Abstract
The basic methodology of rough set theory depends on an equivalence relation induced from the generated partition by the classification of objects. However, the requirements of the equivalence relation restrict the field of applications of this philosophy. To begin, we describe two kinds of closure operators that are based on right and left adhesion neighbourhoods by any binary relation. Furthermore, we illustrate that the suggested techniques are an extension of previous methods that are already available in the literature. As a result of these topological techniques, we propose extended rough sets as an extension of Pawlak’s models. We offer a novel topological strategy for making a topological reduction of an information system for COVID-19 based on these techniques. We provide this medical application to highlight the importance of the offered methodologies in the decision-making process to discover the important component for coronavirus (COVID-19) infection. Furthermore, the findings obtained are congruent with those of the World Health Organization. Finally, we create an algorithm to implement the recommended ways in decision-making.
1. Introduction
General topology [1] has recently been put as a topic in its own right as well as being a significant mathematical instrument with such varied topics as operational research approaches, biochemistry, genetics, and sociology. The construction of topology and its concepts via relations has become a remarkable and hot role in solving many problems such as [2–15]. Closure space has been described by Cech [10] as an extension of classical topology. Pal in [15] defined closure operators through binary relations. Relations represent a mathematical tool that has some real-life data and applications which can be resolved. Moreover, relations are applied in building some topological structures that are used in structure analysis, general view of space-time, biochemistry, biology, dynamics, fuzzy set model, and rough set idea. For more details, see [2–8, 12–14, 16–21]. In the viewpoint of topological structures, topological notions, especially, closure operators with relations, are useful in applications.
In covering-based rough sets, an adhesion-set is introduced [21]. Nawar et al. [22] utilized it to define the j-adhesion neighbourhood formed by any binary relation, and they proposed different types of covering-based rough sets as a result. Furthermore, Atef et al. [23] provided six rough approximations derived by j-adhesion neighbourhoods utilizing j-neighbourhood space [8] which was modified by El-Bably et al. in [5].
In this study, the idea of adhesion neighbourhood is utilized to construct various generalized closure operators via relations. These operators are a generalization of Galton’s [17], Allam’s et al. [18], and El-Bably’s et al. operators [19, 20]. Furthermore, we demonstrated that the new operators are topological properties with no requirements, and we showed some of their features. Rough set theory was established by computer scientist Pawlak [24, 25] based on several difficulties in computer science to overcome this challenge by a modal approximation of a crisp set in the expressions of a pair of sets called the rough approximations of it. Many writers have focused on generalization rough sets [2, 4, 8, 11, 13, 20, 22, 23, 26–39].
The rough set model’s basic subject is the notion of uncertainty areas. It seeks to find the boundary region by expanding the lower approximation and contracting the higher approximation. These approximations were induced by closure operators in this context. The comparison between our techniques with the Yao strategy is studied. Lellis Thivagar and Richard [40] proposed the concept of nano topology. It is mostly based on Pawlak’s preliminary guesses (namely, the lower, upper, and boundary region of a rough set). Nano open sets are the characteristics of a nano topological space.
We construct novel nano topologies from generalized rough approximations using the proposed operators in this research, and we apply them in the decision-making of COVID-19 infection. We reduce the amount of data we collect in order to determine the risk variables for COVID-19 infection. As a result, we assert that remaining at home may reduce the danger. We create an algorithm for decision-making utilizing our ideas towards the conclusion of the article.
2. Basic Concepts and Properties
Some definitions and results from the sequel are provided.
Definition 1 (see [41]). Let be a binary relation from a nonempty set to a nonempty set . can be from to itself and called a relation on . Consequently, if is a binary relation from to , we say that is related to if , sometimes written .
Definition 2 (see [41]). A binary relation on is called as(i)Reflexive if (ii)Symmetric if (iii)Transitive if (iv)Equivalence if conditions (i), (ii), and (iii) are satisfied.
Definition 3 (see [41]). Let be a relation on . After set and foreset of are and , respectively.
Definition 4 (see [1]). A collection of subsets of is a topology on if it contains , finite intersection (resp. arbitrary union) of its elements is closed. is a topological space, each set in is open, and its complement w. r. to is closed. (resp. ) denotes the interior (resp. closure) of in .
Definition 5 (see [27]). The operator , where is the power set of , is called the closure of , where , if satisfies(i)(ii), (iii), . The pair is called a closure space.
Definition 6 (see [17]). Let be a closure space and let . Then,(i), where is the complement of w. r. to (ii) is a neighbourhood of an element if (iii)If , then is closed(iv)if , then is open.For each , is called the interior of [13] and satisfies.(i)(ii), (iii), .
Remark 1. If , then the space is called a closure space.
Definition 7 (see [27]). Let be a relation on . Then, the closure operator on by is .
Lemma 1 (see [17]). Let be a transitive relation on a nonempty finite set . Then, the operator is a topological closure.
Definition 8 (see [24]). Let be a universal set and be an equivalence relation on . are equivalence classes of . is an approximation space. The lower and upper approximations of are and , respectively, for any . is rough set if .
Proposition 1 (see [24]). If is a complement set of in , then represents an empty set. The properties of Pawlak’s rough sets are as follows:
Definition 9 (see [26]). For , the lower and upper approximations of w. r. to neighbourhood of are and , respectively. The properties (L3-L9) and (U1, U2, U4-U9) are satisfied, in general, while some Pawlak’s properties are held in .
Definition 10 (see [40]). In , is a nano topology on w. r. to with a base .
3. A New Closure Operator and Its Equivalences
Throughout this section, some different sorts of closure operators in terms of relations are introduced, and comparisons between them are discussed.
Definition 11 (see [5, 22, 23]). Let be any relation on . The right adhesion and left adhesion set of each are defined, respectively, as follows:(i)r-adhesion: (ii)l-adhesion: It is simple to demonstrate the following lemma, so we omit the proof.
Lemma 2. Let be any relation on . Then, .(i)(ii)(iii) is a partition on .
Lemma 3. For all : if , for any on .
Proof. It is sufficient to prove this lemma for and the others similarly. Firstly, if . Now, it is sufficient to prove . Let , then . Hence, , and this implies . Thus, . Also, is proved similarly.
Proposition 2. For a reflexive relation on and , we get(i)(ii).
Proof. (i) Let , then . But is a reflexive relation on , thus . Hence, , and this implies . (ii) is proved similarly.
Remark 2. The equality of Proposition 5 is not held, in general, as illustrated in Example 1.
Example 1. Let and . Then, we get the following: , , , and . It is clear that , , , and .
Definition 12. For any relation on and , is , .
The following proposition proves that represents a closure operator.
Proposition 3. If is an arbitrary binary relation on . Then, , and the pair is a closure space.
Proof. (i)Clearly, .(ii)According to Lemma 3, Definition 10, and by contradiction, assume that , then there exists at least such that . Thus, such that . But , thus, , and this is a contradiction to the assumption that . Hence, Thus, .We call the pair in Proposition 3, a -closure space.
Definition 13. Suppose that is a -closure space. Then, .(i) is -closed if (ii) is the collection of all -closed in (iii)The complement of -closed is called -open
Proposition 4. Let be a -closure space. Then, , and the -closure is a closure operator in a viewpoint of topology.
Proof. It is sufficient to prove is a topological closure, and is so.
It is clear that . Thus, we only prove that Let . Then, which implies and . Thus, and . Then, which implies .
Theorem 1. Every -closure space, , is topological space.
Proof. Directly by Proposition 4.
Remark 3. According to Theorem 1, we notice that the closure operator , for each , represents a generalization of Galton [17] approach (Definition 7), Allam et al. [18], and El-Bably et al. [19, 20].
Proposition 5. For a reflexive relation on and , .
Proof. From Lemma 4, we have . Thus, if . Then, , and this implies . Hence, by Definition 3, .
Remark 4. The equality of Proposition 5 is not held, in general, as shown in Example 2.
Example 2. Using Example 1, we take . Then, , but .
Now, we also define the interior operation from the -closure operation, which represents a topological interior of the topological space .
Definition 14. If is a -closure space and , then for each , the operator is .
In the following lemmas, we prove that the -interior operator (Abbriv.) is an interior operator on . Also, we prove that and are dual.
Lemma 4. Let be a -closure space and . Then, , where denotes the complement of .
Proof.
Lemma 5. Let be a -closure space and . Then, satisfies the following properties:(i)(ii)(iii)(iv).
Lemma 6. Let be a -closure space, and . Then, is -open if and only if .
4. Generalized Approximations in Terms of Closure Operators
In the current section, we aim to present some topological properties of operators and for each and apply its relationship with the rough set theory. This study is a generalization for Pawlak’s approximations [24]. In addition, a comparison between them is discussed with different sorts of examples.
Definition 15. Let be a -closure space, and . Then, -lower and -upper approximations of are defined, respectively, by
Definition 16. Let be a -closure space, and . The -boundary, -positive, and -negative regions of , for each , are defined, respectively, by
Definition 17. Let be a -closure space, and . Then, is called -exact set if . Otherwise, it is called -rough.
Definition 18. Let be a -closure space, and . Thus, the -accuracy of the approximations of is defined as follows:
, where and refer to the cardinality of .
Remark 5. According to Definitions 15–18, we have .
(i) , for every ,
(ii) If , then is -exact. Otherwise, it is -rough.
Proposition 6. Let be a -closure space, and . Then, we have
Proof. Directly, using the definitions and properties of and , the proof is clear.
Remark 6. (1)If represents an equivalence relation, then the -adhesion set of each represents an equivalence class of ; that is, . Thus, our approximations (-approximations that given in Definitions 15 and 16) are conceded with Pawlak’s approximations.(2)Moreover, according to Proposition 4.6, the -approximations satisfy all properties of the classical rough set model that was introduced by Pawlak [24], using any relation without any restrictions. Therefore, we say that the current methods represent an interesting generalization to the rough set theory.(3)According to Definition 15, we have two different generalized rough approximation operators. Example 3 illustrates that these operators are independent.
Example 3. Let and . Then, we get the following: , , , and . Furthermore, , , , and . Consider , , and . Thus, we compute the approximations for some subsets of as shown in Table 1.
From Table 1, it is clear that the subset is -exact and -exact. However, is -rough although it is -exact, and also the subset is -rough although it is -exact.
Definition 19. Let be a -closure space, and . The generalized lower and upper approximations, the boundary, positive and negative regions, and the accuracy of the approximations of are defined bywhere and refer to the cardinality of .
Definition 20. Let be a -closure space, and . Then, is a generalized exact (shortly, -exact) set if . Otherwise, it is called -rough.
Remark 7. (1), (2) is -exact if . Otherwise, it is a -rough set.
Proposition 7. If is a -closure space, and , then the generalized approximation operators and ) satisfy all Pawlak’s properties (L1-L10) and (U1-U10).
Proof. By using Proposition 6, the proof is obvious.
The main goal of the following results is to introduce the relationships between the -approximations and -approximations. Moreover, they show the best of these approximations.
Theorem 2. Let be a -closure space, and , then(i)(ii).
Proof. It is sufficient to prove (i) and (ii) similarly. Let , then by Definition 19, , and thus, . Hence, which means that, by Definition 19, .
Corollary 1. Let be a -closure space, and . So,(i)(ii).
Corollary 2. Let be a -closure space, and . Then, is -exact if it is -exact.
The opposite of Corollary 2 is not correct in general.
Example 4. (Continuation of Example 3). The approximations of all subsets of are calculated. Thus, Table 2 introduces comparisons between the approximations, boundary, and accurate measure of -approximations and -approximations.
Remark 8. From Table 2, it is noted that(1) and are -exact, but it is neither -exact (-rough) nor -exact (-rough).(2) and are -exact, but it is neither -exact (-rough) nor -rough (-exact).(3) and are -rough. Also, it is -rough and -rough.
Example 5 (Continuation for Example 3). Table 3 represents a comparison between our method and Yao approach [26].
Remark 9. From Table 3, we notice that(1)Our method in Definition 16 represents the best method for computing exactness and the roughness of sets, because the boundary regions are reduced or cancelled, and then, we obtain more accurate measures for approximating sets. On the other hand, in Yao approach, there are rough sets that are not defined, and then, we cannot be able, by Yao [26], to define it or remove the vagueness of it, but it is exact in our approaches.(2)Moreover, in Yao approach, there are sets whose lower (resp., upper) approximation does not belong to the set which represents a contradiction to the original rough set theory given by Pawlak (e.g.,, , , ).
5. Decision Making of COVID–19 as Medical Application
In this part, we provide a realistic example of how our methodologies might be used to make decisions for an information system concerning coronavirus infections (COVID-19). In fact, we have identified the risk factors for COVID-19 infection in people. In this model, the only decisive criteria for infection transmission are gathering, interaction with wounded individuals, and employment in hospitals. We conclude that remaining at home and avoiding contact with people protect against coronavirus infection. The authors of [42] state that human-to-human transmissions have been described with incubation times ranging from 2 to 10 days, allowing the virus to spread through droplets, contaminated hands, or surfaces. The persistence of coronaviruses on diverse inanimate surfaces is seen in Figure 1. We utilized real-life data from an experiment involving six patients in our application, as well as a generic binary relation established by a multi-information system. As a result, Pawlak and some of the other approaches are inapplicable here, and we can conclude that our method broadens the application range of rough sets.
Example 6. Here, we use the notion of nano topology, as defined in Definition 10, to discover the essential elements of “COVID-19” infection via topological reduction of characteristics in a multi-information system. It should be noted that Pawlak’s technique cannot be applied in this case since the used connection is a reflexive relation (not an equivalence relation). Table 4 gives information about six persons of patients and the set of attributes where handled surfaces , protection tools and person status , and decision set of the infected persons with COVID-19 is .
Now, suppose the reflexive relation is . This relation can be applied on Table 4 to get . Thus, the after set of each element in of this relation is , , , , and . Accordingly, the -adhesion neighbourhoods are , , , , and .
Now, from Table 4, there are two types of patients according to infections of COVID-19.
Case 1. (Infection). Using the set , we get and . Hence, by Definition 8, the nano topology of is with a base . The reduction proceeds as follows:
Step 1. is removed. Then, -adhesion neighbourhoods are , , , and . Accordingly, and . Therefore, by Definition 10, . Also, .
Step 2. is removed. So, -adhesion neighbourhoods are , , , and . Accordingly, and . Therefore, by Definition 10, . Also, .
Step 3. is removed. Then, -adhesion neighbourhoods are , , and . Accordingly, and . Therefore, by Definition 10, . In this case, . So, the CORE is the attribute This means that “Person status” is the effective factor for COVID-19 infection.
Case 2. (Noninfection). Using the set , we get and . Hence, by Definition 8, with a base . Here, the reduction proceeds as follows:
Step 4. is removed. Hence, like procedures of Case 1, with a base .
Step 5. is removed. Hence, like procedures of Case 1, with a base .
Step 6. is removed. Then, like procedures of Case 1, with a base . Therefore, the CORE is . This means that “Person status” is the effective factor for COVID-19 infection.
Observation: According to the CORE, “Person status” (whether you remain at home or not) is the most important factor in COVID-19 infection. Proper medical care for those who remain at home may reduce the danger.
We provide an algorithm for decision-making based on our ideas towards the conclusion of the study.
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6. Conclusion and Discussion
The current paper can be divided into three main parts besides the introduction and basic concept sections. The first section examines two kinds of closure operators based on right and left adhesion neighbourhoods formed by a broad binary relation and their features. These operators are extensions of Galton [17], Allam et al. [18], and El-Bably and Fleifel [19, 20], as shown by Theorem 1 with a counterexample. The second part is devoted to the application of the closure operators, proposed in the current paper, in the notion of rough sets. In fact, we have presented three models to approximate the rough sets, which are generalizations of previously presented methods (such as [2, 4, 8, 11, 13, 20, 22, 23, 26–39]). We studied the properties of these approximations, and we were able to demonstrate all of Pawlak’s properties, which were not fulfilled in some other generalizations such as Yao [26] without adding any conditions to the relation. Several comparisons between our methods and previous methods have been presented. Theorem 2 and its results demonstrate that our methods are more accurate and general than other methods of approximations. As a result, we can state that our technique will be beneficial in decision-making for real-world challenges, contributing to the extraction of knowledge from concealed data. Furthermore, the closure operators pave the door for further topological contributions to the rough set theory and applications. Part three of the article offered a medical decision-making application for identifying the effect elements affecting the transmission of the coronavirus (Covid-19) infection. We built a nano topology in this application using a general relation (rather than Pawlak’s approximations, which need an equivalence relation). It is worth mentioning that we have offered algorithms for our decision-making technique, which may be used for any real-world problem.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.