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Comparison between different kinds of approximations by using a family of binary relations
Knowledge-Based Systems 21 (2008) 911–919
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Comparison between different kinds of approximations by using a family of binary relations H.M. Abu-Donia * Department of Mathematics, Faculty of Science, Zagazig University, Egypt
a r t i c l e
i n f o
Article history: Received 17 December 2007 Accepted 30 March 2008 Available online 7 April 2008 Keywords: Lower and upper approximations Right neighborhood Binary relations and rough sets
a b s t r a c t In this paper, we discuss three types of lower and upper approximations of any set with respect to any relation based on right neighborhood. We generalize these three types of approximations into two ways by using a finite number of any binary relations. The first way based on the intersection of the right neighborhoods for a family of binary relations, the second based on the union and intersection of lower and upper approximations of a set according to a family of binary relations fRi : i ¼ 1; 2; . . . ; ng. Also, we make a comparison between these types in tables. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction In early eighties, Pawlak [3–5] introduced the theory of rough sets as an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. It is motivated by the practical needs in concept formation. Incomplete information causes some concepts of the universe can only be approximated. The standard rough set theory starts from an equivalence relation. A universe is divided into a family of disjoint subsets, which is well known in mathematics as a quotient set. The elements in an equivalence class are in indistinguishable by the available knowledge. A pair of operators named lower and upper approximations is introduced. The approximations are expressed in terms of equivalence classes according to their overlaps with sets to be approximated. Many proposals have been made for generalizing and interpreting rough sets [1,2,5–10,12]. In this paper, we discuss three types of upper (lower) approximations depends on right neighborhood by using any relation. We generalize this types by using a family of finite binary relations in two ways. The first generalized definition based on the intersection of these right neighborhoods and the second generalized definition based on the intersection and union of upper and lower approximations of any set with respect to every relation of them. Also, we make a comparison between all these definitions and show these comparisons in tables.
* Tel.: +20 105075372. E-mail address: [email protected] 0950-7051/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2008.03.046
2. Kinds of rough set approximations by using any binary relation In this section, we discuss three kinds of rough set approximations based on the right neighborhood. Yao [11] extend the definition of a rough set to any relation R as follows: Definition 2.1. For the pair ðU; RÞ, where U is a universe, U 6¼ ; and R be any binary relation, let xR be the right neighborhood defined as: xR ¼ fy 2 U : xRyg. Definition 2.2. Let R be any binary relation on a non-empty set U, for any set A U. The lower and upper approximations of A according to R defined as: RðAÞ ¼ fx 2 U : xR Ag; RðAÞ ¼ fx 2 U : xR \ A 6¼ ;g: Obviously, if R is an equivalence relation xR ¼ ½xR and these definitions are equivalent to the original Pawlak’s definitions. We list the properties that are of interest in the theory of rough sets, let A; B U: ðL1 Þ RðAÞ ¼ ½RðAc Þc , where Ac denotes the complement of A in U. ðL2 Þ RðUÞ ¼ U. ðL3 Þ RðA \ BÞ ¼ RðAÞ \ RðBÞ. ðL4 Þ RðA [ BÞ RðAÞ [ RðBÞ. ðL5 Þ A B ) RðAÞ RðBÞ. ðL6 Þ Rð;Þ ¼ ;. ðL7 Þ RðAÞ A.
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ðL8 Þ A RðRðAÞÞ. ðL9 Þ RðAÞ ¼ RðRðAÞÞ. ðL10 Þ RðAÞ ¼ RðRðAÞÞ. ðU 1 Þ RðAÞ ¼ ½RðAc Þc . ðU 2 Þ Rð;Þ ¼ ;. ðU 3 Þ RðA [ BÞ ¼ RðAÞ [ RðBÞ. ðU 4 Þ RðA \ BÞ RðAÞ \ RðBÞ. ðU 5 Þ A B ) RðAÞ RðBÞ. ðU 6 Þ RðUÞ ¼ U. ðU 7 Þ A RðAÞ. ðU 8 Þ A RðRðAÞÞ. ðU 9 Þ RðAÞ ¼ RðRðAÞÞ. ðU 10 Þ RðAÞ ¼ RðRðAÞÞ. (CO) RðAc [ BÞ ðRðAÞÞc [ RðBÞ. (LU) RðAÞ RðAÞ. By using Definition 2.2, we can define the accuracy measure as the following: jA \ RðAÞj ; aany ðAÞ ¼ jA [ RðAÞj where j j denotes the cardinality of the set. Proposition 2.1. For any binary relation R on a non-empty set U, the following conditions hold for every A; B U: ðL1 Þ RðAÞ ¼ ½RðAc Þc . ðL2 Þ RðUÞ ¼ U. ðL3 Þ RðA \ BÞ ¼ RðAÞ \ RðBÞ. ðL4 Þ RðA [ BÞ RðAÞ [ RðBÞ. ðL5 Þ A B ) RðAÞ RðBÞ. ðU 1 Þ RðAÞ ¼ ½RðAc Þc . ðU 2 Þ Rð;Þ ¼ ;. ðU 3 Þ RðA [ BÞ ¼ RðAÞ [ RðBÞ. ðU 4 Þ RðA \ BÞ RðAÞ \ RðBÞ. ðU 5 Þ A B ) RðAÞ RðBÞ. ðCOÞ RðAc [ BÞ ½RðAÞc [ RðBÞ. Proof ðL1 Þ. ½RðAc Þc ¼ fx 2 U : xR \ Ac 6¼ ;gc ¼ fx 2 U : xR \ Ac ¼ ;g ¼ fx 2 U : xR Ag ¼ RðAÞ: ðL2 Þ. As U is the universe set, hence RðUÞ U, 8x 2 U; we have xR U 8i ¼ 1; 2; . . . ; n, then xR U, this implies that x 2 RðUÞ. Thus U RðUÞ. So RðUÞ ¼ U. ðL3 Þ.
We can introduce an example to prove that the equality of L4 , U 4 and CO are not true in general. Example 2.1. Let R1 ¼ fða; bÞ; ða; dÞ; ðb; cÞ; ðb; dÞ; ðc; cÞ; ðd; aÞ; ðd; cÞ; ðd; bÞ; ðc; eÞ; ða; eÞ; ðe; dÞg and R2 ¼ fða; bÞ; ða; cÞ; ðb; cÞ; ðb; dÞ; ðb; bÞ; ðc; aÞ; ðc; dÞ; ðd; aÞ; ðd; cÞ; ðd; eÞ; ða; eÞ; ðe; bÞg are any two binary relations on non-empty set U ¼ fa; b; c; d; eg. Then aR1 ¼ fb; d; eg, bR1 ¼ fc; dg; cR1 ¼ fc; eg, dR1 ¼ fa; b; cg, eR1 ¼ fdg, aR2 ¼ fb; c; eg, bR2 ¼ fb; c; dg, cR2 ¼ fa; dg, dR2 ¼ fa; c; eg and eR2 ¼ fbg. If A ¼ fcg and B ¼ fdg, then R1 ðAÞ ¼ ;, R1 ðBÞ ¼ feg and RðA [ BÞ ¼ fb; eg. So R1 ðA [ BÞ 6¼ R1 ðAÞ [ R1 ðBÞ. If A ¼ fa; dg and B ¼ fb; dg, then R1 ðAÞ ¼ fa; b; d; eg, R1 ðBÞ ¼ fa; b; d; eg and R1 ðA \ BÞ ¼ fa; b; eg. Thus R1 ðAÞ \ R1 ðBÞ 6¼ R1 ðA \ BÞ. If A ¼ fc; d; eg and B ¼ fa; cg, hence R1 ðAc [ BÞ ¼ fdg, ½R1 ðAÞc ¼ fa; dg and R1 ðBÞ ¼ ;, then ½R1 ðAÞc [ R1 ðBÞ ¼ fa; dg. Thus R1 ðAc [ BÞ 6¼ ½R1 ðAÞc [ R1 ðBÞ. Remark 2.1. If R is an any binary relation on a non-empty set U, then the following conditions do not hold for every A; B U. ðL6 Þ Rð;Þ ¼ ;. ðL7 Þ RðAÞ A. ðL8 Þ A RðRðAÞÞ. ðL9 Þ RðAÞ ¼ RðRðAÞÞ. ðL10 Þ RðAÞ ¼ RðRðAÞÞ. ðU 6 Þ RðUÞ ¼ U. ðU 7 Þ A RðAÞ. ðU 8 Þ RðRðAÞÞ A. ðU 9 Þ RðRðAÞÞ ¼ RðAÞ. ðU 10 Þ RðRðAÞÞ ¼ RðAÞ. ðLUÞ RðAÞ RðAÞ. The following examples show Remark 2.1. Example 2.2. Let R1 ¼ fða; bÞ; ðb; cÞg and R2 ¼ fða; cÞ; ðb; cÞg are any two binary relations on non-empty set U ¼ fa; b; cg. Then aR1 ¼ fbg; bR1 ¼ fcg, cR1 ¼ ;, aR2 ¼ fcg; bR2 ¼ fcg and cR2 ¼ ;. Hence we have R1 ð;Þ ¼ fcg and RðUÞ ¼ fa; bg. Consequently L6 and U 6 do not hold. If A ¼ fbg, hence R1 ðAÞ ¼ fa; cg and R1 ðAÞ ¼ fag. Then R1 ðAÞ 6 A, A 6 R1 ðAÞ and R1 ðAÞ 6 R1 ðAÞ, i.e., L7 , U 7 and LU do not hold. If A ¼ fa; bg, hence R1 ðAÞ ¼ fa; dg and R1 ðR1 ðAÞÞ ¼ feg. Thus A 6 R1 ðR1 ðAÞÞ and R1 ðAÞ 6¼ R1 ðR1 ðAÞÞ. Therefore L8 and L10 do not hold. If A ¼ fa; b; c; eg. Hence R1 ðAÞ ¼ fc; dg and R1 ðR1 ðAÞÞ ¼ U. So R1 ðR1 ðAÞÞ 6 A and R1 ðR1 ðAÞÞ 6¼ R1 ðAÞ. Therefore U 8 and U 10 do not hold. If A ¼ fa; b; cg, hence R1 ðAÞ ¼ fdg,R1 ðR1 ðAÞÞ ¼ feg, R1 ðAÞ ¼ fa; b; c; dg and R1 ðR1 ðAÞÞ ¼ U. Then R1 ðAÞ 6¼ R1 ðR1 ðAÞÞ and R1 ðR1 ðAÞÞ 6¼ R1 ðAÞ. Therefore L9 and U 9 do not hold.
() xR A and xR B
Definition 2.3. Let R be any binary relation on a non-empty set U and xR is the right neighborhood of x according to R, for any set A U. The second definition for lower and upper approximations of A according to R is defined [11] as: [ R0 ðAÞ ¼ fxR : xR Ag;
() x 2 RðAÞ and x 2 RðBÞ
R0 ðAÞ ¼ ½R0 ðAc Þc :
x 2 RðA \ BÞ () xR ðA \ BÞ
() x 2 ½RðAÞ \ RðBÞ: Therefore RðA \ BÞ ¼ RðAÞ \ RðBÞ. ðL4 Þ. Let x 2 ½RðAÞ [ RðBÞ, hence x 2 RðAÞ or x 2 RðBÞ. So xR A or xR B and so xR ðA [ BÞ, then x 2 RðA [ BÞ. Therefore RðA [ BÞ RðAÞ [ RðBÞ. ðL5 Þ. Let A B and x 2 RðAÞ. Then xR A but A B. So xR B, hence x 2 RðBÞ. Thus RðAÞ RðBÞ. ðCOÞ. Let x 62 ½RðAÞc [ RðBÞ, then x 62 ½RðAÞc and x 62 RðBÞ, hence x 2 RðAÞ and x 62 RðBÞ, i.e., xR A and xR 6 B, then xR 6 Ac and xR 6 B, hence xR 6 ðAc [ BÞ. Thus x 62 RðAc [ BÞ.i.e., RðAc [ BÞ ½RðAÞc [ RðBÞ. We can prove U 1 ; U 2 ; U 3 ; U 4 and U 5 as the same as L1 ; L2 ; L3 ; L4 and L5 . h
And so the accuracy measure will be defined as: a0any ðAÞ ¼
jA \ R0 ðAÞj ; jA [ R0 ðAÞj
where j j denotes the cardinality of the set. Proposition 2.2. For any binary relation R on a non-empty set U, the following conditions hold for every A; B U: ðL1 Þ R0 ðAÞ ¼ ½R0 ðAc Þc . ðL4 Þ R0 ðA [ BÞ R0 ðAÞ [ R0 ðBÞ. ðL5 Þ A B ) R0 ðAÞ R0 ðBÞ.
H.M. Abu-Donia / Knowledge-Based Systems 21 (2008) 911–919
ðL6 Þ R0 ð;Þ ¼ ;. ðL7 Þ R0 ðAÞ A. ðL9 Þ R0 ðAÞ ¼ R0 ðR0 ðAÞÞ. ðU 1 Þ R0 ðAÞ ¼ ½R0 ðAc Þc . ðU 4 Þ R0 ðA \ BÞ R0 ðAÞ \ R0 ðBÞ. ðU 5 Þ A B ) R0 ðAÞ R0 ðBÞ. ðU 6 Þ R0 ðUÞ ¼ U. ðU 7 Þ A R0 ðAÞ. ðU 9 Þ R0 ðR0 ðAÞÞ ¼ R0 ðAÞ. ðLUÞ R0 ðAÞ R0 ðAÞ. Proof ðL1 Þ. ½R0 ðAc Þc ¼ ½½R0 ðAc Þc c c ¼ R0 ðAÞ. ðL4 Þ. Let x 2 ½R0 ðAÞ [ R0 ðBÞ. Then x 2 R0 ðAÞ or x 2 R0 ðBÞ, then xR A or xR B, this implies that xR ðA [ BÞ. So x 2 R0 ðA [ BÞ. Hence R0 ðA [ BÞ R0 ðAÞ [ R0 ðBÞ. ðL5 Þ. Let A B and x 2 R0 ðAÞ. Then xR A but A B. So xR B, hence x 2 R0 ðBÞ. Thus, R0 ðAÞ R0 ðBÞ. ðL6 Þ. 8x 2 U, we have xR ;, if xR ¼ ;. So R0 ð;Þ ¼ ;. S ðL7 Þ. Since R0 ðAÞ ¼ fxR : xR Ag. So R0 ðAÞ A. S 0 Since R ðAÞ ¼ fxR : xR Ag, hence R0 ðR0 ðAÞÞ ¼ ðL9 Þ. S S S R0 ð fxR : xR AgÞ ¼ fxR : xR ð fxR : xR AgÞg ¼ R0 ðAÞ. Hence R0 ðR0 ðAÞÞ ¼ R0 ðAÞ. We can prove U 1 ; U 4 ; U 5 ; U 6 ; U 7 and U 9 as the same as L1 ; L4 ; L5 ; L6 ; L7 and L9 . We can prove LU from L7 and U 7 . h We can introduce an example to prove that the equality of L4 , U 4 , L7 , U 7 and LU are not true in general. Example 2.3. In Example 2.1, if A ¼ fbg and B ¼ fd; eg, hence R01 ðAÞ ¼ ;, R01 ðBÞ ¼ fdg and R01 ðA [ BÞ ¼ fb; d; eg. Thus, R01 ðA [ BÞ 6¼ R01 ðAÞ [ R01 ðBÞ. If A ¼ fa; c; dg and B ¼ fa; d; eg, then R01 ðAÞ ¼ U, R01 ðBÞ ¼ U and R01 ðA \ BÞ ¼ fa; b; dg. Thus R01 ðAÞ \ R01 ðBÞ 6¼ R01 ðA \ BÞ. If A ¼ fbg, then R01 ðAÞ ¼ ;, R01 ðAÞ ¼ fa; bg, i.e., R01 ðAÞ 6¼ A, R01 ðAÞ 6¼ A and R01 ðAÞ 6¼ R01 ðAÞ. Remark 2.2. If a binary relation R is any relation. Then the following conditions are not hold for every A; B U: ðL2 Þ R0 ðUÞ ¼ U. ðL3 Þ R0 ðA \ BÞ ¼ R0 ðAÞ \ R0 ðBÞ. ðL8 Þ A R0 ðR0 ðAÞÞ. ðL10 Þ R0 ðAÞ ¼ R0 ðR0 ðAÞÞ. ðU 2 Þ R0 ð;Þ ¼ ;. ðU 3 Þ R0 ðA [ BÞ ¼ R0 ðAÞ [ R0 ðBÞ. ðU 8 Þ R0 ðR0 ðAÞÞ A. ðU 10 Þ R0 ðR0 ðAÞÞ ¼ R0 ðAÞ. ðCOÞ R0 ðAc [ BÞ ½R0 ðAÞc [ R0 ðBÞ. The following examples show Remark 2.2. Example 2.4. In Example 2.2, we have R01 ðUÞ ¼ fb; cg and R01 ð;Þ ¼ fag. Thus R01 ðUÞ 6¼ U and R01 ð;Þ 6¼ ;. i.e., L2 and U 2 do not hold. Example 2.5. In Example 2.1, if A ¼ fc; dg and B ¼ fc; eg, hence R01 ðAÞ ¼ fc; dg, R01 ðBÞ ¼ fc; eg and R01 ðA \ BÞ ¼ ;. Thus, R01 ðA \ BÞ 6¼ R01 ðAÞ \ R01 ðBÞ. i.e., L3 does not hold. If A ¼ fa; cg and B ¼ fdg, then R01 ðAÞ ¼ fa; cg, R01 ðBÞ ¼ fdg and R01 ðA [ BÞ ¼ U. Thus, R01 ðAÞ [ R01 ðBÞ 6¼ R01 ðA [ BÞ. i.e., U 3 does not hold. If A ¼ fcg, hence R01 ðAÞ ¼ fa; cg and R01 ðR01 ðAÞÞ ¼ ;. Thus A 6 R01 ðR01 ðAÞÞ and R01 ðAÞ 6¼ R01 ðR01 ðAÞÞ. i.e., L8 and L10 do not hold. If A ¼ fc; dg, hence R01 ðAÞ ¼ fc; dg and R01 ðR01 ðAÞÞ ¼ U. So R01 ðR01 ðAÞÞ 6 Aand R01 ðR01 ðAÞÞ 6¼ R01 ðAÞ. Therefore U 8 and U 10 do not hold. If A ¼ fb; c; eg and B ¼ fb; cg, hence R01 ðAc [ BÞ ¼ fa; b; c; dg, ½R01 ðAÞc ¼ fa; b; dg and R01 ðBÞ ¼ ;, then ½R01 ðAÞc [ R01 ðBÞ ¼ fa; b; dg. Thus R01 ðAc [ BÞ 6¼ ½R01 ðAÞc [ R01 ðBÞ. Hence CO does not hold.
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Definition 2.4. Let R be any binary relation on a non-empty set U and xR is the right neighborhood of x according to R, for any set A U. The third definition for lower and upper approximations of A according to R is defined [11] as the following: R00 ðAÞ ¼ ½R00 ðAc Þc ; [ R00 ðAÞ ¼ fxR : xR \ A 6¼ ;g: And so the accuracy measure will be defined as: a00any ðAÞ ¼
jA \ R00 ðAÞj ; jA [ R00 ðAÞj
where j j denotes the cardinality of the set. Proposition 2.3. For any binary relation R on a non-empty set U the following hold for every A; B U: ðL1 Þ R00 ðAÞ ¼ ½R00 ðAc Þc . ðL2 Þ R00 ðUÞ ¼ U. ðL3 Þ R00 ðA \ BÞ ¼ R00 ðAÞ \ R00 ðBÞ. ðL4 Þ R00 ðA [ BÞ R00 ðAÞ [ R00 ðBÞ. ðL5 Þ A B ) R00 ðAÞ R00 ðBÞ. ðL8 Þ A R00 ðR00 ðAÞÞ. ðU 1 Þ R00 ðAÞ ¼ ½R00 ðAc Þc . ðU 2 Þ R00 ð;Þ ¼ ;. ðU 3 Þ R00 ðA [ BÞ ¼ R00 ðAÞ [ R00 ðBÞ. ðU 4 Þ R00 ðA \ BÞ R00 ðAÞ \ R00 ðBÞ. ðU 5 Þ A B ) RðAÞ RðBÞ. ðU 8 Þ R00 ðR00 ðAÞÞ A. ðCOÞ R00 ðAc [ BÞ ½R00 ðAÞc [ R00 ðBÞ. Proof ðL1 Þ. From Definition 2.4, we have R00 ðAÞ ¼ ½R00 ðAc Þc . S ðL2 Þ. R00 ðUÞ ¼ ½R00 ð/Þc ¼ ½ fxR : xR \ / 6¼ /gc ¼ /c ¼ U. S 00 00 ðL3 Þ. R ðA \ BÞ ¼ ½R ðA \ BÞc c ¼ ½R00 ðAc [ Bc Þc ¼ ½ fxR : xR \ ðAc S c c cTS c c [B Þ 6¼ /g ¼ ½ fxR : xR \ A 6¼ /g ½ fxR : xR \ B 6¼ /gc ¼ ½R00 ðAc Þc \ ½R00 ðBc Þc ¼ R00 ðAÞ \ R00 ðBÞ. ðL4 Þ. Let x 2 R00 ðAÞ [ R00 ðBÞ, hence x 2 R00 ðAÞ or x 2 R00 ðBÞ. So x 2 ½R00 ðAc Þc or x 2 ½R00 ðBc Þc . This implies that x 62 R00 ðAc Þ or x 62 R00 ðBc Þ, then 8s 2 U, we have x 2 sR such that sR \ Ac 6¼ / and sR \ Bc 6¼ /. Consequently, x 62 sR8s 2 U such that sR \ ðAc \ Bc Þ 6¼ /, it is means that x 62 R00 ðA [ BÞc i.e, x 2 R00 ðA [ BÞ. Hence R00 ðAÞ [ R00 ðBÞ R00 ðA [ BÞ. ðL5 Þ. Let A B and x 62 R00 ðBÞ, hence x 62 ½R00 ðBc Þc . So x 2 R00 ðBc Þ and so 9s 2 U such that x 2 sR and sR \ Bc 6¼ / but Ac Bc . This implies that sR \ Ac 6¼ /, it is means that x 2 R00 ðAc Þ and so x 62 ½R00 ðAc Þc ¼ R00 ðAÞ. Therefore R00 ðAÞ R00 ðBÞ. In the same manner we can prove ðU 1 Þ, ðU 2 Þ, ðU 3 Þ, ðU 4 Þ, ðU 5 Þ, ðU 8 Þ, ðL8 Þ and ðCOÞ. h We can introduce an example to show that the equality of L4 , U 4 , U 8 and CO are not true in general. Example 2.6. Let R1 ¼ fðb; dÞ; ðb; cÞ; ðc; cÞ; ðd; aÞ; ðe; bÞ; ðe; eÞg and R2 ¼ fðb; dÞ; ðc; cÞ; ðc; eÞ; ðd; eÞ; ðe; dÞ, ðe; bÞg are any two binary relations on non-empty set U ¼ fa; b; c; d; eg. Then aR1 ¼ ;, bR1 ¼ fc; dg, cR1 ¼ fcg, dR1 ¼ fag, eR1 ¼ fb; eg, aR2 ¼ ;, bR2 ¼ fdg, cR2 ¼ fc; eg, dR2 ¼ feg and eR2 ¼ fb; dg. If A ¼ fa; dg and B ¼ fcg, hence R001 ðAÞ ¼ fag, R001 ðBÞ ¼ ; and R001 ðA [ BÞ ¼ fa; c; dg. Hence R001 ðAÞ [ R001 ðBÞ 6¼ R001 ðA [ BÞ. If A ¼ fcg and B ¼ fdg, hence R001 ðAÞ ¼ fc; dg, R001 ðBÞ ¼ fc; dg and R001 ðA \ BÞ ¼ ;. Thus R001 ðAÞ \ R001 ðBÞ 6¼ R001 ðA \ BÞ. If A ¼ fa; b; c; dg, hence R001 ðAÞ ¼ fa; c; dg and R001 ðR001 ðAÞÞ ¼ fa; c; dg. So R001 ðR001 ðAÞÞ 6¼ A. If A ¼ fa; b; c; dg and B ¼ fa; dg, hence ½R001 ðAÞc ¼ fb; eg, R001 ðBÞ ¼ fag and R001 ðAc [ BÞ ¼ fag. Thus ½R001 ðAÞc [ R001 ðBÞ 6¼ R001 ðAc [ BÞ. We can introduce an example to show that the equality of L8 is not true in general.
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Example 2.7. In Example 2.1, if A ¼ feg, hence R001 ðAÞ ¼ fb; c; d; eg and R001 ðR001 ðAÞÞ ¼ fd; eg. Thus A 6¼ R001 ðR001 ðAÞÞ. Remark 2.3. If R is an any binary relation on a non-empty set U, then the following conditions do not hold for every A; B U. ðL6 Þ R00 ð;Þ ¼ ;. ðL7 Þ R00 ðAÞ A. ðL9 Þ R00 ðAÞ ¼ R00 ðR00 ðAÞÞ. ðL10 Þ R00 ðAÞ ¼ R00 ðR00 ðAÞÞ. ðU 6 Þ R00 ðUÞ ¼ U. ðU 7 Þ A R00 ðAÞ. ðU 9 Þ R00 ðR00 ðAÞÞ ¼ R00 ðAÞ. ðU 10 Þ R00 ðR00 ðAÞÞ ¼ R00 ðAÞ. ðLUÞ R00 ðAÞ R00 ðAÞ.
intersection of the right neighborhoods for a family of any binary relations. Definition 3.1. Let fRi : i ¼ 1; 2; . . . ; ng be a finite family of binary relations on a non-empty set U, for any set A U, we can introduce a definition for n-lower and n-upper approximations of A according to Ri as: ( ! ) n \ x2U: xRi A ; n aprðAÞ ¼ i¼1
( n
aprðAÞ ¼
n \
x2U:
!
) \ A 6¼ ; :
xRi
i¼1
By using Definition 3.1, we can define the accuracy measure of any set A as the following: n aany ðAÞ
The following examples show Remark 2.3. Example 2.8. In Example 2.6, we have R002 ð;Þ ¼ fag and R002 ðUÞ ¼ fb; c; d; eg, i.e., L6 and U 6 do not hold. If A ¼ fb; dg, hence R002 ðAÞ ¼ fa; b; dg and R002 ðAÞ ¼ fb; dg. Thus R2 ðAÞ 6 A and R002 ðAÞ 6 R002 ðAÞ, i.e., L7 and LU do not hold. If A ¼ fa; b; dg, hence R002 ðAÞ ¼ fb; dg. Then A 6 R002 ðAÞ, i.e., U 7 does not hold.
¼
jA\n aprðAÞj : jA[n aprðAÞj
Proposition 3.1. For a family of binary relations Ri ; i ¼ 1; 2; . . . ; n on a non-empty set U the following conditions hold for every A; B U: ðL1 Þ n aprðAÞ ¼ ½n aprðAc Þc ðL2 Þ n aprðUÞ ¼ U. ðL3 Þ n aprðA \ BÞ ¼ n aprðAÞ \ n aprðBÞ. ðL4 Þ n aprðA [ BÞ n aprðAÞ [ n aprðBÞ. ðL5 Þ A B ) n aprðAÞ n aprðBÞ. ðU 1 Þ n aprðAÞ ¼ ½n aprðAc Þc . ðU 2 Þ n aprð;Þ ¼ ;. ðU 3 Þ n aprðA [ BÞ ¼ n aprðAÞ [ n aprðBÞ. ðU 4 Þ n aprðA \ BÞ n aprðAÞ \ n aprðBÞ. ðU 5 Þ A B ) n aprðAÞ n aprðBÞ. ðCOÞ n aprðAc [ BÞ ½n aprðAÞc [ RðBÞ.
Example 2.9. In Example 2.1, if A ¼ fa; b; cg, hence R001 ðAÞ ¼ fag, R001 ðR001 ðAÞÞ ¼ ;. Thus R001 ðAÞ 6¼ R001 ðR001 ðAÞÞ, i.e., L9 does not hold. If A ¼ fag, hence R00 ðAÞ ¼ fa; b; cg, R001 ðR001 ðAÞÞ ¼ U and R001 ðR001 ðAÞÞ ¼ fag. Then R001 ðR001 ðAÞÞ 6¼ R001 ðAÞ and R001 ðAÞ 6¼ R001 ðR001 ðAÞÞ, i.e., U 9 and L10 do R001 ðAÞ ¼ fag and not hold. If A ¼ fa; b; c; eg, hence R001 ðR001 ðAÞÞ ¼ fa; b; cg, so R001 ðR001 ðAÞÞ 6¼ R001 ðAÞ, i.e., U 10 does not hold. Table 1 shows the properties of rough sets by using the lower and upper approximations which is defined in Definitions 2.2, 2.3 and 2.4:
Proof ðL1 Þ.
3. The first kind of generalization In this section, we introduce three approximations which are generalization for a previous three approximations, based on the
n
c
c
½ aprðA Þ ¼ fx 2 U :
n \
!
fx 2 U :
\ ðAc Þ 6¼ ;gc ¼
xRi
i¼1 n \
! \ ðAc Þ ¼ ;g ¼
xRi
i¼1
Table 1 Comparison between the properties of rough sets depending on Definitions 2.2, 2.3 and 2.4 by using any relation R
fx 2 U :
n \
! xRi
Ag ¼n aprðAÞ:
i¼1
Properties
L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 U1 U2 U3 U4 U5 U6 U7 U8 U9 U 10 CO LU
Upper and lower approximations under any relations by using: Definition 2.2
Definition 2.3
Definition 2.4
q q q q q
q
q q q q q
q q q q
q q q q q q q
q
q q q q
q q q q q
q
q q
ðIÞ indicates that the property is satisfied.
x 2 n aprðA \ BÞ () ()
n \ i¼1 n \ i¼1
q q
ðL2 Þ. As U is the universe set, xRi U8i ¼ 1; 2; . . . ; n, then x 2 n aprðUÞ. Thus U n aprðUÞ. ðL3 Þ.
hence n aprðUÞ U, 8x 2 U, we have T ð ni¼1 xRi Þ U, this implies that So n aprðUÞ ¼ U.
xRi A \ B xRi A and
n \
xRi B
i¼1
() x 2 n aprðAÞ and x 2 n aprðBÞ () x 2 ½n aprðAÞ \ n aprðBÞ: Therefore n aprðA \ BÞ ¼ n aprðAÞ \ n aprðBÞ. ðL4 Þ. Let x 2 ½n aprðAÞ [ n aprðBÞ, hence x 2 n aprðAÞ or x 2 n aprðBÞ. T T T So ð ni¼1 xRi Þ A or ð ni¼1 xRi Þ B and ð ni¼1 xRi Þ ðA [ BÞ. Then x 2 n aprðA [ BÞ, i.e., n aprðA [ BÞ n aprðAÞ [ n aprðBÞ. T ðL Þ. Let A B and x 2 n aprðAÞ. Then ð ni¼1 xRi Þ A but A B. So Tn 5 ð i¼1 xRi Þ B, hence x 2 n aprðBÞ. Thus n aprðAÞ n aprðBÞ. ðCOÞ. Let x 62 ½n aprðAÞc [ n aprðBÞ, then x 62 ½n aprðAÞc and Tn x 62 n aprðBÞ, hence x 2 n aprðAÞ and x 62 n aprðBÞ, i.e., i¼1 xRi A Tn Tn Tn c then and hence and i¼1 xRi 6 B, i¼1 xRi 6 A i¼1 xRi 6 B,
H.M. Abu-Donia / Knowledge-Based Systems 21 (2008) 911–919
Tn
c i¼1 ðxÞRi 6 ðA [ BÞ. c ½n aprðAÞ [ n aprðBÞ.
Thus
x 62 n aprðAc [ BÞ,
i.e.,
c n aprðA
ðL4 Þ n apr 0 ðA [ BÞ n apr 0 ðAÞ [ n apr 0 ðBÞ. ðL5 Þ A B ) n apr 0 ðAÞ n apr0 ðBÞ. ðL6 Þ n apr 0 ð;Þ ¼ ;. ðL7 Þ n apr 0 ðAÞ A. ðL9 Þ n apr 0 ðAÞ ¼ n apr0 ðn apr 0 ðAÞÞ. ðU 1 Þ n apr 0 ðAÞ ¼ ½n apr0 ðAc Þc ðU 4 Þ n apr 0 ðA \ BÞ n apr 0 ðAÞ \ n apr 0 ðBÞ. ðU 5 Þ A B ) n apr 0 ðAÞ n apr0 ðBÞ ðU 6 Þ n apr 0 ðUÞ ¼ U ðU 7 Þ A n apr0 ðAÞ. ðU 9 Þ n apr 0 ðn apr 0 ðAÞÞ ¼ n apr 0 ðAÞ. ðLUÞ n apr 0 ðAÞ n apr 0 ðAÞ.
[ BÞ
We can prove U 1 ; U 2 ; U 3 ; U 4 and U 5 as the same as L1 ; L2 ; L3 ; L4 and L5 . h We can introduce an example to prove that the equality of L4 and U 4 are not true in general. T Example 3.1. From Example 2.1, we have got 2i¼1 aRi ¼ fb; eg, T2 T2 T2 T2 i¼1 bRi ¼ fd; cg, i¼1 cRi ¼ ;, i¼1 dRi ¼ fa; cg and i¼1 eRi ¼ ;. If A ¼ fa; bg and B ¼ fc; eg, then 2 aprðAÞ ¼ fc; eg, 2 aprðBÞ ¼ fc; eg, hence 2 aprðAÞ [ 2 aprðBÞ ¼ fc; eg but 2 aprðA [ BÞ ¼ fa; c; d; eg. i.e., 2 aprðA [ BÞ 6¼ 2 aprðAÞ [ 2 aprðBÞ. If A ¼ fag and B ¼ fcg, then 2 2 2 aprðAÞ ¼ fdg, aprðBÞ ¼ fb; dg and aprðA \ BÞ ¼ ;. Thus 2 aprðAÞ \ 2 aprðBÞ 6¼ 2 aprðA \ BÞ. Remark 3.1. For a family of binary relations Ri : i ¼ 1; 2; . . . ; n on a non-empty set U the following conditions do not hold for every A; B U: ðL6 Þ n aprð;Þ ¼ ;. ðL7 Þ n aprðAÞ A. ðL8 Þ A n aprðn aprðAÞÞ. ðL9 Þ n aprðAÞ ¼ n aprðn aprðAÞÞ. ðL10 Þ n aprðAÞ ¼ n aprðn aprðAÞÞ. ðU 6 Þ n aprðUÞ ¼ U. ðU 7 Þ A n aprðAÞ. ðU 8 Þ n aprðn aprðAÞÞ A. ðU 9 Þ n aprðn aprðAÞÞ ¼ n aprðAÞ ðU 10 Þ n aprðn aprðAÞÞ ¼ n aprðAÞ ðLUÞ n aprðAÞ n aprðAÞ The following examples show Remark 3.1. Example 3.2. In Example 2.2, we can see that 2 aprð;Þ ¼ fa; cg and aprðUÞ ¼ fbg. Thus 2 aprð;Þ 6¼ ; and 2 aprðUÞ 6¼ U, i.e., L6 and U 6 do not hold.
2
Example 3.3. In Example 2.1, If A ¼ fag, hence 2 aprðAÞ ¼ fdg, 2 aprð2 aprðAÞÞ ¼ fa; b; dg and 2 aprð2 aprðAÞÞ ¼ 2 aprðAÞ ¼ fc; eg, fbg.Thus 2 aprðAÞ 6 A, A 6 2 aprðAÞ, 2 aprðAÞ 6 2 aprðAÞ, 2 aprð2 aprðAÞÞ 6 A, 2 aprð2 aprðAÞÞ 6¼ 2 aprðAÞ and 2 aprð2 aprðAÞÞ 6¼ 2 aprðAÞ, i.e., L7 , U 7 , LU, U 8 , U 10 and U 9 do not hold.If A ¼ fdg, hence 2 aprðAÞ ¼ fbg and 2 aprð2 aprðAÞÞ ¼ fc; eg. Thus A 6 2 aprð2 aprðAÞÞ and 2 aprðAÞ 6¼ 2 2 aprð aprðAÞÞ, i.e., L8 and L10 do not hold. If A ¼ fa; b; eg, hence aprðAÞ ¼ fa; c; eg and 2 aprð2 aprðAÞÞ ¼ fc; d; eg. Then 2 aprðAÞ6¼2 2 aprð2 aprðAÞÞ. i.e., L9 does not hold.
Proof ðL1 Þ. ½n apr 0 ðAc Þc ¼ ½½2 apr0 ððAc Þc Þc c ¼ 2 apr 0 ðAÞ. ðL4 Þ. Let x 2 ½n apr 0 ðAÞ [ n apr 0 ðBÞ, hence x 2 n apr0 ðAÞ or ST ST x 2 n apr0 ðBÞ, then x 2 ð ni¼1 xRi Þ A or x 2 ð ni¼1 xRi Þ B, this S Tn implies that x 2 ð i¼1 xRi Þ ðA [ BÞ. So x 2 n apr 0 ðA [ BÞ. i.e., 0 0 0 n apr ðA [ BÞ n apr ðAÞ [ n apr ðBÞ. ST ðL5 ). Let A B and x 2 n apr0 ðAÞ. Then x 2 ð ni¼1 xRi Þ A but A B. So ST x 2 ð ni¼1 xRi Þ B, hencex 2 n apr 0 ðBÞ. Thus n apr0 ðAÞ n apr 0 ðBÞ. T T ðL6 Þ. 8x 2 U, we have ð ni¼1 xRi Þ ;, if ð ni¼1 xRi Þ ¼ ;. So 0 n apr ð;Þ ¼ ;. S T T ðL7 Þ. Since n apr 0 ðAÞ ¼ fð ni¼1 xRi Þ : ð ni¼1 xRi Þ Ag. So n apr0 ðAÞ A. S Tn T 0 ðL9 Þ. Since n apr ðAÞ ¼ fð i¼1 xRi Þ : ð ni¼1 xRi Þ Ag, hence n apr 0 S T T S T T ðn apr 0 ðAÞÞ ¼ n apr0 ð fð ni¼1 xRi Þ : ð ni¼1 xRi Þ AgÞ ¼ fð ni¼1 xRi Þ : ð ni¼1 S Tn Tn 0 0 xRi Þ ð fð i¼1 xRi Þ : ð i¼1 xRi Þ AgÞg ¼ n apr ðAÞ. i.e., n apr ðn apr 0 ðAÞÞ ¼ n apr 0 ðAÞ. We can prove U 1 ; U 4 ; U 5 ; U 6 ; U 7 and U 9 as the same as L1 ; L4 ; L5 ; L6 ; L7 and L9 . We can prove LU from L7 and U 7 . h We can introduce an example to prove that the equality of L4 , U 4 , L7 , U 7 and LU does not true in general. Example 3.4. In Example 2.1, if A ¼ fag and B ¼ fcg, hence 0 0 0 and Thus, 2 apr ðAÞ ¼ ;, 2 apr ðBÞ ¼ ; 2 apr ðA [ BÞ ¼ fa; cg. 0 0 0 2 apr ðA [ BÞ 6¼ 2 apr ðAÞ [ 2 apr ðBÞ. If A ¼ fb; cg and B ¼ fc; eg, then 2 apr 0 ðAÞ ¼ U, 2 apr 0 ðBÞ ¼ U and 2 apr0 ðA \ BÞ ¼ fa; c; dg. Thus 2 apr 0 ðAÞ \ 2 apr 0 ðBÞ 6¼ 2 apr 0 ðA \ BÞ. If A ¼ fcg, then 2 apr 0 ðAÞ ¼ ;, 2 0 2 apr 0 ðAÞ ¼ fa; c; dg. Thus apr0 ðAÞ 6¼ A and 2 apr ðAÞ 6¼ A, 0 2 0 apr ðAÞ ¼ 6 apr ðAÞ. 2 Remark 3.2. For a family of binary relations Ri : i ¼ 1; 2; . . . ; n on a non-empty set U the following conditions do not hold for every A; B U: ðL2 Þ n apr 0 ðUÞ ¼ U. ðL3 Þ n apr 0 ðA \ BÞ ¼ n apr 0 ðAÞ \ n apr 0 ðBÞ ðL8 Þ A n apr0 ðn apr 0 ðAÞÞ. ðL10 Þ n apr 0 ðAÞ ¼n apr 0 ðn apr 0 ðAÞÞ. ðU 2 Þ n apr 0 ð;Þ ¼ ;. ðU 3 Þ n apr 0 ðA [ BÞ ¼ n apr 0 ðAÞ [ n apr 0 ðBÞ. ðU 8 Þ n apr 0 ðn apr 0 ðAÞÞ A. ðU 10 Þ n apr 0 ðn apr0 ðAÞÞ ¼ n apr0 ðAÞ. ðCOÞ n apr 0 ðAc [ BÞ ½n apr 0 ðAÞc [ n apr 0 ðBÞ.
Definition 3.2. Let fRi : i ¼ 1; 2; . . . ; ng be a family of binary relations on a non-empty set U, for any set A U. We can introduce a definition for n-lower and n-upper approximations of any set A U according to Ri as ( ! ! ) n n \ [ \ 0 xRi : xRi A ; n apr ðAÞ ¼ i¼1 n
i¼1
apr 0 ðAÞ ¼ ½n apr 0 ðAc Þc :
By using Definition 3.2, we can define the accuracy measure of any set A as the following: jA \ n apr0 ðAÞj 0 : n aany ðAÞ ¼ jA [ n apr0 ðAÞj Proposition 3.2. For a family of binary relations Ri : i ¼ 1; 2; . . . ; n on a non-empty set U, the following conditions hold for every A; B U: ðL1 Þ n apr0 ðAÞ ¼ ½n apr 0 ðAc Þc .
915
The following examples show Remark 3.2. Example 3.5. In Example 2.6, we have 2 apr 0 ðUÞ ¼ fb; c; dg and apr 0 ð;Þ ¼ fa; eg. Thus 2 apr 0 ðUÞ 6¼ U and 2 apr0 ð;Þ 6¼ ;. i.e., L2 and U 2 do not hold. 2
Example 3.6. In Example 2.1, if A ¼ fc; ag and B ¼ fc; dg, hence 0 0 and 2 apr0 ðA \ BÞ ¼ ;. Thus 2 apr ðAÞ ¼ fa; cg, 2 apr ðBÞ ¼ fc; dg 0 0 0 apr ðA \ BÞ ¼ 6 apr ðAÞ \ apr ðBÞ. i.e., L3 does not hold. If 2 2 2
916
H.M. Abu-Donia / Knowledge-Based Systems 21 (2008) 911–919
A ¼ fa; eg and B ¼ fb; dg, then 2 apr 0 ðAÞ ¼ fa; b; eg, 2 apr0 ðBÞ ¼ fb; d; eg and 2 apr0 ðA [ BÞ ¼ U. Thus 2 apr 0 ðAÞ [ 2 apr 0 ðBÞ 6¼ 2 apr0 ðA [ BÞ. i.e., U 3 does not hold. If A ¼ fag, hence 2 apr0 ðAÞ ¼ fag and 2 apr 0 ð2 apr0 ðAÞÞ ¼ ;. Thus A 6 2 apr 0 ð2 apr 0 ðAÞÞand 2 apr 0 ðAÞ 6¼ 0 2 0 2 apr ð apr ðAÞÞ, i.e., L8 and L10 do not hold. If A ¼ fa; cg, hence 0 2 0 0 2 0 0 2 apr ðAÞ¼ fa; cg and apr ð2 apr ðAÞÞ ¼ fa; c; dg, so apr ð2 apr ðAÞÞ 6 A 2 0 0 0 and apr ð2 apr ðAÞÞ 6¼ 2 apr ðAÞ. i.e., U 8 and U 10 do not hold. If A ¼ fa; b; cg and B ¼ fb; cg, hence 2 apr0 ðAc [ BÞ ¼ fb; c; d; eg, ½2 apr0 ðAÞc ¼ fb; d; eg and 2 apr 0 ðBÞ ¼ ;, then ½2 apr0 ðAÞc [ 2 apr0 ðBÞ ¼ fb; d; eg. Thus 2 apr 0 ðAc [ BÞ 6¼ ½2 apr 0 ðAÞc [n apr 0 ðBÞ. i.e., CO does not hold. Definition 3.3. Let fRi : i ¼ 1; 2; . . . ; ng be a family of binary relations on a non-empty set U, for any set A U. We can introduce a definition for n-lower and n-upper approximations of A according to Ri , as the following: n apr
n
00
00
c
c
ðAÞ ¼ ½ apr ðA Þ ; ( ! n [ \ n apr 00 ðAÞ ¼ xRi :
n \
i¼1
! xRi
In the same manner we can prove U 1 , U 2 , U 3 , U 4 , U 5 , U 8 , L8 and CO. h We can introduce an example to show that the equality of L4 , U 4 , L8 , U 8 and CO are not true in general. Example 3.7. In Example 2.1, if A ¼ fag and B ¼ fcg, hence 00 00 00 2 apr 00 ðAÞ ¼ fa; cg, 2 apr ðAÞ ¼ ;, 2 apr ðBÞ ¼ ;, 2 apr ðA [ BÞ ¼ fag, 2 2 2 apr 00 ðBÞ ¼ fa; c; dg and apr00 ðA \ BÞ ¼ ;. Thus apr 00 ðAÞ\ 2 apr 00 ðBÞ6¼2 apr 00 ðA \ BÞ and 2 apr00 ðAÞ [ 2 apr 00 ðBÞ 6¼ 2 apr00 ðA [ BÞ. If A ¼ fcg, hence 2 apr 00 ðAÞ ¼ fa; c; dg, 2 apr00 ð2 apr00 ðAÞÞ ¼ fa; c; dg. Thus A 6¼ 2 apr00 ð2 apr00 ðAÞÞ. If A ¼ fa; c; eg, hence 2 apr00 ðAÞ ¼ fag, and 2 apr 00 ð2 apr 00 ðAÞÞ ¼ fa; cg. Thus, 2 apr00 ð2 apr00 ðAÞÞ 6¼ A. If A ¼ fb; c; eg and B ¼ fb; eg, hence ½2 apr00 ðAÞc ¼ fa; c; dg, 2 apr00 ðBÞ ¼ fb; eg and c c c 00 00 00 00 2 apr ðA [ BÞ ¼ fb; eg. Thus ½2 apr ðAÞ [ 2 apr ðBÞ 6¼ 2 apr ðA [ BÞ. Remark 3.3. For a family of binary relations fRi : i ¼ 1; 2; . . . ; ng on a non-empty set U the following conditions hold for every A; B U:
)
ðL6 Þ n apr 00 ð;Þ ¼ ;. ðL7 Þ n apr 00 ðAÞ A. ðL9 Þ n apr 00 ðAÞ ¼ n apr 00 ðn apr 00 ðAÞÞ. ðL10 Þ n apr00 ðAÞ ¼ n apr00 ðn apr00 ðAÞÞ. ðU 6 Þ n apr 00 ðUÞ ¼ U. ðU 7 Þ A n apr00 ðAÞ. ðU 9 Þ n apr 00 ðn apr 00 ðAÞÞ ¼ n apr00 ðAÞ. ðU 10 Þ n apr00 ðn apr00 ðAÞÞ ¼ n apr 00 ðAÞ. ðLUÞ n apr00 ðAÞ n apr00 ðAÞ.
\ A 6¼ ; :
i¼1
By using Definition 3.3, we can define the accuracy measure of any set A as the following: 00 n aany ðAÞ
¼
jA\n apr00 ðAÞj : jA[n apr00 ðAÞj
Proposition 3.3. For a family of binary relations fRi : i ¼ 1;2; .. .; ng on a non-empty set U the following conditions hold for every A;B U: 00
n
00
c
c
ðL1 Þ n apr ðAÞ ¼ ½ apr ðA Þ . ðL2 Þ n apr00 ðUÞ ¼ U. ðL3 Þ n apr00 ðA \ BÞ ¼ n apr00 ðAÞ \ n apr00 ðBÞ. ðL4 Þ n apr00 ðA [ BÞ n apr00 ðAÞ [ n apr00 ðBÞ. ðL5 Þ A B ) n apr00 ðAÞ n apr 00 ðBÞ. ðL8 Þ A n apr00 ðn apr00 ðAÞÞ. ðU 1 Þ n apr00 ðAÞ ¼ ½n apr 00 ðAc Þc . ðU 2 Þ n apr00 ð;Þ ¼ ;. ðU 3 Þ n apr00 ðA [ BÞ¼n apr 00 ðAÞ [ n apr00 ðBÞ. ðU 4 Þ n apr00 ðA \ BÞ n apr00 ðAÞ \ n apr00 ðBÞ. ðU 5 Þ A B)n apr 00 ðAÞ n apr 00 ðBÞ. ðU 8 Þ n apr00 ðn apr00 ðAÞÞ A. ðCOÞ n apr 00 ðAc [ BÞ ½n apr00 ðAÞc [ n apr00 ðBÞ.
Proof ðL1 Þ. From Definition 3.3, we have n apr00 ðAÞ ¼ ½n apr00 ðAc Þc . S T T ðL2 Þ. n apr 00 ðUÞ ¼ ½n apr 00 ð/Þc ¼ ½ f ni¼1 xRi : ni¼1 xRi \ / 6¼ /gc ¼ /c ¼ U. S Tn c c c c c 00 n 00 n 00 ðL3 Þ. n apr ðA \ BÞ ¼ ½ apr ðA \ BÞ ¼ ½ apr ðA [ B Þ ¼ ½ f i¼1 Tn S Tn Tn T c c c c xRi : i¼1 xRi \ ðA [ B Þ 6¼ /g ¼ ½ f i¼1 xRi : i¼1 xRi \ A 6¼ /gc Tn S Tn c c c c c c ½ f i¼1 xRi : i¼1 xRi \ B 6¼ /g ¼ ½n apr 00 ðA Þ \ ½n apr 00 ðB Þ ¼ n apr 00 ðAÞ \ n apr 00 ðBÞ. ðL4 Þ. Let x2n apr00 ðAÞ[n apr00 ðBÞ, hence x 2 n apr 00 ðAÞ or x 2 n apr 00 ðBÞ. So x 2 ½n apr00 ðAc Þc or x 2 ½n apr 00 ðBc Þc . This implies that x 62 n apr 00 ðAc Þ T or x 62 n apr00 ðBc Þ, then 8s 2 U, we have x 62 ni¼1 sRi such that Tn T Tn c c sR \ B 6¼ /. Consequently, x 62 ni¼1 sRi 8 i¼1 sRi \ A 6¼ / and Tn i¼1 i c c it is means that s 2 U such that i¼1 sRi \ ðA \ B Þ 6¼ /, x62n apr00 ðA [ BÞc i.e, x2n apr 00 ðA [ BÞ. Hence n apr 00 ðAÞ [ n apr00 ðBÞ 00 n apr ðA [ BÞ. ðL5 Þ. Let A B and x 62 n apr00 ðBÞ, hence x 62 ½n apr 00 ðBc Þc . So T x 2 n apr00 ðBc Þ and so 9s 2 U such that x 2 ni¼1 sRi and Tn Tn c c c c i¼1 sRi \ A 6¼ /, i¼1 sRi \ B 6¼ / but A B . This implies that c c c n 00 n 00 it is means that x 2 apr ðA Þ and so x 62 ½ apr ðA Þ ¼ n apr00 ðAÞ. Therefore n apr00 ðAÞ n apr 00 ðBÞ.
The following examples show Remark 3.3. Example 3.8. In Example 2.6, we have 2 apr 00 ð;Þ ¼ fa; cg and apr 00 ðUÞ ¼ fb; c; dg, i.e., L6 and U 6 do not hold. If A ¼ fb; dg, hence 00 00 2 apr ðAÞ ¼ fa; b; d; eg. Thus 2 apr ðAÞ 6 A. i.e., L7 does not hold. If 00 A ¼ fa; eg, hence 2 apr ðAÞ ¼ fag and 2 apr 00 ðAÞ ¼ ;. Thus A 6 2 apr 00 ðAÞ and 2 apr00 ðAÞ 6 2 apr 00 ðAÞ. i.e., U 7 and LU do not hold. 2
Example 3.9. In Example 2.1, if A ¼ fa; cg, hence 2 apr00 ðAÞ ¼ fag, 00 00 00 00 00 2 apr ð2 apr ðAÞÞ ¼ ;. Thus 2 apr ðAÞ 6¼ 2 apr ð2 apr ðAÞÞ, i.e., L9 does 2 00 apr ðAÞ ¼ fb; c; d; eg and not hold. If A ¼ fb; d; eg, hence 2 apr 00 ð2 apr 00 ðAÞÞ ¼ U. Then 2 apr00 ð2 apr00 ðAÞÞ 6¼ 2 apr 00 ðAÞ, i.e., U 9 does 2 apr 00 ðAÞ ¼ fa; cg and not hold. If A ¼ fag, hence 00 2 00 2 00 00 2 00 2 apr ð apr ðAÞÞ ¼ fag. Thus apr ðAÞ 6¼ 2 apr ð apr ðAÞÞ, i.e., L10 does 00 and not hold. If A ¼ fa; b; cg, hence 2 apr ðAÞ ¼ fag 2 00 00 2 00 00 00 apr ð2 apr ðAÞÞ ¼ fa; cg, so apr ð2 apr ðAÞÞ 6¼ 2 apr ðAÞ, i.e., U 10 does not hold. Table 2 shows the properties of rough sets by using the lower and upper approximations which is defined in Definitions 3.1, 3.2 and 3.3. 4. The second kind of generalization In this section, we will introduce another two approximations which are a generalization for approximations defined in Definitions 2.2 and 2.3 by used a family of binary relations. Definition 4.1. Let fRi : i ¼ 1; 2; . . . ; ng be a family of binary relations on a non-empty set U, for any set A U. We can introduce a definition for n-lower and n-upper approximations of A according to Ri as the following: n apr
ðAÞ ¼
n [ i¼1
Ri ðAÞ and n apr ðAÞ ¼
n \
Ri ðAÞ;
i¼1
where Ri ðAÞ and Ri ðAÞ defined in Definition 2.2. we can define the accuracy measure of any set A as the following:
917
H.M. Abu-Donia / Knowledge-Based Systems 21 (2008) 911–919 Table 2 Comparison between the properties of rough sets depending on Definitions 3.1, 3.2 and 3.3 by using any relations Ri : i ¼ 1; 2; . . . ; n Properties
Upper and lower approximations under any relations by using: Definition 3.1
Definition 3.2
Definition 3.3
L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 U1 U2 U3 U4 U5 U6 U7 U8 U9 U 10 CO LU
q q q q q
q
q q q q q
q q q q
Remark 4.1. Let Ri be a family of binary relations. Then the following conditions are not hold for every A; B U: ðL3 Þ n apr ðA \ BÞ ¼ n apr ðAÞ \ n apr ðBÞ. ðL6 Þ n apr ð;Þ ¼ ;. ðL7 Þ n apr ðAÞ A. ðL8 Þ A n apr ðn apr ðAÞÞ. ðL9 Þ n apr ðn apr ðAÞÞ ¼ n apr ðAÞ. ðL10 Þ n apr ðAÞ ¼ n apr ðn apr ðAÞÞ. ðU 3 Þ n apr ðA [ BÞ ¼ n apr ðAÞ [ n apr ðBÞ. ðU 6 Þ n apr ðUÞ ¼ U. ðU 7 Þ A n apr ðAÞ. ðU 8 Þ n apr ðn apr ðAÞÞ A. ðU 9 Þ n apr ðn apr ðAÞÞ ¼ n apr ðAÞ. ðU 10 Þ n apr ðn apr ðAÞÞ ¼ n apr ðAÞ. ðLUÞ n apr ðAÞ n apr ðAÞ. ðCOÞ n apr ðAc [ BÞ ½n apr ðAÞc [ n apr ðBÞ.
q q q q q q q
q
q q q q
q q q q q
The following examples show Remark 4.1.
q q q
q q
ðIÞ indicates that the property is satisfied. n aany ðAÞ
¼
jA \ n apr ðAÞj : jA [ n apr ðAÞj
Proposition 4.1. For a family of any binary relations fRi ; i ¼ 1; 2; . . . ; ng on a non-empty set U, the following conditions hold for every A; B U: ðL1 Þ n apr ðAÞ ¼ ½n apr ðAc Þc ðL2 Þ n apr ðUÞ ¼ U. ðL4 Þ n apr ðA [ BÞ n apr ðAÞ [ n apr ðBÞ. ðL5 Þ A B ) n apr ðAÞ n apr ðBÞ. ðU 1 Þ n apr ðAÞ ¼ ½n apr ðAc Þc . ðU 2 Þ n apr ð;Þ ¼ ;. ðU 4 Þ n apr ðA \ BÞ n apr ðAÞ \ n apr ðBÞ. ðU 5 Þ A B ) n apr ðAÞ n apr ðBÞ.
Example 4.2. In Example 2.1, if A ¼ fa; b; eg and B ¼ fc; d; eg, hence ðAÞ ¼ feg, 2 apr ðBÞ ¼ fb; c; eg and 2 apr ðA \ BÞ ¼ ;. Thus 2 apr ðA \ BÞ 6¼ 2 apr ðAÞ \ 2 apr ðBÞ. i.e., L3 does not hold. If A ¼ fdg and B ¼ feg, then 2 apr ðAÞ ¼ fbg, 2 apr ðBÞ ¼ fag and 2 apr ðA [ BÞ ¼ fa; b; cg. Thus 2 apr ðAÞ [ 2 apr ðBÞ 6¼ 2 apr ðA [ BÞ. i.e., U 3 does not hold. If A ¼ fbg, hence 2 apr ðAÞ ¼ feg and 2 apr ðAÞ ¼ fag, so 2 2 2 apr ðAÞ 6¼ A, apr ðAÞ 6¼ A and 2 apr ðAÞ 6¼ apr ðAÞ, i.e., L7 , U 7 and LU do not hold.If A ¼ feg, hence 2 apr ðAÞ ¼ fag and 2 2 2 apr ð apr ðAÞÞ ¼ ;.Thus A 6 2 apr ð apr ðAÞÞ, i.e., L8 does not hold.If n apr ðAÞ ¼ feg, apr ðn apr ðAÞÞ ¼ fag, A ¼ fdg, hence 2 2 2 n apr ð apr ðAÞÞ ¼ fag.So apr ðn apr ðAÞÞ 6 A, 2 apr ðAÞ 6¼ 2 apr ð2 apr ðAÞÞ and 2 apr ð2 apr ðAÞÞ6¼2 apr ðAÞ, i.e., U 8 , U 9 and U 10 do not hold.If A ¼ fa; b; eg, hence2 apr ðAÞ ¼ feg and 2 apr ð2 apr ðAÞÞ ¼ ;, so 2 apr ðAÞ 6¼ 2 apr ð2 apr ðAÞÞ. i.e., L9 does not hold. If A ¼ fag, hence 2 2 2 apr ðAÞ ¼ fdg and So apr ðAÞ 6¼ 2 apr ð apr ðAÞÞ ¼ feg. 2 2 apr ð apr ðAÞÞ. i.e., L10 does not hold. If A ¼ fc; d; eg and then B ¼ fa; cg, hence ½2 apr ðAÞc ¼ fa; dg, 2 apr ðBÞ ¼ ;, ½2 apr ðAÞc [ 2 apr ðBÞ ¼ fa; dg and 2 apr ðAc [ BÞ ¼ fe; dg. Thus ½2 apr ðAÞc [ 2 apr ðBÞ 6¼ 2 apr ðAc [ BÞ. i.e., CO does not hold. 2 apr
Example 4.3. In Example 2.2, we have apr ðUÞ ¼ fa; bg, i.e., L6 and U 6 do not hold.
2 apr
ð;Þ ¼ fcg and
2
Proof T S ðL1 Þ. ½n apr ðAc Þc ¼ ½ ni¼1 Ri ðAc Þc ¼ ni¼1 apri ðAÞ ¼ n R ðAÞ. ðL2 Þ. As U is the universe set, hence n apr ðUÞ U. 8x 2 U, we have xRi U8i ¼ 1; 2; . . . ; n, so x 2 Ri ðUÞ8i ¼ 1; 2; . . . ; n and so x 2 n apr ðUÞ. Then U n apr ðUÞ. Thus n apr ðUÞ ¼ U. ðL4 Þ. Let x 2 ½n apr ðAÞ [ n apr ðBÞ. Then x 2 n apr ðAÞ or S S x 2 n apr ðBÞ, so x 2 ni¼1 Ri ðAÞ or x 2 ni¼1 Ri ðBÞ, for fixed i 2 f1; 2; . . . ; ng, we have x 2 Ri ðAÞ or x 2 Ri ðBÞ. Then xRi A or xR B, so xRi ðA [ BÞ, hence x 2 Ri ðA [ BÞ but n apr ðA [ BÞ ¼ Sni i¼1 Ri ðA [ BÞ. So x 2 n apr ðA [ BÞ. i.e., n apr ðA [ BÞ n apr ðAÞ [n apr ðBÞ. S ðL5 Þ. Let A B and x 2 n apr ðAÞ. Then x 2 ni¼1 Ri ðAÞ. So there exists i such that x 2 Ri ðAÞ. Then xRi A but A B. So xRi A B, hence x 2 n apr ðBÞ. Thus n apr ðAÞ n apr ðBÞ. We can prove U 1 ; U 2 ; U 4 and U 5 as the same as L1 ; L2 ; L4 and L5 . h We can introduce an example to show that the equality of L4 and U 4 are not true in general. Example 4.1. In Example 2.1, if A ¼ fcg and B ¼ fb; dg, hence and 2 apr ðA [ BÞ ¼ fb; eg. i.e., 2 apr ðAÞ ¼ ;, 2 apr ðBÞ ¼ feg 2 apr ðAÞ [ 2 apr ðBÞ 6¼ 2 apr ðA [ BÞ. If A ¼ fcg and B ¼ fdg, hence 2 apr ðAÞ ¼ fb; dg, 2 apr ðBÞ ¼ fbg and 2 apr ðA \ BÞ ¼ ;. Thus 2 apr ðAÞ \ 2 apr ðBÞ 6¼ 2 apr ðA \ BÞ.
Definition 4.2. Let fRi : i ¼ 1; 2; . . . ; ng be a family of binary relations on a non-empty set U, for any set A U. We can introduce a definition for n-lower and n-upper approximations of A according to Ri as the following: n apr
ðAÞ ¼
n [
Ri ðAÞ
and
i¼1
n
apr ðAÞ ¼
n \
Ri ðAÞ;
i¼1
where R0i ðAÞ and R0i ðAÞ, defined in Definition 2.3. The accuracy measure of any set A will be defined as the following: 0 n aany ðAÞ
¼
jA\n apr 0 ðAÞj : jA [ n apr0 ðAÞj
Proposition 4.2. For a family of any binary relations Ri : i ¼ 1; 2; . . . ; n on a non-empty set U, the following conditions hold for every A; B U: ðL1 Þ ðL4 Þ ðL5 Þ ðL6 Þ ðL7 Þ ðL9 Þ
0 ðAÞ ¼ ½n apr0 ðAc Þc . 0 apr ðA [ BÞ n apr0 ðAÞ [ n apr0 ðBÞ. n A B ) n apr 0 ðAÞ n apr0 ðBÞ. 0 n apr ð;Þ ¼ ;. 0 apr ðAÞ A. n 0 0 0 n apr ðn apr ðAÞÞ¼n apr ðAÞ. n apr
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H.M. Abu-Donia / Knowledge-Based Systems 21 (2008) 911–919
ðU 1 Þ n apr0 ðAÞ ¼ ½n apr0 ðAc Þc . ðU 4 Þ n apr0 ðA \ BÞ n apr 0 ðAÞ \ n apr0 ðBÞ. ðU 5 Þ A B)n apr 0 ðAÞn apr 0 ðBÞ. ðU 6 Þ n apr0 ðUÞ ¼ U. ðU 7 Þ A n apr0 ðAÞ. ðU 9 Þ n apr0 ðn apr 0 ðAÞÞ ¼ n apr0 ðAÞ. ðLUÞ n apr 0 ðAÞ n apr 0 ðAÞ.
Proof T S ðL1 Þ. ½n apr 0 ðAc Þc ¼ ½ ni¼1 Ri ðAc Þc ¼ ni¼1 R0i ðAÞ ¼ n apr 0 ðAÞ. 0 ðL4 Þ. Let x 2 ½n apr ðAÞ [ n apr0 ðBÞ. Then x 2 n apr 0 ðAÞ or S S x 2 n apr0 ðBÞ, so x 2 ni¼1 R0i ðAÞ or x 2 ni¼1 R0i ðBÞ, for fixed S i 2 f1; 2; . . . ; ng, we have x 2 R0i ðAÞ or x 2 R0i ðBÞ. Then x 2 ðsRi Þ A S S or x 2 ðsRi Þ B, where s 2 U. So x 2 ðsRi Þ ðA [ BÞ, hence Sn 0 0 but Thus x 2 R0i ðA [ BÞ n apr ðA [ BÞ ¼ i¼1 Ri ðA [ BÞ. 0 0 0 x 2 n apr ðA [ BÞ.i.e., n apr ðA [ BÞ n apr ðAÞ [ n apr 0 ðBÞ. S ðL5 Þ. Let A B and x 2 n apr 0 ðAÞ, then x 2 ni¼1 R0i ðAÞ, for some i, S we have x 2 R0i ðAÞ, then x 2 ðsRi Þ A, where s 2 U, but A B. So S x 2 ðsRi Þ B, hence x 2 R0i ðBÞ. Thus x 2 n apr 0 ðBÞ. Thus 0 0 n apr ðAÞ n apr ðBÞ. S ðL6 Þ. Since n apr0 ð;Þ ¼ ni¼1 R0i ð;Þ, but we have got that 0 Ri ð;Þ ¼ ;8i ¼ 1; 2; 3; . . . ; n. Thus n apr 0 ð;Þ ¼ ;. S ðL7 Þ. Since n apr 0 ðAÞ ¼ ni¼1 R0i ðAÞ, but we have got that 0 Ri ðAÞ A8i ¼ 1; 2; . . . ; n. Thus n apr 0 ðAÞ A. S ðL9 Þ. Since n apr 0 ðAÞ ¼ ni¼1 R0i ðAÞ, but we have got that R0i ðR0i ðAÞÞ ¼ R0i ðAÞ8i ¼ 1; 2; . . . ; n. Thus n apr0 ðn apr0 ðAÞÞ ¼ n apr 0 ðAÞ. We can prove U 1 ; U 4 ; U 5 ; U 6 , U 7 and U 9 as the same as L1 ; L4 ; L5 ; L6 , L7 and L9 . We can prove LU from L7 and U 7 . h We can introduce an example to show that the equality of L4 , U 4 , L7 , U 7 and LU are not true in general. Example 4.4. In Example 2.1, if A ¼ fcg and B ¼ fb; eg, hence 0 0 and 2 apr 0 ðA [ BÞ ¼ fb; c; eg. i.e., 2 apr ðAÞ ¼ ;, 2 apr ðBÞ ¼ fbg 0 0 0 apr ðAÞ [ apr ðBÞ ¼ 6 apr ðA [ BÞ. If A ¼ fb; cg and B ¼ fc; dg, 2 2 2 hence 2 apr 0 ðAÞ ¼ fb; c; eg, 2 apr 0 ðBÞ ¼ fa; c; d; eg, 2 apr0 ðA \ BÞ ¼ fcg and 2 apr 0 ðAÞ ¼ fbg Thus 2 apr0 ðAÞ \ 2 apr 0 ðBÞ 6¼ 2 apr0 ðA \ BÞ, 0 2 0 0 2 0 2 apr ðAÞ 6¼ A, apr ðAÞ 6¼ A and 2 apr ðAÞ 6¼ apr ðAÞ. Remark 4.2. For a family of any binary relations fRi : i ¼ 1; 2; . . . ; ng on a non-empty set U, the following conditions not hold for every A; B U: ðL2 Þ n apr0 ðUÞ ¼ U. ðL3 Þ n apr0 ðA \ BÞ ¼ n apr 0 ðAÞ \ n apr0 ðBÞ. ðL8 Þ A n apr0 ðn apr 0 ðAÞÞ. ðL10 Þ n apr 0 ðAÞ ¼ n apr 0 ðn apr 0 ðAÞÞ. ðU 2 Þ n apr0 ð;Þ ¼ ;. ðU 3 Þ n apr0 ðA [ BÞ ¼ n apr 0 ðAÞ [ n apr0 ðBÞ. ðU 8 Þ n apr0 ðn apr 0 ðAÞÞ A. ðU 10 Þ n apr 0 ðn apr0 ðAÞÞ ¼ n apr 0 ðAÞ. ðCOÞ n apr 0 ðAc [ BÞ ½n apr 0 ðAÞc [ n apr0 ðBÞ.
2
apr 0 ð2 apr 0 ðAÞÞ ¼ U. So 2 apr0 ð2 apr0 ðAÞÞ 6 A and 2 apr 0 ð2 apr 0 ðAÞÞ 6¼2 apr 0 ðAÞ. i.e., U 8 and U 10 do not hold. If A ¼ fa; b; cg and B ¼ fb; cg, hence 2 apr0 ðAc [ BÞ ¼ fb; c; d; eg, ½2 apr0 ðAÞc ¼ fd; eg and 2 apr 0 ðBÞ ¼ fbg. Thus 2 apr0 ðAc [ BÞ 6¼ ½2 apr 0 ðAÞc [ 2 apr0 ðBÞ. i.e., CO does not hold. Definition 4.3. Let fRi : i ¼ 1; 2; . . . ; ng be a family of binary relations on a non-empty set U, for any set A U. We can introduce a definition for n-lower and n-upper approximations of A according to Ri as the following: n apr
00
ðAÞ ¼
n [
R00i ðAÞ and n apr 00 ðAÞ ¼
i¼1
n \
R00i ðAÞ;
i¼1
where R00i ðAÞ and R00i ðAÞ, defined in Definition 2.4. The accuracy measure of any set A will be defined as the following: 00 n aany ðAÞ
¼
jA \ n apr 00 ðAÞj : jA [ n apr 00 ðAÞj
Proposition 4.3. For a family of any binary relations Ri : i ¼ 1; 2; . . . ; n on a non-empty set U, the following conditions hold for every A; B U: ðL1 Þ n apr 00 ðAÞ ¼ ½n apr00 ðAc Þc . ðL2 Þ n apr 00 ðUÞ ¼ U. ðL4 Þ n apr 00 ðA [ BÞ n apr 00 ðAÞ [ n apr 00 ðBÞ. ðL5 Þ A B ) n apr 00 ðAÞ n apr 00 ðBÞ. ðU 1 Þ n apr 00 ðAÞ ¼ ½n apr00 ðAc Þc . ðU 2 Þ n apr 00 ð;Þ ¼ ;. ðU 4 Þ n apr 00 ðA \ BÞ n apr 00 ðAÞ \ n apr 00 ðBÞ. ðU 5 Þ A B ) n apr 00 ðAÞ n apr 00 ðBÞ. S T Proof. ðU 1 Þ. ½n apr 00 ðAc Þc ¼ ½ ni¼1 R00i ðAc Þc ¼ ni¼1 R00i ðAÞ ¼ n apr 00 ðAÞ. ðU 2 Þ. As xRi \ ; ¼ ;8i ¼ 1; 2; . . . ; n, hence n apr 00 ð;Þ ¼ ;. ðU 4 Þ. Let x 62 ½n apr 00 ðAÞ \ n apr 00 ðBÞ, hence x 62 n apr00 ðAÞ or x 62 n apr 00 ðBÞ. So 9i 2 f1; 2; . . . ; ng, s.t x 62 R00i ðAÞ or x 62 R00i ðBÞ, so x 62 R00i ðAÞ \ R00i ðBÞ, but we have got that R00i ðA \ BÞ R00i ðAÞ \ R00i ðBÞ. So This implies that x 62 n apr00 ðA \ BÞ, i.e., x 62 R00i ðA \ BÞ. n apr 00 ðA \ BÞ n apr00 ðAÞ \ n apr 00 ðBÞ. ðU 5 Þ. Let A B and x 2 n apr00 ðAÞ, hence 8i ¼ 1; 2; . . . ; n, we have x 2 R00i ðAÞ. So there exists s 2 U such that x 2 sRi and sRi \ A 6¼ /8i ¼ 1; 2; . . . ; n. Since A B, then sRi \ B 6¼ /. This implies that x 62 R00i ðBÞ8i ¼ 1; 2; . . . ; n, and so x2n apr00 ðBÞ. Therefore n apr 00 ðAÞ n apr 00 ðBÞ. We can prove L1 ; L2 ; L4 and L5 as the same as U 1 ; U 2 ; U 4 and U5 . h We can introduce an example to prove that the equality of L4 and U 4 is not true in general. Example 4.6. In Example 2.1, if A ¼ fa; bg and B ¼ fcg, hence 00 00 00 ðAÞ ¼ ;, and i.e., 2 apr ðBÞ ¼ ; 2 apr ðA [ BÞ ¼ fag. 00 00 00 2 apr ðAÞ [ 2 apr ðBÞ 6¼ 2 apr ðA [ BÞ. If A ¼ fa; bg and B ¼ fa; cg, hence 2 apr00 ðAÞ ¼ U, 2 apr00 ðBÞ ¼ U and 2 apr 00 ðA \ BÞ ¼ fa; cg. Thus 2 apr 00 ðAÞ \ 2 apr 00 ðBÞ 6¼ 2 apr00 ðA \ BÞ. 2 apr
The following examples show Remark 4.2. Example 4.5. In Example 2.2, we have got 2 apr 0 ðUÞ ¼ fb; cg and apr0 ð;Þ ¼ fag. Thus 2 apr0 ðUÞ 6¼ U and 2 apr0 ð;Þ 6¼ ;. i.e., L2 and U 2 do not hold. If A ¼ fa; dg and B ¼ fa; b; cg, hence 2 apr0 ðAÞ ¼ fa; dg, 0 0 0 2 apr ðBÞ ¼ fa; b; cg and 2 apr ðA \ BÞ ¼ ;. Thus 2 apr ðA \ BÞ 6¼ 2 apr ðAÞ \ 2 apr ðBÞ. i.e., L3 does not hold. If A ¼ fcg and B ¼ fdg, hence 2 apr0 ðAÞ ¼ fcg, 2 apr 0 ðBÞ ¼ fdg2 apr 0 ðA [ BÞ ¼ fa; c; d; eg. So 2 apr0 ðAÞ [ 2 apr 0 ðBÞ 6¼ 2 apr0 ðA [ BÞ. i.e., U 3 does not hold. If A ¼ fag, hence 2 apr 0 ðAÞ ¼ fag and 2 apr 0 ð2 apr 0 ðAÞÞ ¼ ;. Thus A 6 2 apr0 ð2 apr0 ðAÞÞ and 2 apr 0 ðAÞ 6¼ 2 apr0 ð2 apr0 ðAÞÞ. i.e., L8 and L10 do not hold. If A ¼ fb; c; dg, hence 2 apr 0 ðAÞ ¼ fb; c; dg and 2
Remark 4.3. For a family of any binary relations fRi : i ¼ 1; 2; . . . ; ng on a non-empty set U, the following conditions do not hold for every A; B U: ðL3 Þ ðL6 Þ ðL7 Þ ðL8 Þ
ðA \ BÞ ¼ n apr 00 ðAÞ \ n apr 00 ðBÞ. n apr ð;Þ ¼ ;. 00 n apr ðAÞ A. A n apr00 ðn apr 00 ðAÞÞ. n apr
00 00
H.M. Abu-Donia / Knowledge-Based Systems 21 (2008) 911–919 Table 3 Comparison between the properties of rough sets depending on Definitions 4.1, 4.2 and 4.3 by using any relations Ri : i ¼ 1; 2; . . . ; n Properties
L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 U1 U2 U3 U4 U5 U6 U7 U8 U9 U 10 CO LU
Upper and lower approximations under any relations by using: Definition 4.1
Definition 4.2
Definition 4.3
q q
q
q q
q q
q q q q
q q
q q q
q
q q
q q
q q q q
q q
q
q
ðIÞ indicates that the property is satisfied.
ðL9 Þ n apr00 ð2 apr 00 ðAÞÞ ¼ n apr00 ðAÞ. ðL10 Þ n apr 00 ðAÞ ¼ n apr 00 ðn apr00 ðAÞÞ. ðU 3 Þ n apr00 ðA [ BÞ ¼ n apr 00 ðAÞ [ n apr00 ðBÞ. ðU 6 Þ n apr00 ðUÞ ¼ U. ðU 7 Þ A n apr00 ðAÞ. ðU 8 Þ n apr00 ðn apr 00 ðAÞÞ A. ðU 9 Þ n apr00 ðn apr 00 ðAÞÞ ¼ n apr00 ðAÞ. ðU 10 Þ n apr 00 ðn apr 00 ðAÞÞ ¼ n apr 00 ðAÞ. ðLUÞ n apr 00 ðAÞ n apr0 ðAÞ. ðCOÞ n apr 00 ðAc [ BÞ ½n apr00 ðAÞc [ n apr00 ðBÞ. The following examples show Remark 4.3. Example 4.7. In Example 2.1, if A ¼ fa; b; c; dg and B ¼ fa; c; d; eg, hence 2 apr 00 ðAÞ ¼ fa; dg, 2 apr00 ðBÞ ¼ fag and 2 apr00 ðA \ BÞ ¼ ;. Thus 2 apr 00 ðA \ BÞ 6¼ 2 apr 00 ðAÞ \ 2 apr00 ðBÞ. i.e., L3 does not hold. If A ¼ fag and B ¼ fdg, then 2 apr 00 ðAÞ ¼ fa; cg, 2 apr00 ðBÞ ¼ fb; c; dg and 2 apr 00 ðA [ BÞ ¼ U. Thus 2 apr00 ðAÞ [ 2 apr 00 ðBÞ 6¼ 2 apr 00 ðA [ BÞ. i.e., U 3 does not hold. If A ¼ fdg, hence 2 apr00 ðAÞ ¼ fb; c; dg and
919
ð apr00 ðAÞÞ ¼ ;. Thus A 6 2 apr00 ð2 apr 00 ðAÞÞ. i.e., L8 does not hold. If A ¼ fa; b; c; dg, hence 2 apr00 ðAÞ ¼ fa; dg, 2 apr 00 ð2 apr 00 ðAÞÞ ¼ U and 2 apr 00 ð2 apr 00 ðAÞÞ ¼ ;. So 2 apr 00 ð2 apr00 ðAÞÞ 6 A and 00 00 00 2 apr ðAÞ 6¼ 2 apr ð2 apr ðAÞÞ, i.e., U 8 and L9 do not hold. If A ¼ fd; eg, hence 2 apr 00 ðAÞ ¼ fb; c; d; eg and 2 apr00 ð2 apr 00 ðAÞÞ ¼ U. So 2 apr 00 ðAÞ 6¼ 2 apr00 ð2 apr 00 ðAÞÞ. i.e., U 9 does not hold. 2 apr
00 2
Example 4.8. In Example 2.6, we have 2 apr 00 ð;Þ ¼ fag and apr 00 ðUÞ ¼ fb; c; d; eg. Thus 2 apr00 ð;Þ 6¼ ; and 2 apr00 ðUÞ 6¼ U. i.e., L6 and U 6 do not hold. If A ¼ fb; dg, hence 2 apr 00 ðAÞ ¼ fa; b; dg, so 00 i.e., L7 does not hold. If A ¼ fag, hence 2 apr ðAÞ 6¼ A, 2 00 2 00 00 apr 00 ðAÞ ¼ ;, and 2 apr ð apr ðAÞÞ ¼ fag, 2 apr ðAÞ ¼ fag 2 00 apr 00 ð2 apr00 ðAÞÞ ¼ ;, so A 6 2 apr 00 ðAÞ, apr ðAÞ 6 2 apr 00 ðAÞ, 2 2 apr 00 ð2 apr00 ðAÞÞ 6¼ 2 apr 00 ðAÞand 2 apr 00 ðAÞ 6¼ 2 apr00 ð2 apr00 ðAÞÞ, i.e., U 7 , LU, L10 and U 10 do not hold. If A ¼ fa; b; c; eg and B ¼ fa; b; cg, hence ½2 apr 00 ðAÞc ¼ fdg, 2 apr 00 ðBÞ ¼ fag and 2 apr 00 ðAc [ BÞ ¼ fa; b; c; dg. Thus ½2 apr00 ðAÞc [ 2 apr00 ðBÞ 6 2 apr00 ðAc [ BÞ. i.e., CO does not hold. 2
Table 3 shows the properties of rough sets by using the lower and upper approximations which is defined in Definitions 4.1, 4.2 and 4.3. References [1] T.B. Iwinski, Algebraic approach to rough sets, Bull. Polish Acad. Sci. Ser. Sci. Math. 35 (1987) 673–683. [2] E. Orlowska, Z. Pawlak, Measurement and indiscernibility, Bull. Polish Acad. Sci. Ser. Sci. Math. 32 (1984) 617–624. [3] Z. Pawlak, Information systems theoretical foundations, Inform. Syst. 6 (3) (1981) 205–218. [4] Z. Pawlak, Rough sets, Int. J. Comput. Inform. Sci. 11 (1982) 341–356. [5] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data System Theory, Knowledge Engineering and Problem Solving, vol. 9, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. [6] J.A. Pomykala, Approximation operations in approximation space, Bull. Polish Acad. Sci. Ser. Sci. Math. 35 (1987) 653–662. [7] A. Skowron, J. Stepaniuk, Generalized approximations spaces, in: T.Y. Lin, A.M. Wildberger (Eds.), Soft Computing, The Society for Computer Simulation, San Diego, 1995, pp. 18–21. [8] A. Skowron, J. Stepaniuk, Tolerance approximations spaces, Fundame. Inform. 27 (1996) 245–253. [9] U. Wybraniec-Skardowska, On a generalization of approximation space, Bull. Polish Acad. Sci. Ser. Sci. Math. 37 (1989) 51–61. [10] Y.Y. Yao, T.Y. Line, Generalization of rough sets using modal logic, Int. Automation Soft Comput. 2 (1996) 103–120. [11] Y.Y. Yao, On generalized rough set theory, rough sets, fuzzy sets, data mining and granular computing, in: Proc. 9th Int, Conf., 1999, pp. 44–51. [12] W. Zirako (Ed.), Rough sets, fuzzy sets and knowledge discovery (RSKD’93), Workshops in Computing, Springer-Verlag and British Computer Society, London, Berlin, 1994.
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Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
Comparison between different kinds of approximations by using a family of binary relations H.M. Abu-Donia * Department of Mathematics, Faculty of Science, Zagazig University, Egypt
a r t i c l e
i n f o
Article history: Received 17 December 2007 Accepted 30 March 2008 Available online 7 April 2008 Keywords: Lower and upper approximations Right neighborhood Binary relations and rough sets
a b s t r a c t In this paper, we discuss three types of lower and upper approximations of any set with respect to any relation based on right neighborhood. We generalize these three types of approximations into two ways by using a finite number of any binary relations. The first way based on the intersection of the right neighborhoods for a family of binary relations, the second based on the union and intersection of lower and upper approximations of a set according to a family of binary relations fRi : i ¼ 1; 2; . . . ; ng. Also, we make a comparison between these types in tables. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction In early eighties, Pawlak [3–5] introduced the theory of rough sets as an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. It is motivated by the practical needs in concept formation. Incomplete information causes some concepts of the universe can only be approximated. The standard rough set theory starts from an equivalence relation. A universe is divided into a family of disjoint subsets, which is well known in mathematics as a quotient set. The elements in an equivalence class are in indistinguishable by the available knowledge. A pair of operators named lower and upper approximations is introduced. The approximations are expressed in terms of equivalence classes according to their overlaps with sets to be approximated. Many proposals have been made for generalizing and interpreting rough sets [1,2,5–10,12]. In this paper, we discuss three types of upper (lower) approximations depends on right neighborhood by using any relation. We generalize this types by using a family of finite binary relations in two ways. The first generalized definition based on the intersection of these right neighborhoods and the second generalized definition based on the intersection and union of upper and lower approximations of any set with respect to every relation of them. Also, we make a comparison between all these definitions and show these comparisons in tables.
* Tel.: +20 105075372. E-mail address: [email protected] 0950-7051/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2008.03.046
2. Kinds of rough set approximations by using any binary relation In this section, we discuss three kinds of rough set approximations based on the right neighborhood. Yao [11] extend the definition of a rough set to any relation R as follows: Definition 2.1. For the pair ðU; RÞ, where U is a universe, U 6¼ ; and R be any binary relation, let xR be the right neighborhood defined as: xR ¼ fy 2 U : xRyg. Definition 2.2. Let R be any binary relation on a non-empty set U, for any set A U. The lower and upper approximations of A according to R defined as: RðAÞ ¼ fx 2 U : xR Ag; RðAÞ ¼ fx 2 U : xR \ A 6¼ ;g: Obviously, if R is an equivalence relation xR ¼ ½xR and these definitions are equivalent to the original Pawlak’s definitions. We list the properties that are of interest in the theory of rough sets, let A; B U: ðL1 Þ RðAÞ ¼ ½RðAc Þc , where Ac denotes the complement of A in U. ðL2 Þ RðUÞ ¼ U. ðL3 Þ RðA \ BÞ ¼ RðAÞ \ RðBÞ. ðL4 Þ RðA [ BÞ RðAÞ [ RðBÞ. ðL5 Þ A B ) RðAÞ RðBÞ. ðL6 Þ Rð;Þ ¼ ;. ðL7 Þ RðAÞ A.
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ðL8 Þ A RðRðAÞÞ. ðL9 Þ RðAÞ ¼ RðRðAÞÞ. ðL10 Þ RðAÞ ¼ RðRðAÞÞ. ðU 1 Þ RðAÞ ¼ ½RðAc Þc . ðU 2 Þ Rð;Þ ¼ ;. ðU 3 Þ RðA [ BÞ ¼ RðAÞ [ RðBÞ. ðU 4 Þ RðA \ BÞ RðAÞ \ RðBÞ. ðU 5 Þ A B ) RðAÞ RðBÞ. ðU 6 Þ RðUÞ ¼ U. ðU 7 Þ A RðAÞ. ðU 8 Þ A RðRðAÞÞ. ðU 9 Þ RðAÞ ¼ RðRðAÞÞ. ðU 10 Þ RðAÞ ¼ RðRðAÞÞ. (CO) RðAc [ BÞ ðRðAÞÞc [ RðBÞ. (LU) RðAÞ RðAÞ. By using Definition 2.2, we can define the accuracy measure as the following: jA \ RðAÞj ; aany ðAÞ ¼ jA [ RðAÞj where j j denotes the cardinality of the set. Proposition 2.1. For any binary relation R on a non-empty set U, the following conditions hold for every A; B U: ðL1 Þ RðAÞ ¼ ½RðAc Þc . ðL2 Þ RðUÞ ¼ U. ðL3 Þ RðA \ BÞ ¼ RðAÞ \ RðBÞ. ðL4 Þ RðA [ BÞ RðAÞ [ RðBÞ. ðL5 Þ A B ) RðAÞ RðBÞ. ðU 1 Þ RðAÞ ¼ ½RðAc Þc . ðU 2 Þ Rð;Þ ¼ ;. ðU 3 Þ RðA [ BÞ ¼ RðAÞ [ RðBÞ. ðU 4 Þ RðA \ BÞ RðAÞ \ RðBÞ. ðU 5 Þ A B ) RðAÞ RðBÞ. ðCOÞ RðAc [ BÞ ½RðAÞc [ RðBÞ. Proof ðL1 Þ. ½RðAc Þc ¼ fx 2 U : xR \ Ac 6¼ ;gc ¼ fx 2 U : xR \ Ac ¼ ;g ¼ fx 2 U : xR Ag ¼ RðAÞ: ðL2 Þ. As U is the universe set, hence RðUÞ U, 8x 2 U; we have xR U 8i ¼ 1; 2; . . . ; n, then xR U, this implies that x 2 RðUÞ. Thus U RðUÞ. So RðUÞ ¼ U. ðL3 Þ.
We can introduce an example to prove that the equality of L4 , U 4 and CO are not true in general. Example 2.1. Let R1 ¼ fða; bÞ; ða; dÞ; ðb; cÞ; ðb; dÞ; ðc; cÞ; ðd; aÞ; ðd; cÞ; ðd; bÞ; ðc; eÞ; ða; eÞ; ðe; dÞg and R2 ¼ fða; bÞ; ða; cÞ; ðb; cÞ; ðb; dÞ; ðb; bÞ; ðc; aÞ; ðc; dÞ; ðd; aÞ; ðd; cÞ; ðd; eÞ; ða; eÞ; ðe; bÞg are any two binary relations on non-empty set U ¼ fa; b; c; d; eg. Then aR1 ¼ fb; d; eg, bR1 ¼ fc; dg; cR1 ¼ fc; eg, dR1 ¼ fa; b; cg, eR1 ¼ fdg, aR2 ¼ fb; c; eg, bR2 ¼ fb; c; dg, cR2 ¼ fa; dg, dR2 ¼ fa; c; eg and eR2 ¼ fbg. If A ¼ fcg and B ¼ fdg, then R1 ðAÞ ¼ ;, R1 ðBÞ ¼ feg and RðA [ BÞ ¼ fb; eg. So R1 ðA [ BÞ 6¼ R1 ðAÞ [ R1 ðBÞ. If A ¼ fa; dg and B ¼ fb; dg, then R1 ðAÞ ¼ fa; b; d; eg, R1 ðBÞ ¼ fa; b; d; eg and R1 ðA \ BÞ ¼ fa; b; eg. Thus R1 ðAÞ \ R1 ðBÞ 6¼ R1 ðA \ BÞ. If A ¼ fc; d; eg and B ¼ fa; cg, hence R1 ðAc [ BÞ ¼ fdg, ½R1 ðAÞc ¼ fa; dg and R1 ðBÞ ¼ ;, then ½R1 ðAÞc [ R1 ðBÞ ¼ fa; dg. Thus R1 ðAc [ BÞ 6¼ ½R1 ðAÞc [ R1 ðBÞ. Remark 2.1. If R is an any binary relation on a non-empty set U, then the following conditions do not hold for every A; B U. ðL6 Þ Rð;Þ ¼ ;. ðL7 Þ RðAÞ A. ðL8 Þ A RðRðAÞÞ. ðL9 Þ RðAÞ ¼ RðRðAÞÞ. ðL10 Þ RðAÞ ¼ RðRðAÞÞ. ðU 6 Þ RðUÞ ¼ U. ðU 7 Þ A RðAÞ. ðU 8 Þ RðRðAÞÞ A. ðU 9 Þ RðRðAÞÞ ¼ RðAÞ. ðU 10 Þ RðRðAÞÞ ¼ RðAÞ. ðLUÞ RðAÞ RðAÞ. The following examples show Remark 2.1. Example 2.2. Let R1 ¼ fða; bÞ; ðb; cÞg and R2 ¼ fða; cÞ; ðb; cÞg are any two binary relations on non-empty set U ¼ fa; b; cg. Then aR1 ¼ fbg; bR1 ¼ fcg, cR1 ¼ ;, aR2 ¼ fcg; bR2 ¼ fcg and cR2 ¼ ;. Hence we have R1 ð;Þ ¼ fcg and RðUÞ ¼ fa; bg. Consequently L6 and U 6 do not hold. If A ¼ fbg, hence R1 ðAÞ ¼ fa; cg and R1 ðAÞ ¼ fag. Then R1 ðAÞ 6 A, A 6 R1 ðAÞ and R1 ðAÞ 6 R1 ðAÞ, i.e., L7 , U 7 and LU do not hold. If A ¼ fa; bg, hence R1 ðAÞ ¼ fa; dg and R1 ðR1 ðAÞÞ ¼ feg. Thus A 6 R1 ðR1 ðAÞÞ and R1 ðAÞ 6¼ R1 ðR1 ðAÞÞ. Therefore L8 and L10 do not hold. If A ¼ fa; b; c; eg. Hence R1 ðAÞ ¼ fc; dg and R1 ðR1 ðAÞÞ ¼ U. So R1 ðR1 ðAÞÞ 6 A and R1 ðR1 ðAÞÞ 6¼ R1 ðAÞ. Therefore U 8 and U 10 do not hold. If A ¼ fa; b; cg, hence R1 ðAÞ ¼ fdg,R1 ðR1 ðAÞÞ ¼ feg, R1 ðAÞ ¼ fa; b; c; dg and R1 ðR1 ðAÞÞ ¼ U. Then R1 ðAÞ 6¼ R1 ðR1 ðAÞÞ and R1 ðR1 ðAÞÞ 6¼ R1 ðAÞ. Therefore L9 and U 9 do not hold.
() xR A and xR B
Definition 2.3. Let R be any binary relation on a non-empty set U and xR is the right neighborhood of x according to R, for any set A U. The second definition for lower and upper approximations of A according to R is defined [11] as: [ R0 ðAÞ ¼ fxR : xR Ag;
() x 2 RðAÞ and x 2 RðBÞ
R0 ðAÞ ¼ ½R0 ðAc Þc :
x 2 RðA \ BÞ () xR ðA \ BÞ
() x 2 ½RðAÞ \ RðBÞ: Therefore RðA \ BÞ ¼ RðAÞ \ RðBÞ. ðL4 Þ. Let x 2 ½RðAÞ [ RðBÞ, hence x 2 RðAÞ or x 2 RðBÞ. So xR A or xR B and so xR ðA [ BÞ, then x 2 RðA [ BÞ. Therefore RðA [ BÞ RðAÞ [ RðBÞ. ðL5 Þ. Let A B and x 2 RðAÞ. Then xR A but A B. So xR B, hence x 2 RðBÞ. Thus RðAÞ RðBÞ. ðCOÞ. Let x 62 ½RðAÞc [ RðBÞ, then x 62 ½RðAÞc and x 62 RðBÞ, hence x 2 RðAÞ and x 62 RðBÞ, i.e., xR A and xR 6 B, then xR 6 Ac and xR 6 B, hence xR 6 ðAc [ BÞ. Thus x 62 RðAc [ BÞ.i.e., RðAc [ BÞ ½RðAÞc [ RðBÞ. We can prove U 1 ; U 2 ; U 3 ; U 4 and U 5 as the same as L1 ; L2 ; L3 ; L4 and L5 . h
And so the accuracy measure will be defined as: a0any ðAÞ ¼
jA \ R0 ðAÞj ; jA [ R0 ðAÞj
where j j denotes the cardinality of the set. Proposition 2.2. For any binary relation R on a non-empty set U, the following conditions hold for every A; B U: ðL1 Þ R0 ðAÞ ¼ ½R0 ðAc Þc . ðL4 Þ R0 ðA [ BÞ R0 ðAÞ [ R0 ðBÞ. ðL5 Þ A B ) R0 ðAÞ R0 ðBÞ.
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ðL6 Þ R0 ð;Þ ¼ ;. ðL7 Þ R0 ðAÞ A. ðL9 Þ R0 ðAÞ ¼ R0 ðR0 ðAÞÞ. ðU 1 Þ R0 ðAÞ ¼ ½R0 ðAc Þc . ðU 4 Þ R0 ðA \ BÞ R0 ðAÞ \ R0 ðBÞ. ðU 5 Þ A B ) R0 ðAÞ R0 ðBÞ. ðU 6 Þ R0 ðUÞ ¼ U. ðU 7 Þ A R0 ðAÞ. ðU 9 Þ R0 ðR0 ðAÞÞ ¼ R0 ðAÞ. ðLUÞ R0 ðAÞ R0 ðAÞ. Proof ðL1 Þ. ½R0 ðAc Þc ¼ ½½R0 ðAc Þc c c ¼ R0 ðAÞ. ðL4 Þ. Let x 2 ½R0 ðAÞ [ R0 ðBÞ. Then x 2 R0 ðAÞ or x 2 R0 ðBÞ, then xR A or xR B, this implies that xR ðA [ BÞ. So x 2 R0 ðA [ BÞ. Hence R0 ðA [ BÞ R0 ðAÞ [ R0 ðBÞ. ðL5 Þ. Let A B and x 2 R0 ðAÞ. Then xR A but A B. So xR B, hence x 2 R0 ðBÞ. Thus, R0 ðAÞ R0 ðBÞ. ðL6 Þ. 8x 2 U, we have xR ;, if xR ¼ ;. So R0 ð;Þ ¼ ;. S ðL7 Þ. Since R0 ðAÞ ¼ fxR : xR Ag. So R0 ðAÞ A. S 0 Since R ðAÞ ¼ fxR : xR Ag, hence R0 ðR0 ðAÞÞ ¼ ðL9 Þ. S S S R0 ð fxR : xR AgÞ ¼ fxR : xR ð fxR : xR AgÞg ¼ R0 ðAÞ. Hence R0 ðR0 ðAÞÞ ¼ R0 ðAÞ. We can prove U 1 ; U 4 ; U 5 ; U 6 ; U 7 and U 9 as the same as L1 ; L4 ; L5 ; L6 ; L7 and L9 . We can prove LU from L7 and U 7 . h We can introduce an example to prove that the equality of L4 , U 4 , L7 , U 7 and LU are not true in general. Example 2.3. In Example 2.1, if A ¼ fbg and B ¼ fd; eg, hence R01 ðAÞ ¼ ;, R01 ðBÞ ¼ fdg and R01 ðA [ BÞ ¼ fb; d; eg. Thus, R01 ðA [ BÞ 6¼ R01 ðAÞ [ R01 ðBÞ. If A ¼ fa; c; dg and B ¼ fa; d; eg, then R01 ðAÞ ¼ U, R01 ðBÞ ¼ U and R01 ðA \ BÞ ¼ fa; b; dg. Thus R01 ðAÞ \ R01 ðBÞ 6¼ R01 ðA \ BÞ. If A ¼ fbg, then R01 ðAÞ ¼ ;, R01 ðAÞ ¼ fa; bg, i.e., R01 ðAÞ 6¼ A, R01 ðAÞ 6¼ A and R01 ðAÞ 6¼ R01 ðAÞ. Remark 2.2. If a binary relation R is any relation. Then the following conditions are not hold for every A; B U: ðL2 Þ R0 ðUÞ ¼ U. ðL3 Þ R0 ðA \ BÞ ¼ R0 ðAÞ \ R0 ðBÞ. ðL8 Þ A R0 ðR0 ðAÞÞ. ðL10 Þ R0 ðAÞ ¼ R0 ðR0 ðAÞÞ. ðU 2 Þ R0 ð;Þ ¼ ;. ðU 3 Þ R0 ðA [ BÞ ¼ R0 ðAÞ [ R0 ðBÞ. ðU 8 Þ R0 ðR0 ðAÞÞ A. ðU 10 Þ R0 ðR0 ðAÞÞ ¼ R0 ðAÞ. ðCOÞ R0 ðAc [ BÞ ½R0 ðAÞc [ R0 ðBÞ. The following examples show Remark 2.2. Example 2.4. In Example 2.2, we have R01 ðUÞ ¼ fb; cg and R01 ð;Þ ¼ fag. Thus R01 ðUÞ 6¼ U and R01 ð;Þ 6¼ ;. i.e., L2 and U 2 do not hold. Example 2.5. In Example 2.1, if A ¼ fc; dg and B ¼ fc; eg, hence R01 ðAÞ ¼ fc; dg, R01 ðBÞ ¼ fc; eg and R01 ðA \ BÞ ¼ ;. Thus, R01 ðA \ BÞ 6¼ R01 ðAÞ \ R01 ðBÞ. i.e., L3 does not hold. If A ¼ fa; cg and B ¼ fdg, then R01 ðAÞ ¼ fa; cg, R01 ðBÞ ¼ fdg and R01 ðA [ BÞ ¼ U. Thus, R01 ðAÞ [ R01 ðBÞ 6¼ R01 ðA [ BÞ. i.e., U 3 does not hold. If A ¼ fcg, hence R01 ðAÞ ¼ fa; cg and R01 ðR01 ðAÞÞ ¼ ;. Thus A 6 R01 ðR01 ðAÞÞ and R01 ðAÞ 6¼ R01 ðR01 ðAÞÞ. i.e., L8 and L10 do not hold. If A ¼ fc; dg, hence R01 ðAÞ ¼ fc; dg and R01 ðR01 ðAÞÞ ¼ U. So R01 ðR01 ðAÞÞ 6 Aand R01 ðR01 ðAÞÞ 6¼ R01 ðAÞ. Therefore U 8 and U 10 do not hold. If A ¼ fb; c; eg and B ¼ fb; cg, hence R01 ðAc [ BÞ ¼ fa; b; c; dg, ½R01 ðAÞc ¼ fa; b; dg and R01 ðBÞ ¼ ;, then ½R01 ðAÞc [ R01 ðBÞ ¼ fa; b; dg. Thus R01 ðAc [ BÞ 6¼ ½R01 ðAÞc [ R01 ðBÞ. Hence CO does not hold.
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Definition 2.4. Let R be any binary relation on a non-empty set U and xR is the right neighborhood of x according to R, for any set A U. The third definition for lower and upper approximations of A according to R is defined [11] as the following: R00 ðAÞ ¼ ½R00 ðAc Þc ; [ R00 ðAÞ ¼ fxR : xR \ A 6¼ ;g: And so the accuracy measure will be defined as: a00any ðAÞ ¼
jA \ R00 ðAÞj ; jA [ R00 ðAÞj
where j j denotes the cardinality of the set. Proposition 2.3. For any binary relation R on a non-empty set U the following hold for every A; B U: ðL1 Þ R00 ðAÞ ¼ ½R00 ðAc Þc . ðL2 Þ R00 ðUÞ ¼ U. ðL3 Þ R00 ðA \ BÞ ¼ R00 ðAÞ \ R00 ðBÞ. ðL4 Þ R00 ðA [ BÞ R00 ðAÞ [ R00 ðBÞ. ðL5 Þ A B ) R00 ðAÞ R00 ðBÞ. ðL8 Þ A R00 ðR00 ðAÞÞ. ðU 1 Þ R00 ðAÞ ¼ ½R00 ðAc Þc . ðU 2 Þ R00 ð;Þ ¼ ;. ðU 3 Þ R00 ðA [ BÞ ¼ R00 ðAÞ [ R00 ðBÞ. ðU 4 Þ R00 ðA \ BÞ R00 ðAÞ \ R00 ðBÞ. ðU 5 Þ A B ) RðAÞ RðBÞ. ðU 8 Þ R00 ðR00 ðAÞÞ A. ðCOÞ R00 ðAc [ BÞ ½R00 ðAÞc [ R00 ðBÞ. Proof ðL1 Þ. From Definition 2.4, we have R00 ðAÞ ¼ ½R00 ðAc Þc . S ðL2 Þ. R00 ðUÞ ¼ ½R00 ð/Þc ¼ ½ fxR : xR \ / 6¼ /gc ¼ /c ¼ U. S 00 00 ðL3 Þ. R ðA \ BÞ ¼ ½R ðA \ BÞc c ¼ ½R00 ðAc [ Bc Þc ¼ ½ fxR : xR \ ðAc S c c cTS c c [B Þ 6¼ /g ¼ ½ fxR : xR \ A 6¼ /g ½ fxR : xR \ B 6¼ /gc ¼ ½R00 ðAc Þc \ ½R00 ðBc Þc ¼ R00 ðAÞ \ R00 ðBÞ. ðL4 Þ. Let x 2 R00 ðAÞ [ R00 ðBÞ, hence x 2 R00 ðAÞ or x 2 R00 ðBÞ. So x 2 ½R00 ðAc Þc or x 2 ½R00 ðBc Þc . This implies that x 62 R00 ðAc Þ or x 62 R00 ðBc Þ, then 8s 2 U, we have x 2 sR such that sR \ Ac 6¼ / and sR \ Bc 6¼ /. Consequently, x 62 sR8s 2 U such that sR \ ðAc \ Bc Þ 6¼ /, it is means that x 62 R00 ðA [ BÞc i.e, x 2 R00 ðA [ BÞ. Hence R00 ðAÞ [ R00 ðBÞ R00 ðA [ BÞ. ðL5 Þ. Let A B and x 62 R00 ðBÞ, hence x 62 ½R00 ðBc Þc . So x 2 R00 ðBc Þ and so 9s 2 U such that x 2 sR and sR \ Bc 6¼ / but Ac Bc . This implies that sR \ Ac 6¼ /, it is means that x 2 R00 ðAc Þ and so x 62 ½R00 ðAc Þc ¼ R00 ðAÞ. Therefore R00 ðAÞ R00 ðBÞ. In the same manner we can prove ðU 1 Þ, ðU 2 Þ, ðU 3 Þ, ðU 4 Þ, ðU 5 Þ, ðU 8 Þ, ðL8 Þ and ðCOÞ. h We can introduce an example to show that the equality of L4 , U 4 , U 8 and CO are not true in general. Example 2.6. Let R1 ¼ fðb; dÞ; ðb; cÞ; ðc; cÞ; ðd; aÞ; ðe; bÞ; ðe; eÞg and R2 ¼ fðb; dÞ; ðc; cÞ; ðc; eÞ; ðd; eÞ; ðe; dÞ, ðe; bÞg are any two binary relations on non-empty set U ¼ fa; b; c; d; eg. Then aR1 ¼ ;, bR1 ¼ fc; dg, cR1 ¼ fcg, dR1 ¼ fag, eR1 ¼ fb; eg, aR2 ¼ ;, bR2 ¼ fdg, cR2 ¼ fc; eg, dR2 ¼ feg and eR2 ¼ fb; dg. If A ¼ fa; dg and B ¼ fcg, hence R001 ðAÞ ¼ fag, R001 ðBÞ ¼ ; and R001 ðA [ BÞ ¼ fa; c; dg. Hence R001 ðAÞ [ R001 ðBÞ 6¼ R001 ðA [ BÞ. If A ¼ fcg and B ¼ fdg, hence R001 ðAÞ ¼ fc; dg, R001 ðBÞ ¼ fc; dg and R001 ðA \ BÞ ¼ ;. Thus R001 ðAÞ \ R001 ðBÞ 6¼ R001 ðA \ BÞ. If A ¼ fa; b; c; dg, hence R001 ðAÞ ¼ fa; c; dg and R001 ðR001 ðAÞÞ ¼ fa; c; dg. So R001 ðR001 ðAÞÞ 6¼ A. If A ¼ fa; b; c; dg and B ¼ fa; dg, hence ½R001 ðAÞc ¼ fb; eg, R001 ðBÞ ¼ fag and R001 ðAc [ BÞ ¼ fag. Thus ½R001 ðAÞc [ R001 ðBÞ 6¼ R001 ðAc [ BÞ. We can introduce an example to show that the equality of L8 is not true in general.
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Example 2.7. In Example 2.1, if A ¼ feg, hence R001 ðAÞ ¼ fb; c; d; eg and R001 ðR001 ðAÞÞ ¼ fd; eg. Thus A 6¼ R001 ðR001 ðAÞÞ. Remark 2.3. If R is an any binary relation on a non-empty set U, then the following conditions do not hold for every A; B U. ðL6 Þ R00 ð;Þ ¼ ;. ðL7 Þ R00 ðAÞ A. ðL9 Þ R00 ðAÞ ¼ R00 ðR00 ðAÞÞ. ðL10 Þ R00 ðAÞ ¼ R00 ðR00 ðAÞÞ. ðU 6 Þ R00 ðUÞ ¼ U. ðU 7 Þ A R00 ðAÞ. ðU 9 Þ R00 ðR00 ðAÞÞ ¼ R00 ðAÞ. ðU 10 Þ R00 ðR00 ðAÞÞ ¼ R00 ðAÞ. ðLUÞ R00 ðAÞ R00 ðAÞ.
intersection of the right neighborhoods for a family of any binary relations. Definition 3.1. Let fRi : i ¼ 1; 2; . . . ; ng be a finite family of binary relations on a non-empty set U, for any set A U, we can introduce a definition for n-lower and n-upper approximations of A according to Ri as: ( ! ) n \ x2U: xRi A ; n aprðAÞ ¼ i¼1
( n
aprðAÞ ¼
n \
x2U:
!
) \ A 6¼ ; :
xRi
i¼1
By using Definition 3.1, we can define the accuracy measure of any set A as the following: n aany ðAÞ
The following examples show Remark 2.3. Example 2.8. In Example 2.6, we have R002 ð;Þ ¼ fag and R002 ðUÞ ¼ fb; c; d; eg, i.e., L6 and U 6 do not hold. If A ¼ fb; dg, hence R002 ðAÞ ¼ fa; b; dg and R002 ðAÞ ¼ fb; dg. Thus R2 ðAÞ 6 A and R002 ðAÞ 6 R002 ðAÞ, i.e., L7 and LU do not hold. If A ¼ fa; b; dg, hence R002 ðAÞ ¼ fb; dg. Then A 6 R002 ðAÞ, i.e., U 7 does not hold.
¼
jA\n aprðAÞj : jA[n aprðAÞj
Proposition 3.1. For a family of binary relations Ri ; i ¼ 1; 2; . . . ; n on a non-empty set U the following conditions hold for every A; B U: ðL1 Þ n aprðAÞ ¼ ½n aprðAc Þc ðL2 Þ n aprðUÞ ¼ U. ðL3 Þ n aprðA \ BÞ ¼ n aprðAÞ \ n aprðBÞ. ðL4 Þ n aprðA [ BÞ n aprðAÞ [ n aprðBÞ. ðL5 Þ A B ) n aprðAÞ n aprðBÞ. ðU 1 Þ n aprðAÞ ¼ ½n aprðAc Þc . ðU 2 Þ n aprð;Þ ¼ ;. ðU 3 Þ n aprðA [ BÞ ¼ n aprðAÞ [ n aprðBÞ. ðU 4 Þ n aprðA \ BÞ n aprðAÞ \ n aprðBÞ. ðU 5 Þ A B ) n aprðAÞ n aprðBÞ. ðCOÞ n aprðAc [ BÞ ½n aprðAÞc [ RðBÞ.
Example 2.9. In Example 2.1, if A ¼ fa; b; cg, hence R001 ðAÞ ¼ fag, R001 ðR001 ðAÞÞ ¼ ;. Thus R001 ðAÞ 6¼ R001 ðR001 ðAÞÞ, i.e., L9 does not hold. If A ¼ fag, hence R00 ðAÞ ¼ fa; b; cg, R001 ðR001 ðAÞÞ ¼ U and R001 ðR001 ðAÞÞ ¼ fag. Then R001 ðR001 ðAÞÞ 6¼ R001 ðAÞ and R001 ðAÞ 6¼ R001 ðR001 ðAÞÞ, i.e., U 9 and L10 do R001 ðAÞ ¼ fag and not hold. If A ¼ fa; b; c; eg, hence R001 ðR001 ðAÞÞ ¼ fa; b; cg, so R001 ðR001 ðAÞÞ 6¼ R001 ðAÞ, i.e., U 10 does not hold. Table 1 shows the properties of rough sets by using the lower and upper approximations which is defined in Definitions 2.2, 2.3 and 2.4:
Proof ðL1 Þ.
3. The first kind of generalization In this section, we introduce three approximations which are generalization for a previous three approximations, based on the
n
c
c
½ aprðA Þ ¼ fx 2 U :
n \
!
fx 2 U :
\ ðAc Þ 6¼ ;gc ¼
xRi
i¼1 n \
! \ ðAc Þ ¼ ;g ¼
xRi
i¼1
Table 1 Comparison between the properties of rough sets depending on Definitions 2.2, 2.3 and 2.4 by using any relation R
fx 2 U :
n \
! xRi
Ag ¼n aprðAÞ:
i¼1
Properties
L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 U1 U2 U3 U4 U5 U6 U7 U8 U9 U 10 CO LU
Upper and lower approximations under any relations by using: Definition 2.2
Definition 2.3
Definition 2.4
q q q q q
q
q q q q q
q q q q
q q q q q q q
q
q q q q
q q q q q
q
q q
ðIÞ indicates that the property is satisfied.
x 2 n aprðA \ BÞ () ()
n \ i¼1 n \ i¼1
q q
ðL2 Þ. As U is the universe set, xRi U8i ¼ 1; 2; . . . ; n, then x 2 n aprðUÞ. Thus U n aprðUÞ. ðL3 Þ.
hence n aprðUÞ U, 8x 2 U, we have T ð ni¼1 xRi Þ U, this implies that So n aprðUÞ ¼ U.
xRi A \ B xRi A and
n \
xRi B
i¼1
() x 2 n aprðAÞ and x 2 n aprðBÞ () x 2 ½n aprðAÞ \ n aprðBÞ: Therefore n aprðA \ BÞ ¼ n aprðAÞ \ n aprðBÞ. ðL4 Þ. Let x 2 ½n aprðAÞ [ n aprðBÞ, hence x 2 n aprðAÞ or x 2 n aprðBÞ. T T T So ð ni¼1 xRi Þ A or ð ni¼1 xRi Þ B and ð ni¼1 xRi Þ ðA [ BÞ. Then x 2 n aprðA [ BÞ, i.e., n aprðA [ BÞ n aprðAÞ [ n aprðBÞ. T ðL Þ. Let A B and x 2 n aprðAÞ. Then ð ni¼1 xRi Þ A but A B. So Tn 5 ð i¼1 xRi Þ B, hence x 2 n aprðBÞ. Thus n aprðAÞ n aprðBÞ. ðCOÞ. Let x 62 ½n aprðAÞc [ n aprðBÞ, then x 62 ½n aprðAÞc and Tn x 62 n aprðBÞ, hence x 2 n aprðAÞ and x 62 n aprðBÞ, i.e., i¼1 xRi A Tn Tn Tn c then and hence and i¼1 xRi 6 B, i¼1 xRi 6 A i¼1 xRi 6 B,
H.M. Abu-Donia / Knowledge-Based Systems 21 (2008) 911–919
Tn
c i¼1 ðxÞRi 6 ðA [ BÞ. c ½n aprðAÞ [ n aprðBÞ.
Thus
x 62 n aprðAc [ BÞ,
i.e.,
c n aprðA
ðL4 Þ n apr 0 ðA [ BÞ n apr 0 ðAÞ [ n apr 0 ðBÞ. ðL5 Þ A B ) n apr 0 ðAÞ n apr0 ðBÞ. ðL6 Þ n apr 0 ð;Þ ¼ ;. ðL7 Þ n apr 0 ðAÞ A. ðL9 Þ n apr 0 ðAÞ ¼ n apr0 ðn apr 0 ðAÞÞ. ðU 1 Þ n apr 0 ðAÞ ¼ ½n apr0 ðAc Þc ðU 4 Þ n apr 0 ðA \ BÞ n apr 0 ðAÞ \ n apr 0 ðBÞ. ðU 5 Þ A B ) n apr 0 ðAÞ n apr0 ðBÞ ðU 6 Þ n apr 0 ðUÞ ¼ U ðU 7 Þ A n apr0 ðAÞ. ðU 9 Þ n apr 0 ðn apr 0 ðAÞÞ ¼ n apr 0 ðAÞ. ðLUÞ n apr 0 ðAÞ n apr 0 ðAÞ.
[ BÞ
We can prove U 1 ; U 2 ; U 3 ; U 4 and U 5 as the same as L1 ; L2 ; L3 ; L4 and L5 . h We can introduce an example to prove that the equality of L4 and U 4 are not true in general. T Example 3.1. From Example 2.1, we have got 2i¼1 aRi ¼ fb; eg, T2 T2 T2 T2 i¼1 bRi ¼ fd; cg, i¼1 cRi ¼ ;, i¼1 dRi ¼ fa; cg and i¼1 eRi ¼ ;. If A ¼ fa; bg and B ¼ fc; eg, then 2 aprðAÞ ¼ fc; eg, 2 aprðBÞ ¼ fc; eg, hence 2 aprðAÞ [ 2 aprðBÞ ¼ fc; eg but 2 aprðA [ BÞ ¼ fa; c; d; eg. i.e., 2 aprðA [ BÞ 6¼ 2 aprðAÞ [ 2 aprðBÞ. If A ¼ fag and B ¼ fcg, then 2 2 2 aprðAÞ ¼ fdg, aprðBÞ ¼ fb; dg and aprðA \ BÞ ¼ ;. Thus 2 aprðAÞ \ 2 aprðBÞ 6¼ 2 aprðA \ BÞ. Remark 3.1. For a family of binary relations Ri : i ¼ 1; 2; . . . ; n on a non-empty set U the following conditions do not hold for every A; B U: ðL6 Þ n aprð;Þ ¼ ;. ðL7 Þ n aprðAÞ A. ðL8 Þ A n aprðn aprðAÞÞ. ðL9 Þ n aprðAÞ ¼ n aprðn aprðAÞÞ. ðL10 Þ n aprðAÞ ¼ n aprðn aprðAÞÞ. ðU 6 Þ n aprðUÞ ¼ U. ðU 7 Þ A n aprðAÞ. ðU 8 Þ n aprðn aprðAÞÞ A. ðU 9 Þ n aprðn aprðAÞÞ ¼ n aprðAÞ ðU 10 Þ n aprðn aprðAÞÞ ¼ n aprðAÞ ðLUÞ n aprðAÞ n aprðAÞ The following examples show Remark 3.1. Example 3.2. In Example 2.2, we can see that 2 aprð;Þ ¼ fa; cg and aprðUÞ ¼ fbg. Thus 2 aprð;Þ 6¼ ; and 2 aprðUÞ 6¼ U, i.e., L6 and U 6 do not hold.
2
Example 3.3. In Example 2.1, If A ¼ fag, hence 2 aprðAÞ ¼ fdg, 2 aprð2 aprðAÞÞ ¼ fa; b; dg and 2 aprð2 aprðAÞÞ ¼ 2 aprðAÞ ¼ fc; eg, fbg.Thus 2 aprðAÞ 6 A, A 6 2 aprðAÞ, 2 aprðAÞ 6 2 aprðAÞ, 2 aprð2 aprðAÞÞ 6 A, 2 aprð2 aprðAÞÞ 6¼ 2 aprðAÞ and 2 aprð2 aprðAÞÞ 6¼ 2 aprðAÞ, i.e., L7 , U 7 , LU, U 8 , U 10 and U 9 do not hold.If A ¼ fdg, hence 2 aprðAÞ ¼ fbg and 2 aprð2 aprðAÞÞ ¼ fc; eg. Thus A 6 2 aprð2 aprðAÞÞ and 2 aprðAÞ 6¼ 2 2 aprð aprðAÞÞ, i.e., L8 and L10 do not hold. If A ¼ fa; b; eg, hence aprðAÞ ¼ fa; c; eg and 2 aprð2 aprðAÞÞ ¼ fc; d; eg. Then 2 aprðAÞ6¼2 2 aprð2 aprðAÞÞ. i.e., L9 does not hold.
Proof ðL1 Þ. ½n apr 0 ðAc Þc ¼ ½½2 apr0 ððAc Þc Þc c ¼ 2 apr 0 ðAÞ. ðL4 Þ. Let x 2 ½n apr 0 ðAÞ [ n apr 0 ðBÞ, hence x 2 n apr0 ðAÞ or ST ST x 2 n apr0 ðBÞ, then x 2 ð ni¼1 xRi Þ A or x 2 ð ni¼1 xRi Þ B, this S Tn implies that x 2 ð i¼1 xRi Þ ðA [ BÞ. So x 2 n apr 0 ðA [ BÞ. i.e., 0 0 0 n apr ðA [ BÞ n apr ðAÞ [ n apr ðBÞ. ST ðL5 ). Let A B and x 2 n apr0 ðAÞ. Then x 2 ð ni¼1 xRi Þ A but A B. So ST x 2 ð ni¼1 xRi Þ B, hencex 2 n apr 0 ðBÞ. Thus n apr0 ðAÞ n apr 0 ðBÞ. T T ðL6 Þ. 8x 2 U, we have ð ni¼1 xRi Þ ;, if ð ni¼1 xRi Þ ¼ ;. So 0 n apr ð;Þ ¼ ;. S T T ðL7 Þ. Since n apr 0 ðAÞ ¼ fð ni¼1 xRi Þ : ð ni¼1 xRi Þ Ag. So n apr0 ðAÞ A. S Tn T 0 ðL9 Þ. Since n apr ðAÞ ¼ fð i¼1 xRi Þ : ð ni¼1 xRi Þ Ag, hence n apr 0 S T T S T T ðn apr 0 ðAÞÞ ¼ n apr0 ð fð ni¼1 xRi Þ : ð ni¼1 xRi Þ AgÞ ¼ fð ni¼1 xRi Þ : ð ni¼1 S Tn Tn 0 0 xRi Þ ð fð i¼1 xRi Þ : ð i¼1 xRi Þ AgÞg ¼ n apr ðAÞ. i.e., n apr ðn apr 0 ðAÞÞ ¼ n apr 0 ðAÞ. We can prove U 1 ; U 4 ; U 5 ; U 6 ; U 7 and U 9 as the same as L1 ; L4 ; L5 ; L6 ; L7 and L9 . We can prove LU from L7 and U 7 . h We can introduce an example to prove that the equality of L4 , U 4 , L7 , U 7 and LU does not true in general. Example 3.4. In Example 2.1, if A ¼ fag and B ¼ fcg, hence 0 0 0 and Thus, 2 apr ðAÞ ¼ ;, 2 apr ðBÞ ¼ ; 2 apr ðA [ BÞ ¼ fa; cg. 0 0 0 2 apr ðA [ BÞ 6¼ 2 apr ðAÞ [ 2 apr ðBÞ. If A ¼ fb; cg and B ¼ fc; eg, then 2 apr 0 ðAÞ ¼ U, 2 apr 0 ðBÞ ¼ U and 2 apr0 ðA \ BÞ ¼ fa; c; dg. Thus 2 apr 0 ðAÞ \ 2 apr 0 ðBÞ 6¼ 2 apr 0 ðA \ BÞ. If A ¼ fcg, then 2 apr 0 ðAÞ ¼ ;, 2 0 2 apr 0 ðAÞ ¼ fa; c; dg. Thus apr0 ðAÞ 6¼ A and 2 apr ðAÞ 6¼ A, 0 2 0 apr ðAÞ ¼ 6 apr ðAÞ. 2 Remark 3.2. For a family of binary relations Ri : i ¼ 1; 2; . . . ; n on a non-empty set U the following conditions do not hold for every A; B U: ðL2 Þ n apr 0 ðUÞ ¼ U. ðL3 Þ n apr 0 ðA \ BÞ ¼ n apr 0 ðAÞ \ n apr 0 ðBÞ ðL8 Þ A n apr0 ðn apr 0 ðAÞÞ. ðL10 Þ n apr 0 ðAÞ ¼n apr 0 ðn apr 0 ðAÞÞ. ðU 2 Þ n apr 0 ð;Þ ¼ ;. ðU 3 Þ n apr 0 ðA [ BÞ ¼ n apr 0 ðAÞ [ n apr 0 ðBÞ. ðU 8 Þ n apr 0 ðn apr 0 ðAÞÞ A. ðU 10 Þ n apr 0 ðn apr0 ðAÞÞ ¼ n apr0 ðAÞ. ðCOÞ n apr 0 ðAc [ BÞ ½n apr 0 ðAÞc [ n apr 0 ðBÞ.
Definition 3.2. Let fRi : i ¼ 1; 2; . . . ; ng be a family of binary relations on a non-empty set U, for any set A U. We can introduce a definition for n-lower and n-upper approximations of any set A U according to Ri as ( ! ! ) n n \ [ \ 0 xRi : xRi A ; n apr ðAÞ ¼ i¼1 n
i¼1
apr 0 ðAÞ ¼ ½n apr 0 ðAc Þc :
By using Definition 3.2, we can define the accuracy measure of any set A as the following: jA \ n apr0 ðAÞj 0 : n aany ðAÞ ¼ jA [ n apr0 ðAÞj Proposition 3.2. For a family of binary relations Ri : i ¼ 1; 2; . . . ; n on a non-empty set U, the following conditions hold for every A; B U: ðL1 Þ n apr0 ðAÞ ¼ ½n apr 0 ðAc Þc .
915
The following examples show Remark 3.2. Example 3.5. In Example 2.6, we have 2 apr 0 ðUÞ ¼ fb; c; dg and apr 0 ð;Þ ¼ fa; eg. Thus 2 apr 0 ðUÞ 6¼ U and 2 apr0 ð;Þ 6¼ ;. i.e., L2 and U 2 do not hold. 2
Example 3.6. In Example 2.1, if A ¼ fc; ag and B ¼ fc; dg, hence 0 0 and 2 apr0 ðA \ BÞ ¼ ;. Thus 2 apr ðAÞ ¼ fa; cg, 2 apr ðBÞ ¼ fc; dg 0 0 0 apr ðA \ BÞ ¼ 6 apr ðAÞ \ apr ðBÞ. i.e., L3 does not hold. If 2 2 2
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H.M. Abu-Donia / Knowledge-Based Systems 21 (2008) 911–919
A ¼ fa; eg and B ¼ fb; dg, then 2 apr 0 ðAÞ ¼ fa; b; eg, 2 apr0 ðBÞ ¼ fb; d; eg and 2 apr0 ðA [ BÞ ¼ U. Thus 2 apr 0 ðAÞ [ 2 apr 0 ðBÞ 6¼ 2 apr0 ðA [ BÞ. i.e., U 3 does not hold. If A ¼ fag, hence 2 apr0 ðAÞ ¼ fag and 2 apr 0 ð2 apr0 ðAÞÞ ¼ ;. Thus A 6 2 apr 0 ð2 apr 0 ðAÞÞand 2 apr 0 ðAÞ 6¼ 0 2 0 2 apr ð apr ðAÞÞ, i.e., L8 and L10 do not hold. If A ¼ fa; cg, hence 0 2 0 0 2 0 0 2 apr ðAÞ¼ fa; cg and apr ð2 apr ðAÞÞ ¼ fa; c; dg, so apr ð2 apr ðAÞÞ 6 A 2 0 0 0 and apr ð2 apr ðAÞÞ 6¼ 2 apr ðAÞ. i.e., U 8 and U 10 do not hold. If A ¼ fa; b; cg and B ¼ fb; cg, hence 2 apr0 ðAc [ BÞ ¼ fb; c; d; eg, ½2 apr0 ðAÞc ¼ fb; d; eg and 2 apr 0 ðBÞ ¼ ;, then ½2 apr0 ðAÞc [ 2 apr0 ðBÞ ¼ fb; d; eg. Thus 2 apr 0 ðAc [ BÞ 6¼ ½2 apr 0 ðAÞc [n apr 0 ðBÞ. i.e., CO does not hold. Definition 3.3. Let fRi : i ¼ 1; 2; . . . ; ng be a family of binary relations on a non-empty set U, for any set A U. We can introduce a definition for n-lower and n-upper approximations of A according to Ri , as the following: n apr
n
00
00
c
c
ðAÞ ¼ ½ apr ðA Þ ; ( ! n [ \ n apr 00 ðAÞ ¼ xRi :
n \
i¼1
! xRi
In the same manner we can prove U 1 , U 2 , U 3 , U 4 , U 5 , U 8 , L8 and CO. h We can introduce an example to show that the equality of L4 , U 4 , L8 , U 8 and CO are not true in general. Example 3.7. In Example 2.1, if A ¼ fag and B ¼ fcg, hence 00 00 00 2 apr 00 ðAÞ ¼ fa; cg, 2 apr ðAÞ ¼ ;, 2 apr ðBÞ ¼ ;, 2 apr ðA [ BÞ ¼ fag, 2 2 2 apr 00 ðBÞ ¼ fa; c; dg and apr00 ðA \ BÞ ¼ ;. Thus apr 00 ðAÞ\ 2 apr 00 ðBÞ6¼2 apr 00 ðA \ BÞ and 2 apr00 ðAÞ [ 2 apr 00 ðBÞ 6¼ 2 apr00 ðA [ BÞ. If A ¼ fcg, hence 2 apr 00 ðAÞ ¼ fa; c; dg, 2 apr00 ð2 apr00 ðAÞÞ ¼ fa; c; dg. Thus A 6¼ 2 apr00 ð2 apr00 ðAÞÞ. If A ¼ fa; c; eg, hence 2 apr00 ðAÞ ¼ fag, and 2 apr 00 ð2 apr 00 ðAÞÞ ¼ fa; cg. Thus, 2 apr00 ð2 apr00 ðAÞÞ 6¼ A. If A ¼ fb; c; eg and B ¼ fb; eg, hence ½2 apr00 ðAÞc ¼ fa; c; dg, 2 apr00 ðBÞ ¼ fb; eg and c c c 00 00 00 00 2 apr ðA [ BÞ ¼ fb; eg. Thus ½2 apr ðAÞ [ 2 apr ðBÞ 6¼ 2 apr ðA [ BÞ. Remark 3.3. For a family of binary relations fRi : i ¼ 1; 2; . . . ; ng on a non-empty set U the following conditions hold for every A; B U:
)
ðL6 Þ n apr 00 ð;Þ ¼ ;. ðL7 Þ n apr 00 ðAÞ A. ðL9 Þ n apr 00 ðAÞ ¼ n apr 00 ðn apr 00 ðAÞÞ. ðL10 Þ n apr00 ðAÞ ¼ n apr00 ðn apr00 ðAÞÞ. ðU 6 Þ n apr 00 ðUÞ ¼ U. ðU 7 Þ A n apr00 ðAÞ. ðU 9 Þ n apr 00 ðn apr 00 ðAÞÞ ¼ n apr00 ðAÞ. ðU 10 Þ n apr00 ðn apr00 ðAÞÞ ¼ n apr 00 ðAÞ. ðLUÞ n apr00 ðAÞ n apr00 ðAÞ.
\ A 6¼ ; :
i¼1
By using Definition 3.3, we can define the accuracy measure of any set A as the following: 00 n aany ðAÞ
¼
jA\n apr00 ðAÞj : jA[n apr00 ðAÞj
Proposition 3.3. For a family of binary relations fRi : i ¼ 1;2; .. .; ng on a non-empty set U the following conditions hold for every A;B U: 00
n
00
c
c
ðL1 Þ n apr ðAÞ ¼ ½ apr ðA Þ . ðL2 Þ n apr00 ðUÞ ¼ U. ðL3 Þ n apr00 ðA \ BÞ ¼ n apr00 ðAÞ \ n apr00 ðBÞ. ðL4 Þ n apr00 ðA [ BÞ n apr00 ðAÞ [ n apr00 ðBÞ. ðL5 Þ A B ) n apr00 ðAÞ n apr 00 ðBÞ. ðL8 Þ A n apr00 ðn apr00 ðAÞÞ. ðU 1 Þ n apr00 ðAÞ ¼ ½n apr 00 ðAc Þc . ðU 2 Þ n apr00 ð;Þ ¼ ;. ðU 3 Þ n apr00 ðA [ BÞ¼n apr 00 ðAÞ [ n apr00 ðBÞ. ðU 4 Þ n apr00 ðA \ BÞ n apr00 ðAÞ \ n apr00 ðBÞ. ðU 5 Þ A B)n apr 00 ðAÞ n apr 00 ðBÞ. ðU 8 Þ n apr00 ðn apr00 ðAÞÞ A. ðCOÞ n apr 00 ðAc [ BÞ ½n apr00 ðAÞc [ n apr00 ðBÞ.
Proof ðL1 Þ. From Definition 3.3, we have n apr00 ðAÞ ¼ ½n apr00 ðAc Þc . S T T ðL2 Þ. n apr 00 ðUÞ ¼ ½n apr 00 ð/Þc ¼ ½ f ni¼1 xRi : ni¼1 xRi \ / 6¼ /gc ¼ /c ¼ U. S Tn c c c c c 00 n 00 n 00 ðL3 Þ. n apr ðA \ BÞ ¼ ½ apr ðA \ BÞ ¼ ½ apr ðA [ B Þ ¼ ½ f i¼1 Tn S Tn Tn T c c c c xRi : i¼1 xRi \ ðA [ B Þ 6¼ /g ¼ ½ f i¼1 xRi : i¼1 xRi \ A 6¼ /gc Tn S Tn c c c c c c ½ f i¼1 xRi : i¼1 xRi \ B 6¼ /g ¼ ½n apr 00 ðA Þ \ ½n apr 00 ðB Þ ¼ n apr 00 ðAÞ \ n apr 00 ðBÞ. ðL4 Þ. Let x2n apr00 ðAÞ[n apr00 ðBÞ, hence x 2 n apr 00 ðAÞ or x 2 n apr 00 ðBÞ. So x 2 ½n apr00 ðAc Þc or x 2 ½n apr 00 ðBc Þc . This implies that x 62 n apr 00 ðAc Þ T or x 62 n apr00 ðBc Þ, then 8s 2 U, we have x 62 ni¼1 sRi such that Tn T Tn c c sR \ B 6¼ /. Consequently, x 62 ni¼1 sRi 8 i¼1 sRi \ A 6¼ / and Tn i¼1 i c c it is means that s 2 U such that i¼1 sRi \ ðA \ B Þ 6¼ /, x62n apr00 ðA [ BÞc i.e, x2n apr 00 ðA [ BÞ. Hence n apr 00 ðAÞ [ n apr00 ðBÞ 00 n apr ðA [ BÞ. ðL5 Þ. Let A B and x 62 n apr00 ðBÞ, hence x 62 ½n apr 00 ðBc Þc . So T x 2 n apr00 ðBc Þ and so 9s 2 U such that x 2 ni¼1 sRi and Tn Tn c c c c i¼1 sRi \ A 6¼ /, i¼1 sRi \ B 6¼ / but A B . This implies that c c c n 00 n 00 it is means that x 2 apr ðA Þ and so x 62 ½ apr ðA Þ ¼ n apr00 ðAÞ. Therefore n apr00 ðAÞ n apr 00 ðBÞ.
The following examples show Remark 3.3. Example 3.8. In Example 2.6, we have 2 apr 00 ð;Þ ¼ fa; cg and apr 00 ðUÞ ¼ fb; c; dg, i.e., L6 and U 6 do not hold. If A ¼ fb; dg, hence 00 00 2 apr ðAÞ ¼ fa; b; d; eg. Thus 2 apr ðAÞ 6 A. i.e., L7 does not hold. If 00 A ¼ fa; eg, hence 2 apr ðAÞ ¼ fag and 2 apr 00 ðAÞ ¼ ;. Thus A 6 2 apr 00 ðAÞ and 2 apr00 ðAÞ 6 2 apr 00 ðAÞ. i.e., U 7 and LU do not hold. 2
Example 3.9. In Example 2.1, if A ¼ fa; cg, hence 2 apr00 ðAÞ ¼ fag, 00 00 00 00 00 2 apr ð2 apr ðAÞÞ ¼ ;. Thus 2 apr ðAÞ 6¼ 2 apr ð2 apr ðAÞÞ, i.e., L9 does 2 00 apr ðAÞ ¼ fb; c; d; eg and not hold. If A ¼ fb; d; eg, hence 2 apr 00 ð2 apr 00 ðAÞÞ ¼ U. Then 2 apr00 ð2 apr00 ðAÞÞ 6¼ 2 apr 00 ðAÞ, i.e., U 9 does 2 apr 00 ðAÞ ¼ fa; cg and not hold. If A ¼ fag, hence 00 2 00 2 00 00 2 00 2 apr ð apr ðAÞÞ ¼ fag. Thus apr ðAÞ 6¼ 2 apr ð apr ðAÞÞ, i.e., L10 does 00 and not hold. If A ¼ fa; b; cg, hence 2 apr ðAÞ ¼ fag 2 00 00 2 00 00 00 apr ð2 apr ðAÞÞ ¼ fa; cg, so apr ð2 apr ðAÞÞ 6¼ 2 apr ðAÞ, i.e., U 10 does not hold. Table 2 shows the properties of rough sets by using the lower and upper approximations which is defined in Definitions 3.1, 3.2 and 3.3. 4. The second kind of generalization In this section, we will introduce another two approximations which are a generalization for approximations defined in Definitions 2.2 and 2.3 by used a family of binary relations. Definition 4.1. Let fRi : i ¼ 1; 2; . . . ; ng be a family of binary relations on a non-empty set U, for any set A U. We can introduce a definition for n-lower and n-upper approximations of A according to Ri as the following: n apr
ðAÞ ¼
n [ i¼1
Ri ðAÞ and n apr ðAÞ ¼
n \
Ri ðAÞ;
i¼1
where Ri ðAÞ and Ri ðAÞ defined in Definition 2.2. we can define the accuracy measure of any set A as the following:
917
H.M. Abu-Donia / Knowledge-Based Systems 21 (2008) 911–919 Table 2 Comparison between the properties of rough sets depending on Definitions 3.1, 3.2 and 3.3 by using any relations Ri : i ¼ 1; 2; . . . ; n Properties
Upper and lower approximations under any relations by using: Definition 3.1
Definition 3.2
Definition 3.3
L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 U1 U2 U3 U4 U5 U6 U7 U8 U9 U 10 CO LU
q q q q q
q
q q q q q
q q q q
Remark 4.1. Let Ri be a family of binary relations. Then the following conditions are not hold for every A; B U: ðL3 Þ n apr ðA \ BÞ ¼ n apr ðAÞ \ n apr ðBÞ. ðL6 Þ n apr ð;Þ ¼ ;. ðL7 Þ n apr ðAÞ A. ðL8 Þ A n apr ðn apr ðAÞÞ. ðL9 Þ n apr ðn apr ðAÞÞ ¼ n apr ðAÞ. ðL10 Þ n apr ðAÞ ¼ n apr ðn apr ðAÞÞ. ðU 3 Þ n apr ðA [ BÞ ¼ n apr ðAÞ [ n apr ðBÞ. ðU 6 Þ n apr ðUÞ ¼ U. ðU 7 Þ A n apr ðAÞ. ðU 8 Þ n apr ðn apr ðAÞÞ A. ðU 9 Þ n apr ðn apr ðAÞÞ ¼ n apr ðAÞ. ðU 10 Þ n apr ðn apr ðAÞÞ ¼ n apr ðAÞ. ðLUÞ n apr ðAÞ n apr ðAÞ. ðCOÞ n apr ðAc [ BÞ ½n apr ðAÞc [ n apr ðBÞ.
q q q q q q q
q
q q q q
q q q q q
The following examples show Remark 4.1.
q q q
q q
ðIÞ indicates that the property is satisfied. n aany ðAÞ
¼
jA \ n apr ðAÞj : jA [ n apr ðAÞj
Proposition 4.1. For a family of any binary relations fRi ; i ¼ 1; 2; . . . ; ng on a non-empty set U, the following conditions hold for every A; B U: ðL1 Þ n apr ðAÞ ¼ ½n apr ðAc Þc ðL2 Þ n apr ðUÞ ¼ U. ðL4 Þ n apr ðA [ BÞ n apr ðAÞ [ n apr ðBÞ. ðL5 Þ A B ) n apr ðAÞ n apr ðBÞ. ðU 1 Þ n apr ðAÞ ¼ ½n apr ðAc Þc . ðU 2 Þ n apr ð;Þ ¼ ;. ðU 4 Þ n apr ðA \ BÞ n apr ðAÞ \ n apr ðBÞ. ðU 5 Þ A B ) n apr ðAÞ n apr ðBÞ.
Example 4.2. In Example 2.1, if A ¼ fa; b; eg and B ¼ fc; d; eg, hence ðAÞ ¼ feg, 2 apr ðBÞ ¼ fb; c; eg and 2 apr ðA \ BÞ ¼ ;. Thus 2 apr ðA \ BÞ 6¼ 2 apr ðAÞ \ 2 apr ðBÞ. i.e., L3 does not hold. If A ¼ fdg and B ¼ feg, then 2 apr ðAÞ ¼ fbg, 2 apr ðBÞ ¼ fag and 2 apr ðA [ BÞ ¼ fa; b; cg. Thus 2 apr ðAÞ [ 2 apr ðBÞ 6¼ 2 apr ðA [ BÞ. i.e., U 3 does not hold. If A ¼ fbg, hence 2 apr ðAÞ ¼ feg and 2 apr ðAÞ ¼ fag, so 2 2 2 apr ðAÞ 6¼ A, apr ðAÞ 6¼ A and 2 apr ðAÞ 6¼ apr ðAÞ, i.e., L7 , U 7 and LU do not hold.If A ¼ feg, hence 2 apr ðAÞ ¼ fag and 2 2 2 apr ð apr ðAÞÞ ¼ ;.Thus A 6 2 apr ð apr ðAÞÞ, i.e., L8 does not hold.If n apr ðAÞ ¼ feg, apr ðn apr ðAÞÞ ¼ fag, A ¼ fdg, hence 2 2 2 n apr ð apr ðAÞÞ ¼ fag.So apr ðn apr ðAÞÞ 6 A, 2 apr ðAÞ 6¼ 2 apr ð2 apr ðAÞÞ and 2 apr ð2 apr ðAÞÞ6¼2 apr ðAÞ, i.e., U 8 , U 9 and U 10 do not hold.If A ¼ fa; b; eg, hence2 apr ðAÞ ¼ feg and 2 apr ð2 apr ðAÞÞ ¼ ;, so 2 apr ðAÞ 6¼ 2 apr ð2 apr ðAÞÞ. i.e., L9 does not hold. If A ¼ fag, hence 2 2 2 apr ðAÞ ¼ fdg and So apr ðAÞ 6¼ 2 apr ð apr ðAÞÞ ¼ feg. 2 2 apr ð apr ðAÞÞ. i.e., L10 does not hold. If A ¼ fc; d; eg and then B ¼ fa; cg, hence ½2 apr ðAÞc ¼ fa; dg, 2 apr ðBÞ ¼ ;, ½2 apr ðAÞc [ 2 apr ðBÞ ¼ fa; dg and 2 apr ðAc [ BÞ ¼ fe; dg. Thus ½2 apr ðAÞc [ 2 apr ðBÞ 6¼ 2 apr ðAc [ BÞ. i.e., CO does not hold. 2 apr
Example 4.3. In Example 2.2, we have apr ðUÞ ¼ fa; bg, i.e., L6 and U 6 do not hold.
2 apr
ð;Þ ¼ fcg and
2
Proof T S ðL1 Þ. ½n apr ðAc Þc ¼ ½ ni¼1 Ri ðAc Þc ¼ ni¼1 apri ðAÞ ¼ n R ðAÞ. ðL2 Þ. As U is the universe set, hence n apr ðUÞ U. 8x 2 U, we have xRi U8i ¼ 1; 2; . . . ; n, so x 2 Ri ðUÞ8i ¼ 1; 2; . . . ; n and so x 2 n apr ðUÞ. Then U n apr ðUÞ. Thus n apr ðUÞ ¼ U. ðL4 Þ. Let x 2 ½n apr ðAÞ [ n apr ðBÞ. Then x 2 n apr ðAÞ or S S x 2 n apr ðBÞ, so x 2 ni¼1 Ri ðAÞ or x 2 ni¼1 Ri ðBÞ, for fixed i 2 f1; 2; . . . ; ng, we have x 2 Ri ðAÞ or x 2 Ri ðBÞ. Then xRi A or xR B, so xRi ðA [ BÞ, hence x 2 Ri ðA [ BÞ but n apr ðA [ BÞ ¼ Sni i¼1 Ri ðA [ BÞ. So x 2 n apr ðA [ BÞ. i.e., n apr ðA [ BÞ n apr ðAÞ [n apr ðBÞ. S ðL5 Þ. Let A B and x 2 n apr ðAÞ. Then x 2 ni¼1 Ri ðAÞ. So there exists i such that x 2 Ri ðAÞ. Then xRi A but A B. So xRi A B, hence x 2 n apr ðBÞ. Thus n apr ðAÞ n apr ðBÞ. We can prove U 1 ; U 2 ; U 4 and U 5 as the same as L1 ; L2 ; L4 and L5 . h We can introduce an example to show that the equality of L4 and U 4 are not true in general. Example 4.1. In Example 2.1, if A ¼ fcg and B ¼ fb; dg, hence and 2 apr ðA [ BÞ ¼ fb; eg. i.e., 2 apr ðAÞ ¼ ;, 2 apr ðBÞ ¼ feg 2 apr ðAÞ [ 2 apr ðBÞ 6¼ 2 apr ðA [ BÞ. If A ¼ fcg and B ¼ fdg, hence 2 apr ðAÞ ¼ fb; dg, 2 apr ðBÞ ¼ fbg and 2 apr ðA \ BÞ ¼ ;. Thus 2 apr ðAÞ \ 2 apr ðBÞ 6¼ 2 apr ðA \ BÞ.
Definition 4.2. Let fRi : i ¼ 1; 2; . . . ; ng be a family of binary relations on a non-empty set U, for any set A U. We can introduce a definition for n-lower and n-upper approximations of A according to Ri as the following: n apr
ðAÞ ¼
n [
Ri ðAÞ
and
i¼1
n
apr ðAÞ ¼
n \
Ri ðAÞ;
i¼1
where R0i ðAÞ and R0i ðAÞ, defined in Definition 2.3. The accuracy measure of any set A will be defined as the following: 0 n aany ðAÞ
¼
jA\n apr 0 ðAÞj : jA [ n apr0 ðAÞj
Proposition 4.2. For a family of any binary relations Ri : i ¼ 1; 2; . . . ; n on a non-empty set U, the following conditions hold for every A; B U: ðL1 Þ ðL4 Þ ðL5 Þ ðL6 Þ ðL7 Þ ðL9 Þ
0 ðAÞ ¼ ½n apr0 ðAc Þc . 0 apr ðA [ BÞ n apr0 ðAÞ [ n apr0 ðBÞ. n A B ) n apr 0 ðAÞ n apr0 ðBÞ. 0 n apr ð;Þ ¼ ;. 0 apr ðAÞ A. n 0 0 0 n apr ðn apr ðAÞÞ¼n apr ðAÞ. n apr
918
H.M. Abu-Donia / Knowledge-Based Systems 21 (2008) 911–919
ðU 1 Þ n apr0 ðAÞ ¼ ½n apr0 ðAc Þc . ðU 4 Þ n apr0 ðA \ BÞ n apr 0 ðAÞ \ n apr0 ðBÞ. ðU 5 Þ A B)n apr 0 ðAÞn apr 0 ðBÞ. ðU 6 Þ n apr0 ðUÞ ¼ U. ðU 7 Þ A n apr0 ðAÞ. ðU 9 Þ n apr0 ðn apr 0 ðAÞÞ ¼ n apr0 ðAÞ. ðLUÞ n apr 0 ðAÞ n apr 0 ðAÞ.
Proof T S ðL1 Þ. ½n apr 0 ðAc Þc ¼ ½ ni¼1 Ri ðAc Þc ¼ ni¼1 R0i ðAÞ ¼ n apr 0 ðAÞ. 0 ðL4 Þ. Let x 2 ½n apr ðAÞ [ n apr0 ðBÞ. Then x 2 n apr 0 ðAÞ or S S x 2 n apr0 ðBÞ, so x 2 ni¼1 R0i ðAÞ or x 2 ni¼1 R0i ðBÞ, for fixed S i 2 f1; 2; . . . ; ng, we have x 2 R0i ðAÞ or x 2 R0i ðBÞ. Then x 2 ðsRi Þ A S S or x 2 ðsRi Þ B, where s 2 U. So x 2 ðsRi Þ ðA [ BÞ, hence Sn 0 0 but Thus x 2 R0i ðA [ BÞ n apr ðA [ BÞ ¼ i¼1 Ri ðA [ BÞ. 0 0 0 x 2 n apr ðA [ BÞ.i.e., n apr ðA [ BÞ n apr ðAÞ [ n apr 0 ðBÞ. S ðL5 Þ. Let A B and x 2 n apr 0 ðAÞ, then x 2 ni¼1 R0i ðAÞ, for some i, S we have x 2 R0i ðAÞ, then x 2 ðsRi Þ A, where s 2 U, but A B. So S x 2 ðsRi Þ B, hence x 2 R0i ðBÞ. Thus x 2 n apr 0 ðBÞ. Thus 0 0 n apr ðAÞ n apr ðBÞ. S ðL6 Þ. Since n apr0 ð;Þ ¼ ni¼1 R0i ð;Þ, but we have got that 0 Ri ð;Þ ¼ ;8i ¼ 1; 2; 3; . . . ; n. Thus n apr 0 ð;Þ ¼ ;. S ðL7 Þ. Since n apr 0 ðAÞ ¼ ni¼1 R0i ðAÞ, but we have got that 0 Ri ðAÞ A8i ¼ 1; 2; . . . ; n. Thus n apr 0 ðAÞ A. S ðL9 Þ. Since n apr 0 ðAÞ ¼ ni¼1 R0i ðAÞ, but we have got that R0i ðR0i ðAÞÞ ¼ R0i ðAÞ8i ¼ 1; 2; . . . ; n. Thus n apr0 ðn apr0 ðAÞÞ ¼ n apr 0 ðAÞ. We can prove U 1 ; U 4 ; U 5 ; U 6 , U 7 and U 9 as the same as L1 ; L4 ; L5 ; L6 , L7 and L9 . We can prove LU from L7 and U 7 . h We can introduce an example to show that the equality of L4 , U 4 , L7 , U 7 and LU are not true in general. Example 4.4. In Example 2.1, if A ¼ fcg and B ¼ fb; eg, hence 0 0 and 2 apr 0 ðA [ BÞ ¼ fb; c; eg. i.e., 2 apr ðAÞ ¼ ;, 2 apr ðBÞ ¼ fbg 0 0 0 apr ðAÞ [ apr ðBÞ ¼ 6 apr ðA [ BÞ. If A ¼ fb; cg and B ¼ fc; dg, 2 2 2 hence 2 apr 0 ðAÞ ¼ fb; c; eg, 2 apr 0 ðBÞ ¼ fa; c; d; eg, 2 apr0 ðA \ BÞ ¼ fcg and 2 apr 0 ðAÞ ¼ fbg Thus 2 apr0 ðAÞ \ 2 apr 0 ðBÞ 6¼ 2 apr0 ðA \ BÞ, 0 2 0 0 2 0 2 apr ðAÞ 6¼ A, apr ðAÞ 6¼ A and 2 apr ðAÞ 6¼ apr ðAÞ. Remark 4.2. For a family of any binary relations fRi : i ¼ 1; 2; . . . ; ng on a non-empty set U, the following conditions not hold for every A; B U: ðL2 Þ n apr0 ðUÞ ¼ U. ðL3 Þ n apr0 ðA \ BÞ ¼ n apr 0 ðAÞ \ n apr0 ðBÞ. ðL8 Þ A n apr0 ðn apr 0 ðAÞÞ. ðL10 Þ n apr 0 ðAÞ ¼ n apr 0 ðn apr 0 ðAÞÞ. ðU 2 Þ n apr0 ð;Þ ¼ ;. ðU 3 Þ n apr0 ðA [ BÞ ¼ n apr 0 ðAÞ [ n apr0 ðBÞ. ðU 8 Þ n apr0 ðn apr 0 ðAÞÞ A. ðU 10 Þ n apr 0 ðn apr0 ðAÞÞ ¼ n apr 0 ðAÞ. ðCOÞ n apr 0 ðAc [ BÞ ½n apr 0 ðAÞc [ n apr0 ðBÞ.
2
apr 0 ð2 apr 0 ðAÞÞ ¼ U. So 2 apr0 ð2 apr0 ðAÞÞ 6 A and 2 apr 0 ð2 apr 0 ðAÞÞ 6¼2 apr 0 ðAÞ. i.e., U 8 and U 10 do not hold. If A ¼ fa; b; cg and B ¼ fb; cg, hence 2 apr0 ðAc [ BÞ ¼ fb; c; d; eg, ½2 apr0 ðAÞc ¼ fd; eg and 2 apr 0 ðBÞ ¼ fbg. Thus 2 apr0 ðAc [ BÞ 6¼ ½2 apr 0 ðAÞc [ 2 apr0 ðBÞ. i.e., CO does not hold. Definition 4.3. Let fRi : i ¼ 1; 2; . . . ; ng be a family of binary relations on a non-empty set U, for any set A U. We can introduce a definition for n-lower and n-upper approximations of A according to Ri as the following: n apr
00
ðAÞ ¼
n [
R00i ðAÞ and n apr 00 ðAÞ ¼
i¼1
n \
R00i ðAÞ;
i¼1
where R00i ðAÞ and R00i ðAÞ, defined in Definition 2.4. The accuracy measure of any set A will be defined as the following: 00 n aany ðAÞ
¼
jA \ n apr 00 ðAÞj : jA [ n apr 00 ðAÞj
Proposition 4.3. For a family of any binary relations Ri : i ¼ 1; 2; . . . ; n on a non-empty set U, the following conditions hold for every A; B U: ðL1 Þ n apr 00 ðAÞ ¼ ½n apr00 ðAc Þc . ðL2 Þ n apr 00 ðUÞ ¼ U. ðL4 Þ n apr 00 ðA [ BÞ n apr 00 ðAÞ [ n apr 00 ðBÞ. ðL5 Þ A B ) n apr 00 ðAÞ n apr 00 ðBÞ. ðU 1 Þ n apr 00 ðAÞ ¼ ½n apr00 ðAc Þc . ðU 2 Þ n apr 00 ð;Þ ¼ ;. ðU 4 Þ n apr 00 ðA \ BÞ n apr 00 ðAÞ \ n apr 00 ðBÞ. ðU 5 Þ A B ) n apr 00 ðAÞ n apr 00 ðBÞ. S T Proof. ðU 1 Þ. ½n apr 00 ðAc Þc ¼ ½ ni¼1 R00i ðAc Þc ¼ ni¼1 R00i ðAÞ ¼ n apr 00 ðAÞ. ðU 2 Þ. As xRi \ ; ¼ ;8i ¼ 1; 2; . . . ; n, hence n apr 00 ð;Þ ¼ ;. ðU 4 Þ. Let x 62 ½n apr 00 ðAÞ \ n apr 00 ðBÞ, hence x 62 n apr00 ðAÞ or x 62 n apr 00 ðBÞ. So 9i 2 f1; 2; . . . ; ng, s.t x 62 R00i ðAÞ or x 62 R00i ðBÞ, so x 62 R00i ðAÞ \ R00i ðBÞ, but we have got that R00i ðA \ BÞ R00i ðAÞ \ R00i ðBÞ. So This implies that x 62 n apr00 ðA \ BÞ, i.e., x 62 R00i ðA \ BÞ. n apr 00 ðA \ BÞ n apr00 ðAÞ \ n apr 00 ðBÞ. ðU 5 Þ. Let A B and x 2 n apr00 ðAÞ, hence 8i ¼ 1; 2; . . . ; n, we have x 2 R00i ðAÞ. So there exists s 2 U such that x 2 sRi and sRi \ A 6¼ /8i ¼ 1; 2; . . . ; n. Since A B, then sRi \ B 6¼ /. This implies that x 62 R00i ðBÞ8i ¼ 1; 2; . . . ; n, and so x2n apr00 ðBÞ. Therefore n apr 00 ðAÞ n apr 00 ðBÞ. We can prove L1 ; L2 ; L4 and L5 as the same as U 1 ; U 2 ; U 4 and U5 . h We can introduce an example to prove that the equality of L4 and U 4 is not true in general. Example 4.6. In Example 2.1, if A ¼ fa; bg and B ¼ fcg, hence 00 00 00 ðAÞ ¼ ;, and i.e., 2 apr ðBÞ ¼ ; 2 apr ðA [ BÞ ¼ fag. 00 00 00 2 apr ðAÞ [ 2 apr ðBÞ 6¼ 2 apr ðA [ BÞ. If A ¼ fa; bg and B ¼ fa; cg, hence 2 apr00 ðAÞ ¼ U, 2 apr00 ðBÞ ¼ U and 2 apr 00 ðA \ BÞ ¼ fa; cg. Thus 2 apr 00 ðAÞ \ 2 apr 00 ðBÞ 6¼ 2 apr00 ðA \ BÞ. 2 apr
The following examples show Remark 4.2. Example 4.5. In Example 2.2, we have got 2 apr 0 ðUÞ ¼ fb; cg and apr0 ð;Þ ¼ fag. Thus 2 apr0 ðUÞ 6¼ U and 2 apr0 ð;Þ 6¼ ;. i.e., L2 and U 2 do not hold. If A ¼ fa; dg and B ¼ fa; b; cg, hence 2 apr0 ðAÞ ¼ fa; dg, 0 0 0 2 apr ðBÞ ¼ fa; b; cg and 2 apr ðA \ BÞ ¼ ;. Thus 2 apr ðA \ BÞ 6¼ 2 apr ðAÞ \ 2 apr ðBÞ. i.e., L3 does not hold. If A ¼ fcg and B ¼ fdg, hence 2 apr0 ðAÞ ¼ fcg, 2 apr 0 ðBÞ ¼ fdg2 apr 0 ðA [ BÞ ¼ fa; c; d; eg. So 2 apr0 ðAÞ [ 2 apr 0 ðBÞ 6¼ 2 apr0 ðA [ BÞ. i.e., U 3 does not hold. If A ¼ fag, hence 2 apr 0 ðAÞ ¼ fag and 2 apr 0 ð2 apr 0 ðAÞÞ ¼ ;. Thus A 6 2 apr0 ð2 apr0 ðAÞÞ and 2 apr 0 ðAÞ 6¼ 2 apr0 ð2 apr0 ðAÞÞ. i.e., L8 and L10 do not hold. If A ¼ fb; c; dg, hence 2 apr 0 ðAÞ ¼ fb; c; dg and 2
Remark 4.3. For a family of any binary relations fRi : i ¼ 1; 2; . . . ; ng on a non-empty set U, the following conditions do not hold for every A; B U: ðL3 Þ ðL6 Þ ðL7 Þ ðL8 Þ
ðA \ BÞ ¼ n apr 00 ðAÞ \ n apr 00 ðBÞ. n apr ð;Þ ¼ ;. 00 n apr ðAÞ A. A n apr00 ðn apr 00 ðAÞÞ. n apr
00 00
H.M. Abu-Donia / Knowledge-Based Systems 21 (2008) 911–919 Table 3 Comparison between the properties of rough sets depending on Definitions 4.1, 4.2 and 4.3 by using any relations Ri : i ¼ 1; 2; . . . ; n Properties
L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 U1 U2 U3 U4 U5 U6 U7 U8 U9 U 10 CO LU
Upper and lower approximations under any relations by using: Definition 4.1
Definition 4.2
Definition 4.3
q q
q
q q
q q
q q q q
q q
q q q
q
q q
q q
q q q q
q q
q
q
ðIÞ indicates that the property is satisfied.
ðL9 Þ n apr00 ð2 apr 00 ðAÞÞ ¼ n apr00 ðAÞ. ðL10 Þ n apr 00 ðAÞ ¼ n apr 00 ðn apr00 ðAÞÞ. ðU 3 Þ n apr00 ðA [ BÞ ¼ n apr 00 ðAÞ [ n apr00 ðBÞ. ðU 6 Þ n apr00 ðUÞ ¼ U. ðU 7 Þ A n apr00 ðAÞ. ðU 8 Þ n apr00 ðn apr 00 ðAÞÞ A. ðU 9 Þ n apr00 ðn apr 00 ðAÞÞ ¼ n apr00 ðAÞ. ðU 10 Þ n apr 00 ðn apr 00 ðAÞÞ ¼ n apr 00 ðAÞ. ðLUÞ n apr 00 ðAÞ n apr0 ðAÞ. ðCOÞ n apr 00 ðAc [ BÞ ½n apr00 ðAÞc [ n apr00 ðBÞ. The following examples show Remark 4.3. Example 4.7. In Example 2.1, if A ¼ fa; b; c; dg and B ¼ fa; c; d; eg, hence 2 apr 00 ðAÞ ¼ fa; dg, 2 apr00 ðBÞ ¼ fag and 2 apr00 ðA \ BÞ ¼ ;. Thus 2 apr 00 ðA \ BÞ 6¼ 2 apr 00 ðAÞ \ 2 apr00 ðBÞ. i.e., L3 does not hold. If A ¼ fag and B ¼ fdg, then 2 apr 00 ðAÞ ¼ fa; cg, 2 apr00 ðBÞ ¼ fb; c; dg and 2 apr 00 ðA [ BÞ ¼ U. Thus 2 apr00 ðAÞ [ 2 apr 00 ðBÞ 6¼ 2 apr 00 ðA [ BÞ. i.e., U 3 does not hold. If A ¼ fdg, hence 2 apr00 ðAÞ ¼ fb; c; dg and
919
ð apr00 ðAÞÞ ¼ ;. Thus A 6 2 apr00 ð2 apr 00 ðAÞÞ. i.e., L8 does not hold. If A ¼ fa; b; c; dg, hence 2 apr00 ðAÞ ¼ fa; dg, 2 apr 00 ð2 apr 00 ðAÞÞ ¼ U and 2 apr 00 ð2 apr 00 ðAÞÞ ¼ ;. So 2 apr 00 ð2 apr00 ðAÞÞ 6 A and 00 00 00 2 apr ðAÞ 6¼ 2 apr ð2 apr ðAÞÞ, i.e., U 8 and L9 do not hold. If A ¼ fd; eg, hence 2 apr 00 ðAÞ ¼ fb; c; d; eg and 2 apr00 ð2 apr 00 ðAÞÞ ¼ U. So 2 apr 00 ðAÞ 6¼ 2 apr00 ð2 apr 00 ðAÞÞ. i.e., U 9 does not hold. 2 apr
00 2
Example 4.8. In Example 2.6, we have 2 apr 00 ð;Þ ¼ fag and apr 00 ðUÞ ¼ fb; c; d; eg. Thus 2 apr00 ð;Þ 6¼ ; and 2 apr00 ðUÞ 6¼ U. i.e., L6 and U 6 do not hold. If A ¼ fb; dg, hence 2 apr 00 ðAÞ ¼ fa; b; dg, so 00 i.e., L7 does not hold. If A ¼ fag, hence 2 apr ðAÞ 6¼ A, 2 00 2 00 00 apr 00 ðAÞ ¼ ;, and 2 apr ð apr ðAÞÞ ¼ fag, 2 apr ðAÞ ¼ fag 2 00 apr 00 ð2 apr00 ðAÞÞ ¼ ;, so A 6 2 apr 00 ðAÞ, apr ðAÞ 6 2 apr 00 ðAÞ, 2 2 apr 00 ð2 apr00 ðAÞÞ 6¼ 2 apr 00 ðAÞand 2 apr 00 ðAÞ 6¼ 2 apr00 ð2 apr00 ðAÞÞ, i.e., U 7 , LU, L10 and U 10 do not hold. If A ¼ fa; b; c; eg and B ¼ fa; b; cg, hence ½2 apr 00 ðAÞc ¼ fdg, 2 apr 00 ðBÞ ¼ fag and 2 apr 00 ðAc [ BÞ ¼ fa; b; c; dg. Thus ½2 apr00 ðAÞc [ 2 apr00 ðBÞ 6 2 apr00 ðAc [ BÞ. i.e., CO does not hold. 2
Table 3 shows the properties of rough sets by using the lower and upper approximations which is defined in Definitions 4.1, 4.2 and 4.3. References [1] T.B. Iwinski, Algebraic approach to rough sets, Bull. Polish Acad. Sci. Ser. Sci. Math. 35 (1987) 673–683. [2] E. Orlowska, Z. Pawlak, Measurement and indiscernibility, Bull. Polish Acad. Sci. Ser. Sci. Math. 32 (1984) 617–624. [3] Z. Pawlak, Information systems theoretical foundations, Inform. Syst. 6 (3) (1981) 205–218. [4] Z. Pawlak, Rough sets, Int. J. Comput. Inform. Sci. 11 (1982) 341–356. [5] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data System Theory, Knowledge Engineering and Problem Solving, vol. 9, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. [6] J.A. Pomykala, Approximation operations in approximation space, Bull. Polish Acad. Sci. Ser. Sci. Math. 35 (1987) 653–662. [7] A. Skowron, J. Stepaniuk, Generalized approximations spaces, in: T.Y. Lin, A.M. Wildberger (Eds.), Soft Computing, The Society for Computer Simulation, San Diego, 1995, pp. 18–21. [8] A. Skowron, J. Stepaniuk, Tolerance approximations spaces, Fundame. Inform. 27 (1996) 245–253. [9] U. Wybraniec-Skardowska, On a generalization of approximation space, Bull. Polish Acad. Sci. Ser. Sci. Math. 37 (1989) 51–61. [10] Y.Y. Yao, T.Y. Line, Generalization of rough sets using modal logic, Int. Automation Soft Comput. 2 (1996) 103–120. [11] Y.Y. Yao, On generalized rough set theory, rough sets, fuzzy sets, data mining and granular computing, in: Proc. 9th Int, Conf., 1999, pp. 44–51. [12] W. Zirako (Ed.), Rough sets, fuzzy sets and knowledge discovery (RSKD’93), Workshops in Computing, Springer-Verlag and British Computer Society, London, Berlin, 1994.