Abstract
This paper aims to obtain new types of approximations by using topological concepts. Firstly, different kinds of topologies are generated by subset neighborhoods and relationships between them are studied. Then, j-subset approximations based on these topologies are introduced and their basic properties are examined. In addition to this, -near open and -open sets are defined and the connections among them are given. Later, new approximations are presented with the help of the aforementioned sets, and their main properties are investigated. Furthermore, the proposed approximations are compared both with themselves and with the previous one. From this, it is shown that the approximations based on -open sets are more accurate than those based on -open and -near open sets under arbitrary binary relation and than those based on j-open sets under similarity relation. Finally, a real-life problem related to COVID-19 is addressed to highlight the importance of applying the proposed approximations.
1. Introduction
Rough set theory was introduced by Pawlak [1, 2] as a substantial mathematical tool to cope with vagueness and uncertain knowledge. Its central concepts are upper and lower approximations based on equivalence relations. However, the requirement of these relations limits the application fields. Therefore, many researchers have used topological concepts to generalize these approximations and have replaced equivalence relations with binary relations (see [3–5]). Abd El-Monsef et al. [6] defined the concept of -neighborhood space to generalize the classical rough set theory via different topologies induced by an arbitrary relation. Then, new types of neighborhoods such as -adhesion neighborhoods [7], -neighborhoods [8], and containment neighborhoods [9] were introduced and investigated in this space. In addition, initial neighborhoods are proposed by El-Sayed et al. [10]. After then, Al-Shami and Ciucci [11] generalized this concept to subset neighborhoods and gave more properties and results of them. Also, El-Bably [12] and Amer et al. [13] obtained new -near approximations as mathematical instruments modifying and generalizing the -approximations and studied their applications. Hosny [14] and Nawar et al. [15] presented new kinds of approximations by using -open sets, -sets, and -open sets, respectively. On the other hand, a fuzzy rough set as a fuzzy generalization of rough sets was introduced by Dubois and Prade [16]. Thus, varied types of fuzzy rough sets were presented and axiomatic properties of fuzzy rough approximation operators are obtained in many studies (see [17–21]). After then, a new model of fuzzy rough sets called Metric-based fuzzy rough sets [22] was put forward via hemimetric.
This paper aims to find the best approximation and the highest accuracy measure using -near open and -open sets. It is planned as follows: in Section 3, new kinds of topologies based on subset neighborhoods are introduced and the relationships among them are studied. Then, the comparison between these topologies and the previous ones in [6] is given. By using them, new types of approximations, boundary regions, and accuracy measures of a set are defined. Also, the best -subset approximations and highest -subset accuracy measures are obtained for . Moreover, these approximations, boundary regions, and accuracy measures are compared with the previous ones in [6, 11]. In Section 4, the concepts of -near open sets based on the topologies generated by subset neighborhoods are presented and the connections among them are studied. Then, -open sets are defined as the generalization of -near open sets, and the fundamental properties of them are investigated. In Section 5, -subset approximations are generalized to -near approximations based on -near open sets. The main properties of these approximations are studied. Then, -approximations are defined and these approximations are compared with -near approximations, -subset approximations, and -approximations. In Section 6, the real-life problem of COVID-19 is presented to obtain the best approximation and the highest accuracy measures comparing the proposed approaches with Abd El-Monsef et al.’s approach [6] under similarity relation.
2. Preliminaries
This section contains the main ideas about -neighborhood and subset neighborhood cited in [6, 11].
Definition 1 (see [6]). Let be an arbitrary binary relation on a nonempty finite set . The -neighborhood of (briefly, ) for each is defined as follows:(1)-neighborhood: (2)-neighborhood: (3)-neighborhood: (4)-neighborhood: (5)-neighborhood: (6)-neighborhood: (7)-neighborhood: (8)-neighborhood:
Definition 2 (see [6]). Let be a mapping which assigns for each its -neighborhood in . The triple is called -neighborhood space (-NS).
Theorem 1 (see [6]). Let be a -NS. Then, = { for each } is a topology on for every . The elements of are called -open sets and the complement of -open sets are -closed sets.
Definition 3 (see [6]). Let be a -NS, and . Then, the -lower and -upper approximations, -boundary regions, and -accuracy measures of are defined as follows, respectively: and , and where .
If then is called a -exact set. Otherwise, it is called a -rough set.
Definition 4 (see [11]). Let be an arbitrary binary relation on a nonempty finite set . The subset neighborhood of (briefly, ) for each is defined as follows:(1)(2)(3)(4)(5)(6)(7)(8)
Proposition 1 (see [11]). Let be a -NS and . Then,(1) for each (2)If , then for every
Lemma 1 (see [11]). Let be a -NS and be a similarity relation on . Then, for each and for .
Definition 5 (see [11]). Let be a -NS, and . Then, the lower and upper approximations, -boundary regions, and -accuracy measures of are defined as follows, respectively: , , and where .
3. Topologies Based on -Neighborhoods
This section aims to introduce eight different topologies based on -neighborhoods for . The relationships among them are given and comparisons between these topologies and are examined. Then, -subset approximations, -subset boundary regions, and -subset accuracy measures of a set are defined and the connections among them are studied. From here it is seen that the best approximations and the highest accuracy measures are obtained for . Also, the proposed approximations in Definition 7 are compared with the previous ones in Definitions 3 and 5.
Theorem 2. Let be a -NS. Then, = { for each } is a topology on generated by -neighborhood for every .
Proof. (1)(2)Suppose that for each and . Then, there exists such that . From here, . This implies . Hence (3)Let and . Since and , we get and . Thus, . So, we have
Proposition 2. Let be a -NS. Then,(1)(2)(3)(4)
Proof. (1) and (2) Let . Then, for each . Thus, which implies and . Hence, and . Moreover , that is , and so .
(3) and (4) The proofs are done similarly to the proofs of (1) and (2).
Remark 1. and , and , and need not be comparable.
The following example shows that and are not comparable and are not dual topologies. Also, it illustrates that and , and are incomparable.
Example 1. Let and . Then, we have(i) and (ii) and ,(iii) and . Similarly, suitable examples can be found for all other
Proposition 3. Let be a -NS and be a similarity relation on . Then, for .
Proof. It is clear from Lemma 1.
The following example shows that and are not comparable for .
Example 2. Let and . Then, we have = {, , , , } and = {, , , , , , }.
The following example shows that and are not comparable if the relation is not similarity relation.
Example 3. Consider Example 1 and let . Then, we obtain and .
Definition 6. Let be a -NS. is called -open set if , the complement of -open set is said to be -closed set. The family of all -closed sets is defined by .
Definition 7. Let be a -NS, and . Then, the -subset lower and -subset upper approximations are defined as follows, respectively: where and are -interior and -closure of according to , respectively.
If , then is called an -exact set. Otherwise, it is an -rough set.
Remark 2. and provide all the properties of Pawlak approximations.
Proposition 4. Let be a -NS and . Then,(1)(2)(3)(4)
Proof. The proofs are clear from Proposition 2.
Proposition 5. Let be a -NS and . Then,(1)(2)(3)(4)
Proof. The proofs are clear from Proposition 2.
Theorem 3. Let be a -NS, be a similarity relation on and . For , we have .
Proof. It is clear from Proposition 3.
The following example shows that Theorem 3 does not hold for .
Example 4. Consider Example 2. For and , we have , , , . Thus, they are not comparable. Besides, we obtain , , and . That is, they are also not comparable.
Proposition 6. Let be a -NS and . For each , we have(1)(2)
Proof. (1)Let . Then, there exists such that for . Thus, . This implies that . Conversely, let . Then, . Since from Proposition 1, we get .(2)It can be proved similarly.
Definition 8. Let be a -NS, and . Then, the -subset boundary regions and -subset accuracy measures of are defined as follows, respectively: , , where .
The following two corollaries follow from Propositions 4 and 5.
Corollary 1. Let be a -NS and . Then,(1)(2)(3)(4)
Corollary 2. Let be a -NS and . Then,(1)(2)(3)(4)
In Tables 1 and 2, the -subset boundary regions and -subset accuracy measures of all subsets of are calculated for each according to Example 1. Thus, it is seen that the -subset accuracy measures for are more precise than those for .
In Table 3, the algorithm is given to calculate the accuracy measures induced from the topologies generated by subset neighborhoods.
The following two corollaries follow from Theorem 3.
Corollary 3. Let be a -NS, be a similarity relation on and . For , we have(1)(2)
Corollary 4. Let be a -NS, be a similarity relation on and . If is -exact, then it is -exact for .
The converse implication of Corollary 4 need not be true.
Example 5. Let and . Then, is -exact but not -exact.
The following corollary follows from Proposition 6.
Corollary 5. Let be a -NS and . For each , we have(1)(2)
4. Sj-Near Open Sets and -Open Sets
Section 4 aims to present the concepts of -near open set using the topologies . The relationships among these concepts are investigated. From this, it is seen that the concepts of -open and -open sets cannot be compared. Then, the concept of -open set is introduced as a generalization of the -near open sets, and their main properties are studied.
Definition 9. Let be a -NS, and . Then, is called(1)-open if (2)-open if (3)-open if (4)-open if These sets are called -near open sets, the complement of the -near open sets are -near closed sets. The families of -near open sets (-near closed sets) of denoted by for each .
Proposition 7. Let be a -NS and . Then, the following implications hold.(1)Every -open sets are -open(2)Every -open sets are -open(3)Every -open sets are -open(4)Every -open sets are -open(5)Every -open sets are -open
Proof. Straightforward by Definitions 6 and 9.
The converse implications of Proposition 7 need not be true.
Example 6. Let and . Then,(1) is -open but not -open(2) is -open but not -open(3) is -open but not -open(4) is -open but not -open(5) is -open but not -open
Remark 3. The concepts of -open set and -open set are not comparable for . In Example 6, is -open but not -open and is -open but not -open.
Definition 10. Let be a -NS, and . The -closure of is defined by = { for each and }.
Definition 11. Let be a -NS, and . is called -open if . The complement of -open sets are -closed sets and the family of -open (-closed) sets of are denoted by .
Proposition 8. Let be a -NS and . Then, every -open sets are -open.
Proof. Straightforward by definitions of -closure and -closure.
The converse implication of Proposition 8 need not be true.
Example 7. Consider Example 5. Then, is -open but not -open. In that case, it is not -open, -open, -open and -open.
Remark 4. Let and .(1)Proposition 2 is not satisfied in general for (2)Although the union of two -open sets is -open, their intersection need not be -open
Example 8. Let .(1)Consider the Example 1. Then,(i).(ii).(iii).(iv). Thus, , , and . Also, and are -open sets but the intersection of these two sets is not -open.(2)Let and . Then,(i)(ii)(iii)(iv) Thus, and Other examples can be given for that confirms the correctness of Remark 4.
5. Sj-Near Approximations and -Approximations
Section 5 aims to introduce the concepts of -near approximations as a generalization of -subset approximations by using -near open sets. The fundamental properties of them are given. Also, the concepts of -approximations are presented and connections between -approximations, -near approximations, -subset approximations, and -approximations are studied.
Definition 12. Let be a -NS, , and . The -near lower and -near upper approximations, -near boundary regions, and -near accuracy measures of are defined as follows, respectively: = -near interior of , = -near closure of , , where .
If , then is called an -near exact set. Otherwise, it is an -near rough set.
Proposition 9. Let be a -NS, , and . Then,(1)(2)If then and (3) and (4) and (5) and (6) and
Proof. The proofs are clear from the definition of -near approximations.
The converse implications and inclusions of Proposition 9 are not necessarily true.
Example 9. Consider Example 8 (1).(1)Let and . Then, we get and . Thus, . Also, we have and . Hence, . Moreover, and but .(2)Let and . Then, we get and . Thus, . Also, we obtain and . From here, .
Theorem 4. Let be a -NS, , and . Then, .
Proof. By Propositions 7 and 9 (1), the proof is obvious.
The following corollary follows from Theorem 4.
Corollary 6. Let be a -NS, , and . Then,(1)(2)The following two corollaries follow from Theorems 3 and 4.
Corollary 7. Let be a -NS, be a similarity relation on , , and . Then, .
Corollary 8. Let be a -NS, be a similarity relation on , , and . Then,(1)(2)
Definition 13. Let be a -NS, and . The -lower and -upper approximations, -boundary regions, and -accuracy measures of are defined as follows, respectively: = -interior of , = -closure of ,, , where .
If , then is called a -exact set. Otherwise, it is a -rough set.
Proposition 10. Let be a -NS, and . Then,(1)(2)If then and (3) and (4) and (5) and (6) and
Proof. The proofs are clear from the definition of -approximations.
Proposition 11. Let be a -NS, and . Then,(1)(2)(3)(4)
Proof. (1) By Propositions 7 and 8, = = .
The other proofs can be proved by the same propositions in a similar way.
The following corollary follows from Proposition 11.
Corollary 9. Let be a -NS, and . Then,(1)(2)(3)(4)
Theorem 5. Let be a -NS, , and . Then, .
Proof. By Theorem 4, Propositions 10 (1) and 11, the proof is obvious.
The following corollary follows from Theorem 5.
Corollary 10. Let be a -NS, , and . Then,(1)(2)
In Table 4, the boundary regions and accuracy measures of all subsets of are calculated for and according to Example 6. Thus, it is seen that the -accuracy measure is higher than -subset accuracy measure and -near accuracy measures for .
The following corollaries follow from Theorem 5 and Corollary 7.
Corollary 11. Let be a -NS,, be a similarity relation on , , and . Then, .
Corollary 12. Let be a -NS, be a similarity relation on , , and . Then,(1)(2)
6. An Application
Coronavirus disease (COVID-19) is an infectious disease that emerged in 2019 and affects different people differently. Some infected people develop mild to moderate illness, while others can become seriously ill and require medical attention. Therefore, it is very important to make an accurate decision in diagnosis to prevent the spread of the disease. In this section, a real-life example in decision making to COVID-19 is given by applying the suggested two methods to show the importance of higher accuracy measures by using the similarity relation in [9, 11]. Moreover, the given example illustrates comparisons between these two methods and Abd El-Monsef et al.’s method [6].
Example 10. Consider the data in Table 5 gathered from the COVID-19 study conducted by WHO (see [10, 23]). This study was applied to 1000 patients whose most common symptoms of this disease were difficulty in breathing (DB), chest pain (CP), headache (H), dry cough (DC), high temperature (HT), and loss of taste or smell (LTS), then 1000 patients were reduced to 10 patients by El Sayed et al. [10]. Here the set of objects is the set of 10 patients, the set of condition attributes is the set of most common symptoms of COVID-19 and is the set of decision attribute. In Table 6, similarities between symptomatic patients are given using the degree of similarity which is the ratio of equal attributes between any of the two patients where and is the number of condition attributes. Let us take the similarity relation , with and assume .
Consider the two sets of yes and no decisions, that and . From Table 6, the approximations and accuracy measures of and are computed according to Abd El-Monsef et al.’s method in [6] and two proposed methods in Definitions 7 and 13. As seen from Table 7, the accuracy measured by using the proposed methods are higher than the accuracy measured by using Abd El-Monsef et al.’s method. Also, among these three methods, the method using -open sets gives the best approximations and highest accuracy.
7. Conclusion
In this paper, based on the notions of subset neighborhoods, different kinds of topologies were generated and the relationships between the considered types of topologies were given. Then, -subset approximations were defined by using these topologies and it was shown that the best approximations were obtained for . In addition, -near open and -open sets were introduced and the connections among them were investigated. Besides, new types of approximations were defined via these open sets and they were compared with themselves and with the previous one. Thus, it was obtained that -approximations are more precise than -subset approximations and -near approximations for any arbitrary relation and than -approximations for similarity relation. Finally, to confirm the importance of the proposed approaches, a practical problem related to COVID-19 was given and the proposed approaches were compared with Abd El-Monsef et al.’s approaches in [6]. Thus, it was seen that the approaches using -open sets gave the best approximations and the highest accuracy.
In future studies, the methods proposed in this paper can be applied to a variety of real-life problems. In addition, better new approximations and higher accuracy measures can be obtained by using different neighborhood types or set types. Besides, new links between rough set theory and fuzzy set theory based on proposed set types can be established.
Data Availability
The data of Example 10 are taken from [10, 23]. All this information is included in the manuscript.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
The author would like to thank the referees and the editor for their helpful suggestions.