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2022; 22(2): 193-201

Published online June 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.2.193

© The Korean Institute of Intelligent Systems

Properties of Different Types of Rough Approximations Defined by a Family of Dominance Relations

A. S. Salama1, Essam El-Seidy2, and A. K. Salah2

1Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
2Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt

Correspondence to :
A. K. Salah (a.k.salah@sci.asu.edu.eg)

Received: September 20, 2021; Revised: November 17, 2021; Accepted: November 29, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this study, we examine the properties of three types of lower and upper approximations of a non-empty finite set based on the dominance class generated by the dominance relations. We generalize these types using a family of dominance relations {Ri : i = 1, 2,...,n} and studying their properties.

Keywords: Rough sets, Generalized rough sets, Approximation spaces, Lower approximation, Upper approximation, Dominance rough sets, Dominance relations

In 1982, Pawlak [1] published his first paper on rough set theory, a mathematical tool for decision-making and knowledge discovery. The rough set theory depends on equivalence relations. Several extensions of rough sets are suitable for various applications. Fuzzy sets have commonly been used in these extensions [2,3]. Other approaches to fuzzy sets based on fuzzy covering and the comparison of different types of rough sets were introduced, as addressed in [4,5]. The generalizations of fuzzy rough sets constructed using fuzzy covering were studied in [6,7]. New types of generalized rough sets based on neighborhoods with medical applications were studied in [817]. The second research strategy is changing the type of relations from equivalence to tolerance relations to generalize rough sets in [18] and using similarity relations in [19]. Greco et al. [2023] proposed an extension of the rough set theory, the dominance-based rough set approach (DRSA), for solving the ordering problem of objects. Recently, further studies were conducted on DRSA [2426]. In [27], the author developed a new way of comparing different types of approximations using a family of binary relations without any conditions.

The remainder of this paper is organized as follows. In Section 2, we present three rough approximation definitions. These approximations are based on dominance relationships. Section 3 is divided into two parts. The first part discusses the generalization of the previous approximations to a family of n-dominance relations. The second part discusses the properties of these generalized approximations. Section 4 concludes the paper with remarks on forthcoming work.

In this section, we study the properties of three types of rough set approximations constructed on the dominance class generated by a dominance relation.

Definition 1

The relation R on a non-empty set U is called a dominance relation on U if it is reflexive, antisymmetric, and transitive.

Definition 2 [27]

If U is a finite universe and R is a binary relation on U, the right neighborhood of xU is defined as follows:

xR={y:xRyandyU}.

Definition 3

Let R be any dominance relation in a non-empty set U for any set AU. The lower and upper approximations of A based on R are defined as follows:

(A)={xU:xRA},U(A)={xU:xRA}.

This section is noteworthy because of the properties of rough set theory, which appear in [27]

Proposition 1

For any dominance relation R on a non-empty set U, the following conditions hold true for every A, BU:

  • (L1) , where Ac denotes the complement of A in U.

  • (L2) ℒ(U) = U.

  • (L3) ℒ(AB) = ℒ(A)∩ℒ(B).

  • (L4) ℒ(AB) ⊇ ℒ(A)∪ℒ(B).

  • (L5) AB ⇒ ℒ(A) ⊆ ℒ(B).

  • (L6) ℒ(∅︀) = ∅︀.

  • (L7) ℒ(A) ⊆ A.

  • (L9) ℒ(A) = ℒ(ℒ(A)).

  • (U1) .

  • (U2) .

  • (U3) .

  • (U4) .

  • (U5) .

  • (U6)

  • (U7) .

  • (U9) .

  • (CO) ℒ( AcB) ⊆ (ℒ(A))c ∪ ℒ(B).

  • (LU) .

Proof

For L1, L2, L3, L4, L5, U1, U2, U3, U4, U5 and CO. See [27].

  • (L6). This is because R is reflexive.

  • (L7). Let x ∈ ℒ(A) ⇒ xRA and xxRAxA.

  • (L9).

    • (i) ℒ(A) ⊇ ℒ(ℒ(A)).

      The proof of (i) is clear from L7.

    • (ii) ℒ(A) ⊆ ℒ(ℒ(A)).

      Let x ∈ ℒ(A) ⇒ xRA and y be any element in xR; then, we have yRxRA, Hence x ∈ ℒ(ℒ(A)).

  • (U6), (U7). Clearly,

  • (U9).

    • (i) .

      The proof of (i) is clear from U7.

    • (ii) .

      Let and y be any element in xR; then, we have yRxR. Hence, yRA = ∅︀. Therefore, , and .

  • (LU) This is evident from L7 and U7.

Remark 1

For any dominance relation R on a non-empty set U, the following conditions do not hold true for every AU:

  • (L8) .

  • (L10) .

  • (U8) .

  • (U10) .

We show Remark 1 in the following example:

Example 1

Let U = {x1, x2, x3, x4, x5, x6, x7, x8, x9, x10} and let the dominance relation R = {(x1, x8), (x1, x5), (x1, x1), (x1, x7), (x2, x10), (x2, x1), (x2, x7), (x2, x2), (x2, x6), (x2, x9), (x2, x8), (x2, x3), (x2, x5), (x3, x1), (x3, x6), (x3, x7), (x3, x8), (x3, x5), (x3, x3), (x4, x8), (x4, x4), (x4, x9), (x4, x7), (x4, x6), (x4, x10), (x4, x2), (x4, x1), (x4, x3), (x4, x5), (x5, x5), (x6, x8), (x6, x6), (x6, x5), (x7, x5), (x7, x7), (x7, x8), (x8, x8), (x9, x9), (x10, x10)}. Thus, the right neighborhoods are as follows:

x1R={x1,x5,x7,x8},x2R={x1,x2,x3,x5,x6,x7,x8,x9,x10},x3R={x1,x3,x5,x6,x7,x8},x4R=U,x5R={x5},x6R={x5,x6,x8},x7R={x5,x7,x8},x8R={x8},x9R={x9},x10R={x10}.

If A = {x1, x2, x4}, and . Thus, and . Therefore, L8 and L10 do not hold true. If B = {x5, x7, x8}, ℒ(B) = B and . Thus, and . Therefore, U8 and U10 do not hold true.

Definition 4

Let R be any dominance relation in a non-empty set U, And let xR be the right neighborhood of x according to R for any set AU. The second definition of the lower and upper approximations of A according to R is as follows:

*(A)={xR:xRA},U*(A)=[*(Ac)]c.

Proposition 2

For any dominance relation R on a non-empty set U, the following conditions hold true for every A, BU.

  • (L1) .

  • (L2) ℒ*(U) = U.

  • (L3) ℒ*(AB) = ℒ*(A)∩ℒ*(B).

  • (L4) ℒ*(AB) ⊇ ℒ*(A)∪ℒ*(B).

  • (L5) AB ⇒ ℒ*(A) ⊆ ℒ*(B).

  • (L6) ℒ*(∅︀) = ∅︀.

  • (L7) ℒ*(A) ⊆ A.

  • (L9) ℒ*(A) = ℒ*(ℒ*(A)).

  • (U1) .

  • (U2) .

  • (U3) .

  • (U4) .

  • (U5) .

  • (U6) .

  • (U7) .

  • (U9) .

  • (CO) ℒ*(AcB) ⊆ (ℒ*(A))c ∪ℒ*(B).

  • (LU) .

Proof

For L1, L4, L5, L6, L7, L9, U1, U4, U5, U6, U7, U9LU. See [27].

  • (L2).

  • (L3).

    • (i) ℒ*(AB) ⊆ ℒ*(A)∩ℒ*(B).

      Let x ∈ ℒ*(AB) ⇒ there exist yU s.t yRAByRA and yRBx ∈ ℒ*(A) and ℒ*(B).

    • (ii) ℒ*(AB) ⊇ ℒ*(A)∩ℒ*(B).

      Let x ∈ ℒ*(A) ∩ ℒ*(B) ⇒ x ∈ ℒ*(A) and x ∈ ℒ*(B) ⇒ ; then, there exists yU s.t xyRA and zU s.t xzRB. We have xRyRA and xRzRBxRA and xRBxRABxR ⊆ ℒ*(AB); thus, x ∈ ℒ*(AB).

  • (U2).

  • (U3).

    U*(AB)=[*((AB)c)]c=[*(Acbc)]c=[*(Ac)*(Bc)]c=[*(Ac)]c[*(Bc)]c=U*(A)U*(B).

  • (CO) Let x ∉ [ℒ*(A)]c ∪ ℒ*(B), thus x ∉ [ℒ*(A)]c and x ∉ ℒ*(B), thus x ∈ [ℒ*(A)] ⊆ A and x ∉ ℒ*(B) ⊆ B. Therefore, xAcB and then x ∉ ℒ*(AcB).

Remark 2

For any dominance relation R in a non-empty set U, the the following conditions do not hold true for every AU:

  • (L8) .

  • (L10) .

  • (U8) .

  • (U10) .

We show Remark 2 in the following example:

Example 2

In Example 1, if A = {x7, x8, x9, x10} then Ac = {x1, x2, x3, x4, x5, x6}, ℒ*(Ac) = {x5}, and which is not contains A. Thus, L8 and L10 are invalid. If B = {x5, x7, x8} then, ℒ*(B) = B, ℒ*(Bc) = {x9, x10}, hence [ℒ*(Bc)]cB. Thus, U8 and U10 are invalid.

Definition 5

Let R be any dominance relation in a non-empty set U and xR is the right neighborhood of x according to R for any set AU. The third definition of the lower and upper approximations of A according to R is as follows:

**(A)=[U**(Ac)]c,U**(A)={xR:xRA}

Proposition 3

For any dominance relation R in a non-empty set U, the following conditions hold true for every A,BU:

  • (L1) .

  • (L2) ℒ**(U) = U.

  • (L3) ℒ**(AB) = ℒ**(A)∩ℒ**(B).

  • (L4) ℒ**(AB) ⊇ ℒ**(A)∪ℒ**(B).

  • (L5) AB ⇒ ℒ**(A) ⊆ ℒ**(B).

  • (L6) ℒ**(∅︀) = ∅︀.

  • (L7) ℒ**(A) ⊆ A.

  • (L8) .

  • (U1) .

  • (U2) .

  • (U3) .

  • (U4) .

  • (U5) .

  • (U6) .

  • (U7) .

  • (U8) .

  • (CO) ℒ**(AcB) ⊆ (ℒ**(A))c ∪ℒ**(B).

  • (LU) .

Proof

For L1,L2, L3L4, L5, L8, U1, U2, U3, U4, U5, U8, CO. See [27].

  • (U6, L6, U7, L7). Clear.

  • (LU). This is straightforward from L7 and U7.

Remark 3

For any dominance relation R in a non-empty set U, the the following conditions do not hold true for every AU:

  • (L9) ℒ**(A) = ℒ**(ℒ**(A)).

  • (U9) .

  • (L10) .

  • (U10) .

We show Remark 3 in the following example:

Example 3

Let U = {1, 2, 3, 4} and the dominance relation R = {(1, 1), (2, 2), (3, 1), (3, 2), (3, 3), (4, 1), (4, 4)}; then, we have 1R = {1}, 2R = {2}, 3R = {1, 2, 3} and 4R = {1, 4}. If A = {4}, then , and . Thus, U9 and L10 are invalid. If B = {1, 2, 3}, then ℒ**(B) = {2, 3}, ℒ**(ℒ**(B)) = ∅︀, and . Hence, L9 and U10 are invalid.

In this section, we introduce the generalization of the previous three definitions of approximations using a family of dominance relations.

Definition 6

Let U be a non-empty set, and let R1, R2, … , Rn be n dominance relations on U for any set AU. The n-lower and n-upper approximations of A are defined as follows:

n(A)={xU:(i=1nxRi)A},Un(A)={xU:(i=1nxRi)A}.

Proposition 4

Let U be a non-empty set, and let R1, R2, … , Rn be n dominance relations on U. Then, the following conditions hold true for every A, BU:

  • (L1) .

  • (L2) ℒn(U) = U.

  • (L3) ℒn(AB) = ℒn(A)∩ℒn(B).

  • (L4) ℒn(AB) ⊇ ℒn(A)∪ℒn(B).

  • (L5) AB ⇒ ℒn(A) ⊆ ℒn(B).

  • (L6) ℒn(∅︀) = ∅︀.

  • (L7) ℒn(A) ⊆ A.

  • (L9) ℒn(A) = ℒn(ℒn(A)).

  • (U1) .

  • (U2) .

  • (U3) .

  • (U4) .

  • (U5) .

  • (U6) .

  • (U7) .

  • (U9) .

  • (CO) ℒn(AcB) ⊆ (ℒn(A))c ∪ℒn(B).

  • (LU) .

Proof

For L1, L2, L3, L4, L5, U1, U2, U3, U4, U5 and CO. See [27].

  • (L6). Clearly, Ri for i from 1 to n is reflexive.

  • (L7). Let xn(A)(i=1nxRi)A, and let x(i=1nxRi)AxA.

  • (L9).

    • (i) ℒn(A) ⊇ ℒn(ℒn(A)).

      The proof of (i) is clear from L7.

    • (ii) ℒn(A) ⊆ ℒn(ℒn(A)).

      Let xn(A)(i=1nxRi)A and y be any element in (i=1nxRi); then, we have yRixRi for all i = 1, 2, … , n and (i=1nyRi)(i=1nxRi)A. Thus, y ∈ ℒn(A) and x ∈ ℒn(ℒn(A)).

  • (U6), (U7).

  • (U9).

    • (i) .

      The proof of (i) can be easily obtained from U7.

    • (ii) .

      Let xUn(A)(i=1nxRi)A=, and let y be any element in (i=1nxRi); thus, yRixRi for all i = 1, 2, … , n and (i=1nyRi)(i=1nxRi). Hence, (i=1nyRi)A=. Therefore, (i=1nxRi)[Un(A)]c and .

  • (LU) This is evident from L7 and U7.

Remark 4

For any n dominance relations R1, R2, … , Rn in a non-empty set U, the following conditions do not hold true for every AU:

  • (L8) .

  • (L10) .

  • (U8) .

  • (U10) .

We show Remark 4 in the following example:

Example 4

Let U = {1, 2, 3, 4} and let two dominance relations R1 = {(1, 1), (2, 2), (3, 1), (3, 2), (3, 3), (4, 1), (4, 4)} and R2 = {(1, 1), (2, 2), (3, 1), (3, 3), (4, 1), (4, 2), (4, 4)}; then, we have 1R1 = {1}, 2R1 = {2}, 3R1 = {1, 2, 3}, 4R1 = {1, 4}, 1R2 = {1}, 2R2 = {2}, 3R2 = {1, 3}, and 4R2 = {1, 2, 4}. If A = {4}, . Thus, L8 and L10 are invalid. If B = {1, 4}, ; hence, U8 and U10 are invalid.

Definition 7

Let U be a non-empty set, and let R1, R2, … , Rn be n dominance relations on U for any set AU. The n-lower and n-upper approximations of A are defined as follows:

n*(A)={(i=1nxRi):(i=1nxRi)A},Un*(A)=[n*(Ac)]c.

Proposition 5

Let U be a non-empty set, and let R1, R2, … , Rn be n dominance relations on U. The following conditions hold true for every A,BU:

  • (L1) n*(A)=[Un*(Ac)]c.

  • (L2) n*(U)=U.

  • (L3) n*(AB)=n*(A)n*(B).

  • (L4) n*(AB)n*(A)n*(B).

  • (L5) ABn*(A)n*(B).

  • (L6) n*()=.

  • (L7) n*(A)A.

  • (L9) n*(A)=n*(n*(A)).

  • (U1) Un*(A)=[n*(Ac)]c.

  • (U2) Un*()=.

  • (U3) Un*(AB)=Un*(A)Un*(B).

  • (U4) Un*(AB)Un*(A)Un*(B).

  • (U5) ABUn*(A)Un*(B).

  • (U6) Un*(U)=U.

  • (U7) AUn*(A).

  • (U9) Un*(A)=Un*(Un*(A)).

  • (CO) n*(AcB)(n*(A))cn*(B).

  • (LU) n*(A)Un*(A).

Proof

For L1, L4, L5, L6, L7, L9, U1, U4, U5, U6, U7, U9, and LU. See [27].

  • (L2). Let xn*(U). Then, there exists yU:xi=1nyRiU. Therefore, xU. On the other hand, let xU. Then, i=1n(xRi)U. Therefore, xn*(U).

  • (L3).

    • (i) inline-formula>n*(AB)n*(A)n*(B).

      Let xn*(AB). Then, there exists yU:xi=1nyRiAB. Hence, i=1nyRiA and i=1nyRiB. Therefore, xn*(A) and xn*(B).

    • (ii) n*(AB)n*(A)n*(B).

      Let xn*(A)n*(B). Then, xn*(A) and xn*(B), and there exist yU:xi=1nyRiA and zU:xi=1nzRiB. As a result, we have xRiyRi and xRizRi for all i to n. Therefore, i=1nxRii=1nyRiA and i=1nxRii=1nzRiB. Hence, xn*(AB).

  • (U2). Obvious

  • (U3).

    Un*(AB)=[n*(AB)c]c=[n*(AcBc)]c=[n*(Ac)n*(Bc)]c=[n*(Ac)]c[n*(Bc)]c=Un*(A)Un*(B)

  • (CO) Let x[n*(Ac)]cn*(B). Then, xn*(A)A and xn*(B)B. Therefore, x ∉ (AcB) and xn*(AcB).

Remark 5

For any n dominance relations R1, R2, … , Rn in a non-empty set U, the following conditions do not hold true for every AU:

  • (L8) An*(Un*(A)).

  • (L10) Un*(A)=n*(Un*(A)).

  • (U8) AUn*(n*(A)).

  • (U10) n*(A)=Un*(n*(A)).

We show Remark 5 in the following example:

Example 5

In Example 4, if A = {1, 4}, Ac = {2, 3}, then 2*(A)={1,4},2*(Ac)={2}, and U2*(2*(A))={1,3,4}A; hence, U8 and U10 are invalid. If B = {2, 3, 4} and U2*(B)=B,2*(B)={2}, then 2*(U2*(B))={2}B. Therefore, L8 and L10 are invalid.

Definition 8

Let U be a non-empty set, and let R1, R2, … , Rn be n dominance relations on U for any set AU. The n-lower and n-upper approximations of A are defined as follows:

n**(A)=[Un**(Ac)]c,Un**(A)={(i=1nxRi):(i=1nxRi)A}.

Proposition 6

Let U be a non-empty set, and let R1, R2, … , Rn be n dominance relations on U. The following conditions hold true for every A,BU.

  • (L1) n**(A)=[Un**(Ac)]c.

  • (L2) n**(U)=U.

  • (L3) n**(AB)=n**(A)n**(B).

  • (L4) n**(AB)n**(A)n**(B).

  • (L5) ABn**(A)n**(B).

  • (L6) n**()=.

  • (L7) n**(A)A.

  • (L8) An**(Un**(A)).

  • (U1) Un**(A)=[n**(Ac)]c.

  • (U2) Un**()=.

  • (U3) Un**(AB)=Un**(A)Un**(B).

  • (U4) Un**(AB)Un**(A)Un**(B).

  • (U5) ABUn**(A)Un**(B).

  • (U6) Un**(U)=U.

  • (U7) AUn**(A).

  • (U8) AUn**(n**(A)).

  • (CO) n**(AcB)(n**(A))cn**(B).

  • (LU) n**(A)Un**(A).

Proof

For L1, L2, L3L4, L5, L8, U1, U2, U3, U4, U5, U8, and CO. See [27].

  • (U6).

  • (U7). xA(i=1nxRi)AxUn**(A).

  • (L6). This can be obtained directly from U6.

  • (L7). We have AcUn**(Ac) from U7 and [Un**(Ac)]c(Ac)c. Therefore n**(A)A.

  • (LU) This is clear from L7 and U7.

Remark 6

For any n dominance relations R1, R2, … , Rn in a non-empty set U, the following conditions do not hold true for every AU:

  • (L9) n**(A)=n**(n**(A)).

  • (L10) Un**(A)=n**(Un**(A)).

  • (U9) Un**(A)=Un**(Un**(A)).

  • (U10) n**(A)=Un**(n**(A)).

We show Remark 6 in the following example:

Example 6

In Example 4, if A = {1, 2, 3}, then 2**(A)={2,3},2**(2**(A))={2}; hence, U9 is not valid. If B = {4}, C = {1, 4}, then we have U2**(B)=C,2**(C)=B and U2**(U2**(B))={1,3,4}; thus, we have U2**(2**(C))=CB and 2**(U2**(B))=BC. Therefore, U9, L10 and U10 are invalid.

We defined several upper and lower approximations and studied the properties of the rough sets according to these approximations. We found that the changes in the relationship were reflected in the satisfactory properties. Furthermore, changing the approximation definition for the same relationship can change the properties that are satisfied. Future work will focus on changing the type of relationship.

A . S. Salama received the B.S. degree in mathematics from Tanta University, Tanta, Egypt, in 1998 and the M.S. degree in Topological Rough Sets from the University of Tanta, Egypt, in 2002. He worked at Tanta university from 1998 to 2008. He received his Ph.D. degree at 2004 from Tanta University, Tanta, Egypt, in Topology and Information Systems. He was a professor in the Department of Mathematics at Faculty of Science in Tanta University, from 2019 until now. His research interests include artificial intelligence, rough set, data mining, topology, fuzzy sets, information systems. He published about 42 papers in different journals in the felid of topology and its applications.

E-mail: asalama@science.tanta.edu.eg


Essam El-Seidy received the B.Sc. degree in mathematics from Ain Shams University, Egypt, in 1983 and the M.Sc. degree from the University of Ain Shams, Egypt, in 1988. He received his Ph.D. degree in 1994 in Operations Research “Game Theory”. He is a professor in the Department of Mathematics at Faculty of Science in Ain Shams University, from 2019 until now. He is now Head of Mathematics Department, Faculty of Science, Ain Shams University. His research interests include population game dynamic, symmetric and asymmetric games, differential games, data classifications. He published about 48 papers in different journals.

E-mail: esam_elsedy@hotmail.com


A. K. Salah received the B.Sc. degree in pure mathematics and computer science from Ain Shams University, Egypt, in 2016. He is a master’s student in Topological Approaches for Data Classifications at Ain Shams University, Egypt. He works as a teaching assistant at Ain Shams University. His research interests include rough set, topology, information systems, data classification.

E-mail: a.k.salah@sci.asu.edu.eg


Article

Original Article

2022; 22(2): 193-201

Published online June 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.2.193

Copyright © The Korean Institute of Intelligent Systems.

Properties of Different Types of Rough Approximations Defined by a Family of Dominance Relations

A. S. Salama1, Essam El-Seidy2, and A. K. Salah2

1Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
2Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt

Correspondence to:A. K. Salah (a.k.salah@sci.asu.edu.eg)

Received: September 20, 2021; Revised: November 17, 2021; Accepted: November 29, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this study, we examine the properties of three types of lower and upper approximations of a non-empty finite set based on the dominance class generated by the dominance relations. We generalize these types using a family of dominance relations {Ri : i = 1, 2,...,n} and studying their properties.

Keywords: Rough sets, Generalized rough sets, Approximation spaces, Lower approximation, Upper approximation, Dominance rough sets, Dominance relations

1. Introduction

In 1982, Pawlak [1] published his first paper on rough set theory, a mathematical tool for decision-making and knowledge discovery. The rough set theory depends on equivalence relations. Several extensions of rough sets are suitable for various applications. Fuzzy sets have commonly been used in these extensions [2,3]. Other approaches to fuzzy sets based on fuzzy covering and the comparison of different types of rough sets were introduced, as addressed in [4,5]. The generalizations of fuzzy rough sets constructed using fuzzy covering were studied in [6,7]. New types of generalized rough sets based on neighborhoods with medical applications were studied in [817]. The second research strategy is changing the type of relations from equivalence to tolerance relations to generalize rough sets in [18] and using similarity relations in [19]. Greco et al. [2023] proposed an extension of the rough set theory, the dominance-based rough set approach (DRSA), for solving the ordering problem of objects. Recently, further studies were conducted on DRSA [2426]. In [27], the author developed a new way of comparing different types of approximations using a family of binary relations without any conditions.

The remainder of this paper is organized as follows. In Section 2, we present three rough approximation definitions. These approximations are based on dominance relationships. Section 3 is divided into two parts. The first part discusses the generalization of the previous approximations to a family of n-dominance relations. The second part discusses the properties of these generalized approximations. Section 4 concludes the paper with remarks on forthcoming work.

2. Different Types of Rough Set Approximations Using Any Dominance Relation

In this section, we study the properties of three types of rough set approximations constructed on the dominance class generated by a dominance relation.

Definition 1

The relation R on a non-empty set U is called a dominance relation on U if it is reflexive, antisymmetric, and transitive.

Definition 2 [27]

If U is a finite universe and R is a binary relation on U, the right neighborhood of xU is defined as follows:

xR={y:xRyandyU}.

Definition 3

Let R be any dominance relation in a non-empty set U for any set AU. The lower and upper approximations of A based on R are defined as follows:

(A)={xU:xRA},U(A)={xU:xRA}.

This section is noteworthy because of the properties of rough set theory, which appear in [27]

Proposition 1

For any dominance relation R on a non-empty set U, the following conditions hold true for every A, BU:

  • (L1) , where Ac denotes the complement of A in U.

  • (L2) ℒ(U) = U.

  • (L3) ℒ(AB) = ℒ(A)∩ℒ(B).

  • (L4) ℒ(AB) ⊇ ℒ(A)∪ℒ(B).

  • (L5) AB ⇒ ℒ(A) ⊆ ℒ(B).

  • (L6) ℒ(∅︀) = ∅︀.

  • (L7) ℒ(A) ⊆ A.

  • (L9) ℒ(A) = ℒ(ℒ(A)).

  • (U1) .

  • (U2) .

  • (U3) .

  • (U4) .

  • (U5) .

  • (U6)

  • (U7) .

  • (U9) .

  • (CO) ℒ( AcB) ⊆ (ℒ(A))c ∪ ℒ(B).

  • (LU) .

Proof

For L1, L2, L3, L4, L5, U1, U2, U3, U4, U5 and CO. See [27].

  • (L6). This is because R is reflexive.

  • (L7). Let x ∈ ℒ(A) ⇒ xRA and xxRAxA.

  • (L9).

    • (i) ℒ(A) ⊇ ℒ(ℒ(A)).

      The proof of (i) is clear from L7.

    • (ii) ℒ(A) ⊆ ℒ(ℒ(A)).

      Let x ∈ ℒ(A) ⇒ xRA and y be any element in xR; then, we have yRxRA, Hence x ∈ ℒ(ℒ(A)).

  • (U6), (U7). Clearly,

  • (U9).

    • (i) .

      The proof of (i) is clear from U7.

    • (ii) .

      Let and y be any element in xR; then, we have yRxR. Hence, yRA = ∅︀. Therefore, , and .

  • (LU) This is evident from L7 and U7.

Remark 1

For any dominance relation R on a non-empty set U, the following conditions do not hold true for every AU:

  • (L8) .

  • (L10) .

  • (U8) .

  • (U10) .

We show Remark 1 in the following example:

Example 1

Let U = {x1, x2, x3, x4, x5, x6, x7, x8, x9, x10} and let the dominance relation R = {(x1, x8), (x1, x5), (x1, x1), (x1, x7), (x2, x10), (x2, x1), (x2, x7), (x2, x2), (x2, x6), (x2, x9), (x2, x8), (x2, x3), (x2, x5), (x3, x1), (x3, x6), (x3, x7), (x3, x8), (x3, x5), (x3, x3), (x4, x8), (x4, x4), (x4, x9), (x4, x7), (x4, x6), (x4, x10), (x4, x2), (x4, x1), (x4, x3), (x4, x5), (x5, x5), (x6, x8), (x6, x6), (x6, x5), (x7, x5), (x7, x7), (x7, x8), (x8, x8), (x9, x9), (x10, x10)}. Thus, the right neighborhoods are as follows:

x1R={x1,x5,x7,x8},x2R={x1,x2,x3,x5,x6,x7,x8,x9,x10},x3R={x1,x3,x5,x6,x7,x8},x4R=U,x5R={x5},x6R={x5,x6,x8},x7R={x5,x7,x8},x8R={x8},x9R={x9},x10R={x10}.

If A = {x1, x2, x4}, and . Thus, and . Therefore, L8 and L10 do not hold true. If B = {x5, x7, x8}, ℒ(B) = B and . Thus, and . Therefore, U8 and U10 do not hold true.

Definition 4

Let R be any dominance relation in a non-empty set U, And let xR be the right neighborhood of x according to R for any set AU. The second definition of the lower and upper approximations of A according to R is as follows:

*(A)={xR:xRA},U*(A)=[*(Ac)]c.

Proposition 2

For any dominance relation R on a non-empty set U, the following conditions hold true for every A, BU.

  • (L1) .

  • (L2) ℒ*(U) = U.

  • (L3) ℒ*(AB) = ℒ*(A)∩ℒ*(B).

  • (L4) ℒ*(AB) ⊇ ℒ*(A)∪ℒ*(B).

  • (L5) AB ⇒ ℒ*(A) ⊆ ℒ*(B).

  • (L6) ℒ*(∅︀) = ∅︀.

  • (L7) ℒ*(A) ⊆ A.

  • (L9) ℒ*(A) = ℒ*(ℒ*(A)).

  • (U1) .

  • (U2) .

  • (U3) .

  • (U4) .

  • (U5) .

  • (U6) .

  • (U7) .

  • (U9) .

  • (CO) ℒ*(AcB) ⊆ (ℒ*(A))c ∪ℒ*(B).

  • (LU) .

Proof

For L1, L4, L5, L6, L7, L9, U1, U4, U5, U6, U7, U9LU. See [27].

  • (L2).

  • (L3).

    • (i) ℒ*(AB) ⊆ ℒ*(A)∩ℒ*(B).

      Let x ∈ ℒ*(AB) ⇒ there exist yU s.t yRAByRA and yRBx ∈ ℒ*(A) and ℒ*(B).

    • (ii) ℒ*(AB) ⊇ ℒ*(A)∩ℒ*(B).

      Let x ∈ ℒ*(A) ∩ ℒ*(B) ⇒ x ∈ ℒ*(A) and x ∈ ℒ*(B) ⇒ ; then, there exists yU s.t xyRA and zU s.t xzRB. We have xRyRA and xRzRBxRA and xRBxRABxR ⊆ ℒ*(AB); thus, x ∈ ℒ*(AB).

  • (U2).

  • (U3).

    U*(AB)=[*((AB)c)]c=[*(Acbc)]c=[*(Ac)*(Bc)]c=[*(Ac)]c[*(Bc)]c=U*(A)U*(B).

  • (CO) Let x ∉ [ℒ*(A)]c ∪ ℒ*(B), thus x ∉ [ℒ*(A)]c and x ∉ ℒ*(B), thus x ∈ [ℒ*(A)] ⊆ A and x ∉ ℒ*(B) ⊆ B. Therefore, xAcB and then x ∉ ℒ*(AcB).

Remark 2

For any dominance relation R in a non-empty set U, the the following conditions do not hold true for every AU:

  • (L8) .

  • (L10) .

  • (U8) .

  • (U10) .

We show Remark 2 in the following example:

Example 2

In Example 1, if A = {x7, x8, x9, x10} then Ac = {x1, x2, x3, x4, x5, x6}, ℒ*(Ac) = {x5}, and which is not contains A. Thus, L8 and L10 are invalid. If B = {x5, x7, x8} then, ℒ*(B) = B, ℒ*(Bc) = {x9, x10}, hence [ℒ*(Bc)]cB. Thus, U8 and U10 are invalid.

Definition 5

Let R be any dominance relation in a non-empty set U and xR is the right neighborhood of x according to R for any set AU. The third definition of the lower and upper approximations of A according to R is as follows:

**(A)=[U**(Ac)]c,U**(A)={xR:xRA}

Proposition 3

For any dominance relation R in a non-empty set U, the following conditions hold true for every A,BU:

  • (L1) .

  • (L2) ℒ**(U) = U.

  • (L3) ℒ**(AB) = ℒ**(A)∩ℒ**(B).

  • (L4) ℒ**(AB) ⊇ ℒ**(A)∪ℒ**(B).

  • (L5) AB ⇒ ℒ**(A) ⊆ ℒ**(B).

  • (L6) ℒ**(∅︀) = ∅︀.

  • (L7) ℒ**(A) ⊆ A.

  • (L8) .

  • (U1) .

  • (U2) .

  • (U3) .

  • (U4) .

  • (U5) .

  • (U6) .

  • (U7) .

  • (U8) .

  • (CO) ℒ**(AcB) ⊆ (ℒ**(A))c ∪ℒ**(B).

  • (LU) .

Proof

For L1,L2, L3L4, L5, L8, U1, U2, U3, U4, U5, U8, CO. See [27].

  • (U6, L6, U7, L7). Clear.

  • (LU). This is straightforward from L7 and U7.

Remark 3

For any dominance relation R in a non-empty set U, the the following conditions do not hold true for every AU:

  • (L9) ℒ**(A) = ℒ**(ℒ**(A)).

  • (U9) .

  • (L10) .

  • (U10) .

We show Remark 3 in the following example:

Example 3

Let U = {1, 2, 3, 4} and the dominance relation R = {(1, 1), (2, 2), (3, 1), (3, 2), (3, 3), (4, 1), (4, 4)}; then, we have 1R = {1}, 2R = {2}, 3R = {1, 2, 3} and 4R = {1, 4}. If A = {4}, then , and . Thus, U9 and L10 are invalid. If B = {1, 2, 3}, then ℒ**(B) = {2, 3}, ℒ**(ℒ**(B)) = ∅︀, and . Hence, L9 and U10 are invalid.

3. Generalizations of Rough Set Approximations Using n-Dominance Relations

In this section, we introduce the generalization of the previous three definitions of approximations using a family of dominance relations.

Definition 6

Let U be a non-empty set, and let R1, R2, … , Rn be n dominance relations on U for any set AU. The n-lower and n-upper approximations of A are defined as follows:

n(A)={xU:(i=1nxRi)A},Un(A)={xU:(i=1nxRi)A}.

Proposition 4

Let U be a non-empty set, and let R1, R2, … , Rn be n dominance relations on U. Then, the following conditions hold true for every A, BU:

  • (L1) .

  • (L2) ℒn(U) = U.

  • (L3) ℒn(AB) = ℒn(A)∩ℒn(B).

  • (L4) ℒn(AB) ⊇ ℒn(A)∪ℒn(B).

  • (L5) AB ⇒ ℒn(A) ⊆ ℒn(B).

  • (L6) ℒn(∅︀) = ∅︀.

  • (L7) ℒn(A) ⊆ A.

  • (L9) ℒn(A) = ℒn(ℒn(A)).

  • (U1) .

  • (U2) .

  • (U3) .

  • (U4) .

  • (U5) .

  • (U6) .

  • (U7) .

  • (U9) .

  • (CO) ℒn(AcB) ⊆ (ℒn(A))c ∪ℒn(B).

  • (LU) .

Proof

For L1, L2, L3, L4, L5, U1, U2, U3, U4, U5 and CO. See [27].

  • (L6). Clearly, Ri for i from 1 to n is reflexive.

  • (L7). Let xn(A)(i=1nxRi)A, and let x(i=1nxRi)AxA.

  • (L9).

    • (i) ℒn(A) ⊇ ℒn(ℒn(A)).

      The proof of (i) is clear from L7.

    • (ii) ℒn(A) ⊆ ℒn(ℒn(A)).

      Let xn(A)(i=1nxRi)A and y be any element in (i=1nxRi); then, we have yRixRi for all i = 1, 2, … , n and (i=1nyRi)(i=1nxRi)A. Thus, y ∈ ℒn(A) and x ∈ ℒn(ℒn(A)).

  • (U6), (U7).

  • (U9).

    • (i) .

      The proof of (i) can be easily obtained from U7.

    • (ii) .

      Let xUn(A)(i=1nxRi)A=, and let y be any element in (i=1nxRi); thus, yRixRi for all i = 1, 2, … , n and (i=1nyRi)(i=1nxRi). Hence, (i=1nyRi)A=. Therefore, (i=1nxRi)[Un(A)]c and .

  • (LU) This is evident from L7 and U7.

Remark 4

For any n dominance relations R1, R2, … , Rn in a non-empty set U, the following conditions do not hold true for every AU:

  • (L8) .

  • (L10) .

  • (U8) .

  • (U10) .

We show Remark 4 in the following example:

Example 4

Let U = {1, 2, 3, 4} and let two dominance relations R1 = {(1, 1), (2, 2), (3, 1), (3, 2), (3, 3), (4, 1), (4, 4)} and R2 = {(1, 1), (2, 2), (3, 1), (3, 3), (4, 1), (4, 2), (4, 4)}; then, we have 1R1 = {1}, 2R1 = {2}, 3R1 = {1, 2, 3}, 4R1 = {1, 4}, 1R2 = {1}, 2R2 = {2}, 3R2 = {1, 3}, and 4R2 = {1, 2, 4}. If A = {4}, . Thus, L8 and L10 are invalid. If B = {1, 4}, ; hence, U8 and U10 are invalid.

Definition 7

Let U be a non-empty set, and let R1, R2, … , Rn be n dominance relations on U for any set AU. The n-lower and n-upper approximations of A are defined as follows:

n*(A)={(i=1nxRi):(i=1nxRi)A},Un*(A)=[n*(Ac)]c.

Proposition 5

Let U be a non-empty set, and let R1, R2, … , Rn be n dominance relations on U. The following conditions hold true for every A,BU:

  • (L1) n*(A)=[Un*(Ac)]c.

  • (L2) n*(U)=U.

  • (L3) n*(AB)=n*(A)n*(B).

  • (L4) n*(AB)n*(A)n*(B).

  • (L5) ABn*(A)n*(B).

  • (L6) n*()=.

  • (L7) n*(A)A.

  • (L9) n*(A)=n*(n*(A)).

  • (U1) Un*(A)=[n*(Ac)]c.

  • (U2) Un*()=.

  • (U3) Un*(AB)=Un*(A)Un*(B).

  • (U4) Un*(AB)Un*(A)Un*(B).

  • (U5) ABUn*(A)Un*(B).

  • (U6) Un*(U)=U.

  • (U7) AUn*(A).

  • (U9) Un*(A)=Un*(Un*(A)).

  • (CO) n*(AcB)(n*(A))cn*(B).

  • (LU) n*(A)Un*(A).

Proof

For L1, L4, L5, L6, L7, L9, U1, U4, U5, U6, U7, U9, and LU. See [27].

  • (L2). Let xn*(U). Then, there exists yU:xi=1nyRiU. Therefore, xU. On the other hand, let xU. Then, i=1n(xRi)U. Therefore, xn*(U).

  • (L3).

    • (i) inline-formula>n*(AB)n*(A)n*(B).

      Let xn*(AB). Then, there exists yU:xi=1nyRiAB. Hence, i=1nyRiA and i=1nyRiB. Therefore, xn*(A) and xn*(B).

    • (ii) n*(AB)n*(A)n*(B).

      Let xn*(A)n*(B). Then, xn*(A) and xn*(B), and there exist yU:xi=1nyRiA and zU:xi=1nzRiB. As a result, we have xRiyRi and xRizRi for all i to n. Therefore, i=1nxRii=1nyRiA and i=1nxRii=1nzRiB. Hence, xn*(AB).

  • (U2). Obvious

  • (U3).

    Un*(AB)=[n*(AB)c]c=[n*(AcBc)]c=[n*(Ac)n*(Bc)]c=[n*(Ac)]c[n*(Bc)]c=Un*(A)Un*(B)

  • (CO) Let x[n*(Ac)]cn*(B). Then, xn*(A)A and xn*(B)B. Therefore, x ∉ (AcB) and xn*(AcB).

Remark 5

For any n dominance relations R1, R2, … , Rn in a non-empty set U, the following conditions do not hold true for every AU:

  • (L8) An*(Un*(A)).

  • (L10) Un*(A)=n*(Un*(A)).

  • (U8) AUn*(n*(A)).

  • (U10) n*(A)=Un*(n*(A)).

We show Remark 5 in the following example:

Example 5

In Example 4, if A = {1, 4}, Ac = {2, 3}, then 2*(A)={1,4},2*(Ac)={2}, and U2*(2*(A))={1,3,4}A; hence, U8 and U10 are invalid. If B = {2, 3, 4} and U2*(B)=B,2*(B)={2}, then 2*(U2*(B))={2}B. Therefore, L8 and L10 are invalid.

Definition 8

Let U be a non-empty set, and let R1, R2, … , Rn be n dominance relations on U for any set AU. The n-lower and n-upper approximations of A are defined as follows:

n**(A)=[Un**(Ac)]c,Un**(A)={(i=1nxRi):(i=1nxRi)A}.

Proposition 6

Let U be a non-empty set, and let R1, R2, … , Rn be n dominance relations on U. The following conditions hold true for every A,BU.

  • (L1) n**(A)=[Un**(Ac)]c.

  • (L2) n**(U)=U.

  • (L3) n**(AB)=n**(A)n**(B).

  • (L4) n**(AB)n**(A)n**(B).

  • (L5) ABn**(A)n**(B).

  • (L6) n**()=.

  • (L7) n**(A)A.

  • (L8) An**(Un**(A)).

  • (U1) Un**(A)=[n**(Ac)]c.

  • (U2) Un**()=.

  • (U3) Un**(AB)=Un**(A)Un**(B).

  • (U4) Un**(AB)Un**(A)Un**(B).

  • (U5) ABUn**(A)Un**(B).

  • (U6) Un**(U)=U.

  • (U7) AUn**(A).

  • (U8) AUn**(n**(A)).

  • (CO) n**(AcB)(n**(A))cn**(B).

  • (LU) n**(A)Un**(A).

Proof

For L1, L2, L3L4, L5, L8, U1, U2, U3, U4, U5, U8, and CO. See [27].

  • (U6).

  • (U7). xA(i=1nxRi)AxUn**(A).

  • (L6). This can be obtained directly from U6.

  • (L7). We have AcUn**(Ac) from U7 and [Un**(Ac)]c(Ac)c. Therefore n**(A)A.

  • (LU) This is clear from L7 and U7.

Remark 6

For any n dominance relations R1, R2, … , Rn in a non-empty set U, the following conditions do not hold true for every AU:

  • (L9) n**(A)=n**(n**(A)).

  • (L10) Un**(A)=n**(Un**(A)).

  • (U9) Un**(A)=Un**(Un**(A)).

  • (U10) n**(A)=Un**(n**(A)).

We show Remark 6 in the following example:

Example 6

In Example 4, if A = {1, 2, 3}, then 2**(A)={2,3},2**(2**(A))={2}; hence, U9 is not valid. If B = {4}, C = {1, 4}, then we have U2**(B)=C,2**(C)=B and U2**(U2**(B))={1,3,4}; thus, we have U2**(2**(C))=CB and 2**(U2**(B))=BC. Therefore, U9, L10 and U10 are invalid.

4. Conclusion and Future Work

We defined several upper and lower approximations and studied the properties of the rough sets according to these approximations. We found that the changes in the relationship were reflected in the satisfactory properties. Furthermore, changing the approximation definition for the same relationship can change the properties that are satisfied. Future work will focus on changing the type of relationship.

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