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
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)
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 [8–17]. 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. [20–23] 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 [24–26]. 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
In this section, we study the properties of three types of rough set approximations constructed on the dominance class generated by a dominance relation.
The relation
If
Let
This section is noteworthy because of the properties of rough set theory, which appear in [27]
For any dominance relation
(, where
(
(
(
(
(
(
(
(.
(.
(.
(.
(.
(
(.
(.
(
(.
For
(
(
(
(i) ℒ(
The proof of (
(ii) ℒ(
Let
(
(
(i) .
The proof of (
(ii) .
Let and
, and
.
(
For any dominance relation
(.
(.
(.
(.
We show Remark 1 in the following example:
Let
If and
. Thus,
and
. Therefore,
. Thus,
and
. Therefore,
Let
For any dominance relation
(.
(
(
(
(
(
(
(
(.
(.
(.
(.
(.
(.
(.
(.
(
(.
For
(
(
(i) ℒ*(
Let
(ii) ℒ*(
Let
(
(
(CO) Let
For any dominance relation
(.
(.
(.
(.
We show Remark 2 in the following example:
In Example 1, if and
which is not contains
Let
For any dominance relation
(.
(
(
(
(
(
(
(.
(.
(.
(.
(.
(.
(.
(.
(.
(
(.
For
(
(
For any dominance relation
(
(.
(.
(.
We show Remark 3 in the following example:
Let , and
. Thus,
. Hence,
In this section, we introduce the generalization of the previous three definitions of approximations using a family of dominance relations.
Let
Let
(.
(
(
(
(
(
(
(
(.
(.
(.
(.
(.
(.
(.
(.
(
(.
For
(
(
(
(i) ℒ
The proof of (
(ii) ℒ
Let
(
(
(i) .
The proof of (
(ii) .
Let .
(
For any n dominance relations
(.
(.
(.
(.
We show Remark 4 in the following example:
Let . Thus,
; hence,
Let
Let
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
For
(
(
(i) inline-formula>
Let
(ii)
Let
(
(
(
For any
(
(
(
(
We show Remark 5 in the following example:
In Example 4, if
Let
Let
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
For
(
(
(
(
(
For any
(
(
(
(
We show Remark 6 in the following example:
In Example 4, if
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.
No potential conflict of interest relevant to this article was reported.
E-mail: asalama@science.tanta.edu.eg
E-mail: esam_elsedy@hotmail.com
E-mail: a.k.salah@sci.asu.edu.eg
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.
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)
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 [8–17]. 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. [20–23] 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 [24–26]. 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
In this section, we study the properties of three types of rough set approximations constructed on the dominance class generated by a dominance relation.
The relation
If
Let
This section is noteworthy because of the properties of rough set theory, which appear in [27]
For any dominance relation
(, where
(
(
(
(
(
(
(
(.
(.
(.
(.
(.
(
(.
(.
(
(.
For
(
(
(
(i) ℒ(
The proof of (
(ii) ℒ(
Let
(
(
(i) .
The proof of (
(ii) .
Let and
, and
.
(
For any dominance relation
(.
(.
(.
(.
We show Remark 1 in the following example:
Let
If and
. Thus,
and
. Therefore,
. Thus,
and
. Therefore,
Let
For any dominance relation
(.
(
(
(
(
(
(
(
(.
(.
(.
(.
(.
(.
(.
(.
(
(.
For
(
(
(i) ℒ*(
Let
(ii) ℒ*(
Let
(
(
(CO) Let
For any dominance relation
(.
(.
(.
(.
We show Remark 2 in the following example:
In Example 1, if and
which is not contains
Let
For any dominance relation
(.
(
(
(
(
(
(
(.
(.
(.
(.
(.
(.
(.
(.
(.
(
(.
For
(
(
For any dominance relation
(
(.
(.
(.
We show Remark 3 in the following example:
Let , and
. Thus,
. Hence,
In this section, we introduce the generalization of the previous three definitions of approximations using a family of dominance relations.
Let
Let
(.
(
(
(
(
(
(
(
(.
(.
(.
(.
(.
(.
(.
(.
(
(.
For
(
(
(
(i) ℒ
The proof of (
(ii) ℒ
Let
(
(
(i) .
The proof of (
(ii) .
Let .
(
For any n dominance relations
(.
(.
(.
(.
We show Remark 4 in the following example:
Let . Thus,
; hence,
Let
Let
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
For
(
(
(i) inline-formula>.
Let
(ii)
Let
(
(
(
For any
(
(
(
(
We show Remark 5 in the following example:
In Example 4, if
Let
Let
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
For
(
(
(
(
(
For any
(
(
(
(
We show Remark 6 in the following example:
In Example 4, if
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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