ðAÞÞ A and R0 ðR0 ðAÞÞ R0 ðAÞ; ) U 8 and U10 do not hold.If A = {a, b, d} and B = {a, b}, then R0 (A) = {b, d}, R0 (B) = / andR0 (Ac [ B) = {a, b, c}, hence [R0 (A)]c [ R0 (B) = {a, c}, thus, R0 (Ac [ B)6 # [R0 (A)]c [ R0 (B), ) K do not hold. Definition 2.4. Let R be any reflexive relation on a nonempty set U. For any subset A # U, the lower and the upper approximations of A according to R are then defined as
R00 ðAÞ ¼
[ fxR : xR \ A–/g;
R00 ðAÞ ¼ ½R00 ðAc Þc : And so the accuracy measure will be defined as:
l00R ðAÞ ¼
jR00 ðAÞj jR00 ðAÞj
:
Proposition 2.3. For any reflexive relation R on a nonempty set U and for every A; B # U the properties L1–L8, U1–U8, K and LU hold according to Definition 2.4. Proof. (U1–U5) see [2] (U6) Obviously, R00 ðUÞ # U. For all x 2 U, we have x 2 xR, then U # R00 ðUÞ and so R00 ðUÞ ¼ U. (U7) Since R is reflexive, then x 2 xR for all x 2 A, hence A # R00 ðAÞ. (U8) Assume that y 2 R00 ðR00 ðAÞÞ, then there exists x 2 U such that y 2 xR and xR \ R00 (A) – /, hence there exists z 2 U such that z 2 xR and z 2 R00 (A), so z R R00 ðAc Þ, thus xR \ Ac = /, then xR # A, hence y 2 A, therefore R00 ðR00 ðAÞÞ # A.. (K) Assume that y 2 R00 ðAc [ BÞ ¼ ½R00 ðAc [ BÞc c , then y R R00 ððAc [ BÞc , so for all x 2 U such that y 2 xR we get xR \ (Ac [ B)c = /, that implies xR # (Ac [ B), then xR \ Ac – / or xR \ B – /, hence y 2 R00 ðAc Þ or y R R00 ðBc Þ, thus y R R00 (A) or y 2 R00 (B) and so y 2 [R00 (A)]c [ R00 (B). Therefore R00 (Ac [ B) # [R00 (A)]c [ R00 (B). We can prove L1–L8 as the same as U1–U8. Also we can prove LU from L7 and U7. h The following example shows that the inverse in L4, L7, L8, U4, U7 and U8 in Proposition 2.3 is not true in general. Example 2.5. In Example 2.1, if A = {b, c} and B = {c, d}, then R00 (A) = /, R00 (B) = /, R00 (A [ B) = {d}, ) R00 (A [ B) – R00 (A) [ R00 (B). If A = {a, b} and B = {a, d}, then R00 ðAÞ ¼ R00 ðBÞ ¼ U and R00 ðA \ BÞ ¼ fa; b; cg; ) R00 ðA \ BÞ–R00 ðAÞ \ R00 ðBÞ. If A = {a, b}, then R00 ðAÞ ¼ /; R00 ðAÞ ¼ U; R00 ðR00 ðAÞÞ ¼ U and R00 ðR00 ðAÞÞ ¼ /; ) R00 ðAÞ–A–R00 ðAÞ; A–R00 ðR00 ðAÞÞ and A–R00 ðR00 ðAÞÞ.
Remark. Let R be any reflexive relation, then for every A # U the properties L9, L10, U9 and U10 do not hold. Example 2.6 (Examples of the above properties do not hold). In Example 2.1., If A = {a, b, c}, then R00 (A) = {a}, R00 (R00 (A)) = / and R00 ðR00 ðAÞÞ ¼ fa; b; cg; ) R00 ðAÞ R00 ðR00 ðAÞÞ and R00 ðR00 ðAÞÞ R00 ðAÞ; ) L9 and U10 do not hold. If A = {d}, then R00 ðAÞ ¼ fb; c; dg; R00 ðR00 ðAÞÞ ¼ U and R00 ðR00 ðAÞÞ ¼ fdg, so R00 ðR00 ðAÞÞ R00 ðAÞ and R00 ðAÞ R00 ðR00 ðAÞÞ; ) U 9 and L10 do not hold.
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3. Our approaches to generalized rough sets In this section, we introduce three types of generalized rough approximations with respect to any reflexive relation. All of those approximations are based on the intersection of the right neighborhoods. We can extend the Pawlak’s definitions of a rough set to any reflexive relation R as follows: Definition 3.1 [1]. For the pair (U, R), a set hpiR is the intersection of all right neighborhoods containing p, i.e.,
8 > < \p2xR ðxRÞ if 9x : p 2 xR; hpiR ¼ / otherwise: > :
Lemma 3.1 [1]. For any binary relation R on U if x 2 hyiR, then hxiR # hyiR. Proof. Let z 2 hxiR = \x2wR(wR). Then z is contained in every wR which contains x, and since also x is contained in every uR which contains y, then z is contained in every uR which contains y, i.e., z 2 hyiR. Then hxiR # hyiR. Definition 3.2. Let R be any reflexive relation on a nonempty set U. For any subset A # U, the lower and the upper approximations of A according to R are then defined as
aprðAÞ ¼ fx 2 U : hxiR # Ag; aprðAÞ ¼ fx 2 U : hxiR \ A–/g: Obviously, if R is an equivalence relation, hxi R = [x]R and these definitions are equivalent to the original Pawlak’s definitions. By using Definition 3.2 we can define the accuracy measure of any set A as the following:
lapr ðAÞ ¼
japrðAÞj : japrðAÞj
Proposition 3.1. For any reflexive relation R on a nonempty set U and for every A, B # U the properties L1–L7, L9, U1–U7, U9, K, and LU hold according to Definition 3.2. Proof c
c
c
c
c
(L1) ½aprðA Þ ¼ fx 2 U : hxiR \ A –/g ¼ fx 2 U : hxiR \ A ¼ /g ¼ fx 2 U : hxiR # Ag ¼ aprðAÞ: (L2) Since hxiR # U for all x 2 U, hence x 2 apr(U). Then U # apr(U). Also, since apr(U) # U. Thus, apr(U) = U. (L3) aprðA \ BÞ ¼ fx 2 U : hxiR # A \ Bg ¼ fx 2 U : hxiR # A ^ hxiR # Bg ¼ fx 2 U : hxiR # Ag \ fx 2 U : hxiR # Bg
¼ aprðAÞ \ aprðBÞ: (L4) Assume that x 2 apr(A) [ apr(B), then x 2 apr(A) or x 2 apr(B), hence hxiR # A or hxiR # B, so hxiR # A [ B, thus x 2 apr(A [ B), therefore apr(A [ B) apr(A) [ apr(B). (L5) Let A # B and x 2 apr(A), then hxiR # A and so hxi R # B, hence x 2 apr(B). Thus apr(A) # apr(B). (L6) Since R is a reflexive relation on U, then x 2 hxiR for all x 2 U, also there does not exists x 2 U such that hxiR # /, hence apr(/) = /. (L7) Assume that x 2 apr(A), then hxiR # A. Since R is a reflexive relation on U, then x 2 hxiR for all x 2 U and there does not exists y 2 U A such that hyiR # A. Thus x 2 A and so apr(A) # A. (L9) Let x 2 apr(A), we will show that hxiR # apr(A). Since x 2 apr(A), then hxiR # A. Let y 2 hxiR then hyiR # h xiR for all y 2 hxiR, hence h yiR # A for all y 2 hxiR. Thus y 2 apr(A) for all y 2 hxiR and so hxiR # apr(A), that implies x 2 apr(apr(A)). Hence, apr(A) # apr(apr(A)).(K) Assume that x R [apr(A)]c [ apr(B), then x R [apr(A)]c and x R apr(B), hence x 2 apr(A) and x R apr(B), that implies hxiR # A and hxi R6 # B, then hxiR6 # Ac and h xiR6 # B, hence hxiR6 # Ac [ B, thus x R apr(Ac [ B), therefore apr(Ac [ B) # [apr(A)]c [ apr(B). We can prove U1–U7 and U9 as the same as L1–L7 and L9. Also we can prove LU from L7 and U7. h The following example shows that the inverse in L4, L7, U4 and U7 in proposition 3.1 is not true in general. Example 3.1. In Example 2.1, we have haiR = {a, b, c}, h biR = {b}, hciR = {c} and hdi R = {d}. If A = {a, b} and B = {c}, then aprðAÞ ¼ fbg; aprðBÞ ¼ fcg; aprðAÞ ¼ fa; bg; aprðBÞ ¼ fa; cg; aprðA [ BÞ ¼ fa; b; cg and aprðA \ BÞ ¼ /; ) aprðA [ BÞ–aprðAÞ[ aprðBÞ and aprðA \ BÞ–aprðAÞ \ aprðBÞ. If A = {a}, then apr(A) = / and so A – apr(A). If A = {b}, then aprðAÞ ¼ fa; bg and so A–aprðAÞ.
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Remark. Let R be any reflexive relation, then for every A # U the properties L8, L10, U8, and U10 do not hold. Example 3.2. (Examples of the above properties do not hold). From Examples 2.1 and 3.1, if A = {a, b}, then aprðAÞ ¼ fa; bg and aprðaprðAÞÞ ¼ fbg, hence A aprðaprðAÞÞ and aprðAÞ aprðaprðAÞÞ; ) L8 and L10 do not hold. If A = {b}, then apr(A) = {b} and aprðaprðAÞÞ ¼ fa; bg, hence aprðaprðAÞÞ A and aprðaprðAÞÞ aprðAÞ, ) U8 and U10 do not hold. Definition 3.3. Let R be any reflexive relation on a nonempty set U. For any subset A # U, the lower and the upper approximations of A according to R are then defined as
[ fhxiR : hxiR # Ag;
apr0 ðAÞ ¼
apr0 ðAÞ ¼ ½apr0 ðAc Þc : By using Definition 3.3 we can define the accuracy measure of any set A as the following:
l0apr ðAÞ ¼
japr0 ðAÞj japr0 ðAÞj
:
Proposition 3.2. For any reflexive relation R on a nonempty set U and for every A, B U the properties L1–L7, L9, U1–U7, U9, K, and LU hold according to Definition 3.3. Proof (L1) ½apr0 ðAc Þc ¼ ½fapr 0 ðAc Þc gc c ¼ apr0 ðAÞ. S (L2) Since for all x 2 U, we get x 2 hxiR # {hxiR:hxi R # U} = apr0 (U), then U # apr0 (U), but apr0 (U) # U, therefore 0 U = apr (U). (L3) Assume that y 2 apr0 (A \ B), there exists x 2 U such that y 2 hxi R # (A \ B), so y 2 hxiR # A and y 2 hxiR # B, then y 2 apr0 (A) \ apr0 (B), therefore apr0 (A \ B) # apr0 (A) \ apr0 (B). Conversely, if y 2 apr0 (A) \ apr0 (B), then y 2 apr0 (A) and y 2 apr0 (B), there exist x1, x2 2 U such that y 2 hx1iR # A and y 2 h x2iR # B, hence y 2 hyi R # hx1iR # A and y 2 hyi R # hx2iR # B, so y 2 hyi R # (A \ B), thus y 2 apr0 (A \ B), therefore apr0 (A) \ apr0 (B) # apr0 (A \ B). (L4) Assume that y 2 apr0 (A) [ apr0 (B), then y 2 apr0 (A) or y 2 apr0 (B), hence there exist x1 or x2 2 U such that y 2 hx1iR # A or y 2 hx2iR # B, so y 2 hyi R # A [ B, thus y 2 apr0 (A [ B), therefore apr0 (A [ B) apr0 (A) [ apr0 (B). (L5) Let A # B and y 2 apr0 (A), then there exists x 2 U such that y 2 hxiR # A, so y 2 hxi R # B, hence y 2 apr0 (B), therefore apr0 (A) # apr0 (B). (L6) Since for all x 2 U we get hxi R – /, thus apr0 (/) = /. S (L7) Since apr0 (A) = {h xiR:hxiR # A}, then apr0 (A) # A. S (L9) Assume that y 2 apr0 (A), then there exists x 2 U such that y 2 hxiR # A, hence y 2 hxiR # {hxi R:hxiR # A} # A, so y 2 h xiR # apr0 (A) # A, then y 2 apr0 (apr0 (A)), therefore apr0 (A) # apr0 (apr0 (A)).(K) Assume that y 2 apr0 (Ac [ B), there exists x 2 U such that y 2 hxiR # (Ac [ B), so y 2 hyiR # (Ac) or y 2 h yiR # (B), then y 2 hyiR6 # (A) or y 2 apr0 (B), hence y R apr0 (A) or y 2 apr0 (B), that implies y 2 [apr0 (A)]c [ apr0 (B). Therefore, apr0 (Ac [ B) # [apr0 (A)]c [ apr0 (B ). We can prove U1–U7 and U9 as the same as L1–L7 and L9. Also we can prove LU from L7 and U7. h The following example shows that the inverse in L4, L7, U4 and U7 in Proposition 3.2 is not true in general. Example 3.3. From Examples 2.1 and 3.1, if A = {a, b} and B = {c, d}, then apr0 ðAÞ ¼ fbg; apr 0 ðBÞ ¼ fc; dg; apr0 ðA [ BÞ ¼ U; apr0 ðAÞ ¼ fa; bg; apr0 ðBÞ ¼ fa; c; dg and apr0 ðA \ BÞ ¼ /; ) apr 0 ðA [ BÞ–apr 0 ðAÞ [ apr0 ðBÞ and apr0 ðA \ BÞ–apr0 ðAÞ \ apr 0 ðBÞ. If A = {a, b}, then apr0 (A) = {b} and so apr0 (A) – A. If A = {c, d}, then apr0 ðAÞ ¼ fa; c; dg, hence A–apr 0 ðAÞ. Remark. Let R be any reflexive relation, then for every A, B U the properties L8, L10, U8, and U10 do not hold. Example 3.4. (Examples of the above properties do not hold). From Examples 2.1 and 3.1, if A = {a, b}, then apr 0 ðAÞ ¼ fa; bg and apr0 ðapr0 ðAÞÞ ¼ fbg, hence A apr 0 ðapr 0 ðAÞÞ and apr0 ðAÞ apr 0 ðapr0 ðAÞÞ; ) L8 and L10 do not hold. If A = {b}, then apr0 (A) = {b} and apr0 ðapr 0 ðAÞÞ ¼ fa; bg, thus apr0 ðapr 0 ðAÞÞ A and apr0 ðapr 0 ðAÞÞ apr 0 ðAÞ; ) U 8 and U10 do not hold. Definition 3.4. Let R be any reflexive relation on a nonempty set U. For any subset A # U, the lower and the upper approximations of A according to R are then defined as
apr00 ðAÞ ¼
[ fhxiR : hxiR \ A–/g;
apr00 ðAÞ ¼ ½apr00 ðAc Þc :
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By using Definition 3.4 we can define the accuracy measure of any set A as the following:
l00apr ðAÞ ¼
japr00 ðAÞj japr00 ðAÞj
:
Proposition 3.3. For any reflexive relation R on a nonempty set U and for every A, B # U the properties L1–L8, U1–U8, K, and LU hold according to Definition 3.4. Proof (U1) ½apr00 ðAc Þc ¼ ½fapr00 ðAc Þc gc c ¼ apr00 ðAÞ. (U2) Since for all x 2 U we get hxi R \ / = /, thus apr 00 ð/Þ ¼ /. (U3)
apr00 ðA [ BÞ ¼
[
fhxiR : hxiR \ ðA [ BÞ–/g ¼
[ [ [ fhxiR : ðhxiR \ AÞ [ ðhxiR \ BÞ–/g ¼ fhxiR : hxiR \ A–/g or fhxiR
: hxiR \ B–/g ¼ apr00 ðAÞ [ apr00 ðBÞ: (U4) Assume that y 2 apr 00 ðA \ BÞ, then there exists x 2 U such that y 2 hxiR and hxiR \ (A \ B) – /, so hxi R \ A – / and hxiR \ B – /, hence y 2 apr00 ðAÞ and y 2 apr 00 ðBÞ, thus y 2 apr00 ðAÞ \ apr00 ðBÞ. Therefore, apr00 ðA \ BÞ # apr 00 ðAÞ \ apr 00 ðBÞ. (U5) Assume that A # B and y 2 apr 00 ðAÞ, then there exists x 2 U such that y 2 hxiR and hxiR \ A – /, so hxiR \ B – /, hence y 2 apr00 ðBÞ, therefore apr 00 ðAÞ # apr00 ðBÞ. (U6) Obviously, apr 00 ðUÞ # U. Also, for all x 2 U, we have x 2 hxiR, then U # apr 00 ðUÞ and so apr00 ðUÞ ¼ U. (U7) For all x 2 A, we have x 2 hxi R, then A # apr00 ðAÞ. (U8) Assume that y 2 apr00 ðapr 00 ðAÞÞ, then there exists x 2 U such that y 2 hxiR and h xiR \ apr00 (A) – /, hence there exists z 2 U such that z 2 hxiR and z 2 apr00 (A), so z R apr00 ðAc Þ, thus hxiR \ Ac = /, then hxiR # A, hence y 2 A, therefore apr00 ðapr 00 ðAÞÞ # A.(K) Assume that y 2 apr 00 ðAc [ BÞ ¼ ½apr 00 ðAc [ BÞc c , then y R apr00 ðAc [ BÞc , so for all x 2 U such that y 2 hxiR we get hxiR \ (Ac [ B)c = /, so y 2 hyiR # h xiR # (Ac [ B), hence hyiR \ Ac – / or hyiR \ B – /, thus y 2 apr00 ðAc Þ or y R apr00 ðBc Þ, then y R apr00 (A) or y 2 apr00 (B), that implies y 2 [apr00 (A)]c [ apr00 (B). Therefore R00 (Ac [ B) # [R00 (A)]c [ R00 (B). We can prove L1–L8 as the same as U1–U8. Also we can prove LU from L7 and U7. The following example shows that the inverse in L4, L7, L8, U4, U7 and U8 in Proposition 3.3 is not true in general. Example 3.5. Let R = {(a, a), (b, b), (c, c), (d, d), (a, b), (b, a), (c, b)} be a reflexive relation on a set U = {a, b, c, d}. Then aR = {a, b}, bR = {a, b}, cR = {b, c} and dR = {d}, ) hai R = {a, b}, hbiR = {b}, hciR = {b, c} and hdiR = {d}. If A = {a, b} and B = {a, c}, then apr 00 ðAÞ ¼ fag; apr00 ðBÞ ¼ /; apr 00 ðA [ BÞ ¼ fa; b; cg; apr 00 ðAÞ ¼ fa; b; cg; apr00 ðBÞ ¼ fa; b; cg; apr00 ðA \ BÞ ¼ fa; bg; apr00 ðapr 00 ðAÞÞ ¼ fa; b; cg and apr 00 ðapr 00 ðBÞÞ ¼ /; ) apr 00 ðA [ BÞ–apr00 ðAÞ [ apr 00 ðBÞ; apr 00 ðA \ BÞ–apr 00 ðAÞ \ apr 00 ðBÞ; apr00 ðAÞ–A–apr 00 ðAÞ; A– 00 00 apr ðapr ðAÞÞ and B–apr 00 ðapr 00 ðBÞÞ.
Remark. Let R be any reflexive relation, then for every A # U the properties L9, L10, U9 and U10 do not hold. Example 3.6. (Examples of the above properties do not hold). In Example 3.5., If A = {a, d}, then apr 00 ðAÞ ¼ fa; b; dg; apr00 ðapr 00 ðAÞÞ ¼ U and apr 00 ðapr00 ðAÞÞ ¼ fa; dg; ) apr 00 ðapr00 ðAÞÞ apr00 ðAÞ and apr 00 ðapr00 ðAÞÞ apr00 ðAÞ; ) U 9 and L10 do not hold. If A = {a, b}, then apr00 (A) = {a}, apr00 (apr00 (A)) = / and apr 00 ðapr00 ðAÞÞ ¼ fa; bg, so apr00 (A)6 # apr00 (apr00 (A)) and apr 00 ðAÞ apr00 ðapr 00 ðAÞÞ; ) L9 and U10 do not hold. In Table 1 we summarize the properties of the above definitions of lower and upper approximation operators. Remark. For any reflexive relation R on a finite nonempty set U, we observe that:
L9 and U9 hold according to Definition 3.2 but it does not hold according to Definition 2.2. L3, U3 and K hold according to Definition 3.3 but it does not hold according to Definition 2.3. Definition 2.4 is the same as Definition 3.4. 4. The relationship between the two approaches For the sake of illustration, we study the relationship between Definitions 2.2 and 3.2. Below we present a simple example, using a nonsymmetric and nontransitive relation R, where each definition gives a different result. We can find the relationship between the lower (upper) approximation in Definitions 2.2 and 3.2 as follows:
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Table 1 The comparison between the properties of rough sets depending on Definitions 2.2, 2.3, 2.4, Definitions 3.2, 3.3 and 3.4 by using any reflexive relation R. A cross () indicates that the property is satisfied. Property
Def. 2.2
Def. 2.3
Def. 2.4
Def. 3.2
Def. 3.3
Def. 3.4
L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 K LU
Lemma 4.1. For any reflexive relation R on a nonempty set U and for every A # U, then R(A) # apr(A) and aprðAÞ # RðAÞ. Proof. Assume that a 2 R(A), then aR # A and so haiR # A, hence a 2 apr(A), therefore R(A) # apr(A). Also, suppose that a 2 aprðAÞ, then hai R \ A – /, hence aR \ A – /, and so a 2 RðAÞ, that implies aprðAÞ # RðAÞ.
h
Lemma 4.2. For any transitive relation R on a nonempty set U. If x 2 yR, then xR # yR. Proof. Assume that x 2 yR and z 2 xR, then yRx and xRz, hence by transitive yRz, and so z 2 yR, that implies xR # yR. h The following theorem gives us the relationship between the right neighborhood of any element x 2 U and the intersection of the right neighborhoods containing this element. Theorem 4.1. For any reflexive and transitive relation R on a nonempty set U. xR = hxiR for all x 2 U. Proof. The proof is immediately derived from Lemma 4.2.
h
In Table 2 we will use the relation R which comes from the Example 2.1 to compare the properties of the rough sets depending on Definitions 2.2 and 3.2. From Lemma 4.1 and Theorem 4.1 we can introduce the final theorem as follows. Theorem 4.2. For any reflexive relation R on a nonempty set U, then
POSR(A) # POSapr(A). NEGR(A) # NEGapr(A). BNDapr(A) # BNDR(A). lR(A) 6 lapr(A). If R is transitive, then our approach is equivalent to Yao’s approach.
One can study the relationship between Definition 2.3 (Definition 2.4) and Definition 3.3 (Definition 3.4). Also, one can illustrate the differences between them on the analogical example to the one which was showed above. It should also be noted that the family of neighborhoods produced by a similarity relation forms coverings of the universe. This property relates this study to many recent studies on the covering-based rough set model (see [29,31,32]). Abu-Donia [2] generalized the rough set approximations by using a family of binary relations. If this family consists of exactly single relation, then this generalization of approximations is equivalent to the Yao’s approaches. Additionally, our
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Table 2 The comparison between the boundary of the set and the accuracy measure of the set depending on Definitions 2.2 and 3.2 by using a reflexive relation R. A
R
R
apr
apr
BNDR
BNDapr
lR
lapr
{a} {b}
/ /
{a} {a, b, d}
/ {b}
{a} {a, b}
{a} {a, b, d}
{a} {a}
0 0
0
{c}
/
{a, b, c}
{c}
{a, c}
{a, b, c}
{a}
0
{d} {a, b}
/ /
{c, d} {a, b, d}
{d} {b}
{d} {a, b}
{c, d} {a, b, d}
/ {a}
0 0
{a, c}
/
{a, b, c}
{c}
{a, c}
{a, b, c}
{a}
0
{a, d}
/
{a, c, d}
{d}
{a, d}
{a, c, d}
{a}
0
{b, c}
{b}
U
{b, c}
{a, b, c}
{a, c, d}
{a}
{b, d}
{d}
U
{b, d}
{a, b, d}
{a, b, c}
{a}
{c, d}
{c}
U
{c, d}
{a, c, d}
{a, b, d}
{a}
{a, b, c}
{a, b}
U
{a, b, c}
{a, b, c}
{c, d}
/
{a, b, d}
{d}
U
{b, d}
{a, b, d}
{a, b, c}
{a}
{a, c, d}
{c}
U
{c, d}
{a, c, d}
{a, b, d}
{a}
{b, c, d}
{b, c, d}
U
{b, c, d}
U
{a}
{a}
1 4 1 4 1 4 1 2 1 4 1 4 3 4
1 2 1 2
1 1 2 1 2 1 2 2 3 2 3 2 3
1 2 3 2 3 3 4
approach and those generalizations are independent. Moreover, if the relations of this family are equivalence relations, then all generalizations of approximations are equivalent to the original Pawlak’s definitions. 5. Conclusion The theory of rough sets is typically studied based on the notion of an approximation space and the induced lower and upper approximations of subsets of a universe. In this paper, we introduce new definitions of lower and upper approximations, which are basic concepts of the rough set theory. These definitions follow naturally from the base of topological space {hxiR: x 2 U}. In the study, we use a reflexive relation in order to get hxiR – /. These new definitions, which differ from the classical definitions of approximations, are the only ones to properly characterize the set of lower approximation and the set of upper approximation when the similarity relation is reflexive, but not necessarily symmetric and transitive. This is why we suggest using our new definitions even when the relation is symmetric and transitive. The work presented in this paper utilized the approximations generated by similarity relation to get a specific type of the right neighborhoods and to apply them in order to get more accurate approximations. This approach is more general than that of Yao’s. Acknowledgements The author thanks the anonymous referees and the Editor-in-Chief, Professor Witold Pedrycz for their valuable suggestions in improving this paper. References [1] E.A. Abo-Tabl, Some Topological Concepts and their Applications from the Relations Point of View, PhD. Thesis-Assiut University, Egypt, 2008. [2] H.M. Abu-Donia, Comparison between different kinds of approximations by using a family of binary relations, Knowledge-Based Systems 21 (2008) 911–919. [3] W. Bartol, J. Miro, K. Pioro, F. Rossello, On the coverings by tolerance classes, Information Sciences 166 (14) (2004) 193–211. [4] Z. Bonikowski, E. Bryniarski, U. Wybraniec-Skardowska, Extensions and intentions in the rough set theory, Information Sciences 107 (1998) 149–167. [5] K. Chakrabarty, R. Biswas, S. Nanda, Fuzziness in rough sets, Fuzzy Sets and Systems 110 (2000) 247–251. [6] J. Dai, Rough 3-valued algebras, Information Sciences 178 (8) (2008) 1986–1996. [7] U. Hoehle, Quotients with respect to similarity relations, Fuzzy Sets and Systems 27 (1988) 31–44. [8] M. Kryszkiewicz, Rough set approach to incomplete information systems, Information Sciences 112 (1998) 39–49. [9] T.Y. Lin, Q. Liu, Rough approximate operators: axiomatic rough set theory, in: W. Ziarko (Ed.), Rough sets, Fuzzy Sets and Knowledge Discovery, Springer, Berlin, 1994, pp. 256–260. [10] G.L. Liu, Y. Sai, A comparison of two types of rough sets induced by coverings, International Journal of Approximate Reasoning 50 (2009) 521–528. [11] G.L. Liu, W. Zhu, The algebraic structures of generalized rough set theory, Information Sciences 178 (21) (2008) 4105–4113. [12] H.T. Nguyen, Some mathematical structures for computational information, Information Sciences 128 (2000) 67–89. [13] Z. Pawlak, Rough sets, International Journal of Computer and Information Science 11 (1982) 341–356. [14] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data, Kluwer Academic Publishers, Boston, 1991. [15] Z. Pawlak, A. Skowron, Rudiments of rough sets, Information Sciences 177 (1) (2007) 3–27. [16] Z. Pawlak, A. Skowron, Rough sets: some extensions, Information Sciences 177 (1) (2007) 28–40. [17] Z. Pawlak, A. Skowron, Rough sets and Boolean reasoning, Information Sciences 177 (1) (2007) 41–73. [18] J.A. Pomykala, Approximation operations in approximation space, Bulletin of the Polish Academy of sciences: Mathematics 35 (1987) 653–662. [19] M. Quafafou, A-RST: a generalization of rough set theory, Information Sciences 124 (2000) 301–316. [20] A. Skowron, J. Stepaniuk, Tolerance approximation spaces, Fundamenta Informaticae 27 (1996) 245–253. [21] R. Slowinski, D. Vanderpooten, Similarity relation as a basis for rough approximations, in: P.P. Wang (Ed.), Advances in machine intelligence and softcomputing, Department of Electrical Engineering, Duke University, Durham, NC. USA, 1997, pp. 17–33. [22] R. Slowinski, D. Vanderpooten, A generalized definition of rough approximations based on similarity, IEEE Transactions on Knowledge and Data Engineering 12 (2) (2000) 331–336.
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