# Topological Models of Rough Sets and Decision Making of COVID-19

Authors: Mostafa A. El-Gayar; Abd El Fattah El Atik
Journal: Complexity
 (2022)

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## 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.


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## Body

## 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]).
LetR be a binary relation from a nonempty set X to a nonempty set Y. R can be from X to itself and called a relation on X. Consequently, if R is a binary relation from X to Y, we say that a∈X is related to b∈Y if a,b∈R, sometimes written aRb.Definition 2 (see [41]).
A binary relationR on X is called as(i)
Reflexive ifaRa,∀a∈X(ii)
Symmetric ifaRb⟶bRa,∀a,b∈X(iii)
Transitive ifaRb∧bRc⟶aRc,∀a,b,c∈X(iv)
Equivalence if conditions (i), (ii), and (iii) are satisfied.Definition 3 (see [41]).
LetR be a relation on X. After set and foreset of x∈X are xR=y∈X:xRy and Rx=y∈X:yRx, respectively.Definition 4 (see [1]).
A collectionτ of subsets of U is a topology on U if it contains U,∅, finite intersection (resp. arbitrary union) of its elements is closed. U,τ is a topological space, each set in τ is open, and its complement w. r. to U is closed. intτX (resp. clτX) denotes the interior (resp. closure) of X in U,τ.Definition 5 (see [27]).
The operatorCl:PU⟶PU, where PU is the power set of U, is called the closure of X, where ClX⊆U, if satisfies(i)
Cl∅=∅(ii)
X⊆ClX, ∀X⊆U(iii)
ClX∪Y=clX∪clY, ∀X,Y⊆U. The pair U,Cl is called a closure space.Definition 6 (see [17]).
LetU,cl be a closure space and let X⊆U. Then,(i)
IntX=ClXcc, where Xc is the complement of X w. r. to U(ii)
X is a neighbourhood of an element x∈X if x∈IntX(iii)
IfB=ClB, then B is closed(iv)
ifA=IntA, then A is open.
For eachU,Cl, Int:PU⟶PU is called the interior of X [13] and satisfies.(i)
IntU=U(ii)
IntX⊆X, ∀X⊆U(iii)
IntX∩Y=IntX∩IntY, ∀X,Y⊆U.Remark 1.
IfClClX=ClX, then the space is called a closure space.Definition 7 (see [27]).
LetR be a relation on X. Then, the closure operator on U by R is clRX=X∪y∈U:∃x∈X:yRx.Lemma 1 (see [17]).
LetR be a transitive relation on a nonempty finite set U. Then, the operator clR is a topological closure.Definition 8 (see [24]).
LetU be a universal set and R be an equivalence relation on U. U/R=xR:x∈U are equivalence classes of R. U,R is an approximation space. The lower and upper approximations of X are R¯X=x∈X:xR⊆X and R¯X=x∈U:xR∩X≠∅, respectively, for any X⊆U. X is rough set if R¯X≠R¯X.Proposition 1 (see [24]).
IfXc is a complement set of X in U, then ∅ represents an empty set. The properties of Pawlak’s rough sets are as follows:L1R¯X⊆XU1X⊆R¯XL2R¯∅=∅U2R¯∅=∅L3R¯U=UU3R¯U=UL4R¯X∩Y=R¯X∩R¯YU4R¯X∪Y=R¯X∪R¯YL5IfX⊆Y,thenR¯X⊆R¯YU5IfX⊆Y,thenR¯X⊆R¯YL6R¯X∪R¯Y⊆R¯X∪YU6R¯X∩R¯Y⊇R¯X∩YL7R¯Xc=R¯XcU7R¯Xc=R¯XcL8R¯R¯X=R¯XU8R¯R¯X=R¯XL9R¯R¯Xc=R¯XcU9R¯R¯Xc=R¯XcL10∀K∈U/R⇒R¯K=KU10∀K∈U/R⇒R¯K=KDefinition 9 (see [26]).
ForA⊆U, the lower and upper approximations of A w. r. to neighbourhood nx of x are apr¯A=x∈U:nx⊆A and apr¯A=x∈U:nx∩A≠∅, respectively. The properties (L3-L9) and (U1, U2, U4-U9) are satisfied, in general, while some Pawlak’s properties are held in nx.Definition 10 (see [40]).
InU,R, τR=U,∅,R¯X,R¯X,BX is a nano topology on U w. r. to X with a base ℬ=U,R¯X,BX.


## 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]).
LetR be any relation on U. The right adhesion and left adhesion set of each y∈U are defined, respectively, as follows:(i)
r-adhesion:Arx=y∈U:xR=yR(ii)
l-adhesion: Alx=y∈U:Rx=Ry
It is simple to demonstrate the following lemma, so we omit the proof.Lemma 2.
LetR be any relation on U. Then, ∀j∈r,l.(i)
Ajx≠∅(ii)
x∈Ajx(iii)
AjU=Ajx:x∈U is a partition on U.Lemma 3.
For allj∈r,l: y∈Ajx if Ajy=Ajx, for any R on U.Proof.
It is sufficient to prove this lemma forj=r and the others similarly. Firstly, if ∈Arx⇒yR=xR. Now, it is sufficient to prove Ary⊆Arx. Let z∈Ary, then zR=yR. Hence, zR=xR, and this implies z∈Arx. Thus, Ary⊆Arx. Also, Ary⊇Arx is proved similarly.Proposition 2.
For a reflexive relationR on U and x∈U, we get(i)
Arx⊆xR(ii)
Alx⊆Rx.Proof.
(i) Lety∈Arx, then yR=xR. But R is a reflexive relation on U, thus x∈xR,∀x∈U. Hence, y∈xR, and this implies Arx⊆xR,∀x∈U. (ii) is proved similarly.Remark 2.
The equality of Proposition5 is not held, in general, as illustrated in Example 1.Example 1.
LetU=a,b,c,d and R=a,a,a,b,b,b,c,c,c,a,d,d. Then, we get the following: aR=a,b, bR=b, cR=a,c, and dR=d. It is clear that Ara=a, Arb=b, Arc=c, and Ard=d.Definition 12.
For any relationR on U and X⊆U, Clj:PU⟶PU is CljX=x∈U:Ajx∩X≠∅, ∀j∈r,l.
The following proposition proves thatClj represents a closure operator.Proposition 3.
IfR is an arbitrary binary relation on U. Then, ∀j∈r,l, and the pair U,Clj is a closure space.Proof.
(i)
Clearly,Clj∅=∅.(ii)
According to Lemma3, Definition 10, and by contradiction, assume that X⊄CljX, then there exists at least x∈X such that x∉CljX. Thus, x∈X such that Ajx∩X=∅. But x∈Ajx,∀x∈U, thus, Ajx∩X≠∅, and this is a contradiction to the assumption that x∉CljX. Hence, X⊆CljX.(1)CljX∪Y=x∈U:Ajx∩X∪Y≠Φ=x∈U:Ajx∩X≠Φorx∈U:Ajx∩Y≠Φ=x∈U:Ajx∩X≠Φ∪x∈U:Ajx∩Y≠Φ. Thus, CljX∪Y=CljX∪CljY.
We call the pairU,Clj in Proposition 3, a j-closure space.Definition 13.
Suppose thatU,Clj is a j-closure space. Then, ∀j∈r,l.(i)
X⊆U is j-closed if X=CljX(ii)
Γj=X⊆U:X=CljX is the collection of all j-closed in U(iii)
The complement ofj-closed is called j-openProposition 4.
LetU,Clj be a j-closure space. Then, ∀j∈r,l, and the j-closure Clj is a closure operator in a viewpoint of topology.Proof.
It is sufficient to proveClr is a topological closure, and Cll is so.
It is clear thatClrX⊆ClrClrX. Thus, we only prove that ClrClrX⊆ClrX. Let x∈ClrClrX. Then, Arx∩ClrX≠∅ which implies ∃y∈Arx and y∈ClrX. Thus, Ary=Arx and Ary∩X≠∅. Then, Arx∩X≠∅ which implies x∈ClrX.Theorem 1.
Everyj-closure spaceU,Clj, ∀j∈r,l, is topological space.Proof.
Directly by Proposition4.Remark 3.
According to Theorem1, we notice that the closure operator Clj, for each j∈r,l, 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 relationR on U and X⊆U, CljX⊆clRX,∀j∈r,l.Proof.
From Lemma4, we have Arx⊆xR. Thus, if x∈CljX. Then, Arx∩X≠∅, and this implies xR∩X≠∅. Hence, by Definition 3, x∈clRX.Remark 4.
The equality of Proposition5 is not held, in general, as shown in Example 2.Example 2.
Using Example1, we take X=a,b,d. Then, ClrX=a,b,d=X, but clRX=U.
Now, we also define the interior operation from thej-closure operation, which represents a topological interior of the topological space U,Clj.Definition 14.
IfU,Clj is a j-closure space and ∀j∈r,l, then for each X⊆U, the operator Intj:PU⟶PU is IntjX=x∈U:Ajx⊆X.In the following lemmas, we prove that thej-interior operator (Abbriv.Intj) is an interior operator on U. Also, we prove that Clj and Intj are dual.Lemma 4.
LetU,Clj be a j-closure space and ∀j∈r,l. Then, intjX=CljXcc, where Xc denotes the complement of X.Proof.
CljXcc=x∈U:Ajx∩Xc≠∅c=x∈U:Ajx∩Xc≠∅c.=x∈U:Ajx∩Xc=∅=x∈U:Ajx∩Xc=∅Lemma 5.
LetU,Clj be a j-closure space and ∀j∈r,l. Then, IntjX satisfies the following properties:(i)
IntjU=U(ii)
IntjX⊆X,foreachX⊆U(iii)
IntjX∩Y=IntjX∩IntjY(iv)
IntjIntjX=IntjX.Lemma 6.
LetU,Clj be a j-closure space, ∀j∈r,l and X⊆U. Then, X is j-open if and only if IntjX=X.


## 4. Generalized Approximations in Terms of Closure Operators

In the current section, we aim to present some topological properties of operatorsClj and Intj for each j∈r,l 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.
LetU,Clj be a j-closure space, ∀j∈r,l and X⊆U. Then, j-lower and j-upper approximations of X are defined, respectively, by(2)R¯jX=x∈U:Ajx⊆X=IntjX,andR¯jX=x∈U:Ajx∩X≠∅=CljX.Definition 16.
LetU,Clj be a j-closure space, ∀j∈r,l and X⊆U. The j-boundary, j-positive, and j-negative regions of X, for each j∈r,l, are defined, respectively, by(3)BjX=R¯jX−R¯jX,POSjX=R¯jX,andNEGjX=U−R¯jX.Definition 17.
LetU,Clj be a j-closure space, ∀j∈r,l and X⊆U. Then, X is called j-exact set if R¯jX=R¯jX=X. Otherwise, it is called j-rough.Definition 18.
LetU,Clj be a j-closure space, ∀j∈r,l and X⊆U. Thus, the j-accuracy of the approximations of X is defined as follows:
αjX=R¯jA/R¯jA, where R¯jX≠0 and X refer to the cardinality of X.Remark 5.
According to Definitions15–18, we have ∀j∈r,l.
(i)0≤αjX≤1, for every X⊆U, ∀j∈r,l
(ii) IfαjX=1andℬjX=∅, then X is j-exact. Otherwise, it is j-rough.Proposition 6.
LetU,Clj be a j-closure space, ∀j∈r,l and X,Y⊆U. Then, we haveL1R¯jX⊆XU1X⊆R¯jXL2R¯j∅=∅U2R¯j∅=∅L3R¯jU=UU3R¯jU=UL4R¯jX∩Y=R¯jX∩R¯jYU4R¯jX∪Y=R¯jX∪R¯jYL5IfX⊆Y,thenR¯jX⊆R¯jYU5IfX⊆Y,thenR¯X⊆R¯YL6R¯jX∪R¯jY⊆R¯jX∪YU6R¯jX∩R¯jY⊇R¯jX∩YL7R¯jXc=(R¯jX)cU7R¯jXc=(R¯jX)cL8R¯jR¯jX=R¯jXU8R¯jR¯jX=R¯jXL9R¯jR¯jXc=(R¯jX)cU9R¯jR¯jXc=(R¯jX)cL10∀K∈AjU⇒R¯jK=KU10∀K∈AjU⇒R¯K=KProof.
Directly, using the definitions and properties ofClj and Intj, the proof is clear.Remark 6.
(1)
IfR represents an equivalence relation, then the j-adhesion set Ajx of each x∈U represents an equivalence class of x; that is, Ajx=xR. Thus, our approximations (j-approximations that given in Definitions 15 and 16) are conceded with Pawlak’s approximations.(2)
Moreover, according to Proposition 4.6, thej-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 Definition15, we have two different generalized rough approximation operators. Example 3 illustrates that these operators are independent.Example 3.
LetU=a,b,c,d and R=a,a,a,b,b,b,c,a,d,b. Then, we get the following: Ara=a, Arb=b,d, Arc=c, and Ard=b,d. Furthermore, Ala=a, Alb=b, Alc=c,d, and Ald=c,d. Consider X=a, Y=a,b, and Z=a,c. Thus, we compute the approximations for some subsets of U as shown in Table 1.
From Table1, it is clear that the subset X is r-exact and l-exact. However, Y is r-rough although it is l-exact, and also the subset Z is l-rough although it is r-exact.Table 1
Comparison betweenr-approximations and l-approximations.
Ar-approximationsl-approximationsR¯rAR¯rABrXαrAR¯lAR¯lABlAαjAaaa∅1aa∅1a,baa,b,db,d1/3a,ba,b∅1a,ca,ca,c∅1aa,c,dc,d1/3Definition 19.
LetU,Clj be a j-closure space, ∀j∈r,l and X⊆U. The generalized lower and upper approximations, the boundary, positive and negative regions, and the accuracy of the approximations of X are defined by(4)G¯X=R¯rX∪R¯lX,G¯X=R¯rX∩R¯lX,BX=G¯X−G¯X,POSX=G¯X,NEGX=U−G¯XandμX=G¯XG¯Xwhere G¯X≠0 and X refer to the cardinality of X.Definition 20.
LetU,Clj be a j-closure space, ∀j∈r,l and X⊆U. Then, X is a generalized exact (shortly, g-exact) set if G¯X=G¯X=X. Otherwise, it is called g-rough.Remark 7.
(1)
0≤μX≤1, ∀X⊆U(2)
X is g-exact if μX=1. Otherwise, it is a g-rough set.Proposition 7.
IfU,Clj is a j-closure space, ∀j∈r,l and X,Y⊆U, then the generalized approximation operators G¯X and G¯X) satisfy all Pawlak’s properties (L1-L10) and (U1-U10).Proof.
By using Proposition6, the proof is obvious.
The main goal of the following results is to introduce the relationships between theg-approximations and j-approximations. Moreover, they show the best of these approximations.Theorem 2.
LetU,Clj be a j-closure space, ∀j∈r,l and X⊆U, then(i)
R¯rX⊆G¯X⊆X⊆G¯X⊆R¯rX(ii)
R¯lX⊆G¯X⊆X⊆G¯X⊆R¯lX.Proof.
It is sufficient to prove (i) and (ii) similarly. Letx∈R¯rX, then by Definition 19, x∈G¯X, and thus, x∈X. Hence, x∈G¯X which means that, by Definition 19, x∈R¯rX.Corollary 1.
LetU,Clj be a j-closure space, ∀j∈r,l and Y⊆U. So,(i)
BY⊆BjY(ii)
αjY≤μY.Corollary 2.
LetU,Clj be a j-closure space, ∀j∈r,l and X⊆U. Then, X is g-exact if it is j-exact.
The opposite of Corollary2 is not correct in general.Example 4.
(Continuation of Example3). The approximations of all subsets of U are calculated. Thus, Table 2 introduces comparisons between the approximations, boundary, and accurate measure of j-approximations and g-approximations.Table 2
Comparison betweenj-approximations and g-approximations.
Aj-approximationsg-approximationsr-approximationsl-approximationsR¯rAR¯rABrXαrAR¯lAR¯lABlAαlAG¯XG¯XℬXμXaaa∅1aa∅1aa∅1b∅b,db,d0bb∅1bb∅1ccc∅1∅c,dc,d0cc∅1d∅b,db,d0∅c,dc,d0∅dd0a,baa,b,db,d1/3a,ba,b∅1a,ba,b∅1a,ca,ca,c∅1aa,c,dc,d1/3a,ca,c∅1a,daa,b,db,d1/3aa,c,dc,d1/3aa,dd1/2b,ccb,c,db,d1/3bb,c,dc,d1/3b,cb,c,dd2/3b,db,db,d∅1bb,c,dc,d1/3b,db,d∅1c,dcb,c,db,d1/3c,dc,d∅1c,dc,d∅1a,b,ca,cUb,d1/2a,bUc,d1/2a,b,cUd3/4a,b,da,b,da,b,d∅1a,bUc,d1/2a,b,da,b,d∅1a,c,da,cUb,d1/2a,c,da,c,d∅1a,c,da,c,d∅1b,c,db,c,db,c,d∅1b,c,db,c,d∅1b,c,db,c,d∅1UUU∅1UU∅1UU∅1Remark 8.
From Table2, it is noted that(1)
a,b and c,d are g-exact, but it is neither r-exact (r-rough) nor l-exact (notl-rough).(2)
b,d and a,b,d are g-exact, but it is neither r-exact (notl-rough) nor l-rough (notl-exact).(3)
d and b,c are g-rough. Also, it is r-rough and l-rough.Example 5 (Continuation for Example3).
Table3 represents a comparison between our method and Yao approach [26].Table 3
The boundary and accuracy of approximations for Yao Method 2.9 [26] and current method in Definition 4.2.
AYao MethodMethod in Definition16apr¯Aapr¯ABAπAG¯AG¯ABAμAaca,c,da,d1/3aa∅1bb,da,b,db2/3bb∅1c∅∅∅Not definedcc∅1d∅∅∅Not defined∅dd0a,bUU∅1a,ba,b∅1a,cca,ca1/2a,ca,c∅1a,dca,ca1/2aa,dd1/2b,cb,da,b,da2/3b,cb,c,dd2/3b,db,da,b,da2/3b,db,d∅1c,d∅∅∅Not definedc,dc,d∅1a,b,cUU∅1a,b,cUd3/4a,b,dUU∅1a,b,da,b,d∅1a,c,dca,ca1/2a,c,da,c,d∅1b,c,db,da,b,da2/3b,c,db,c,d∅1UUU∅1UU∅1Remark 9.
From Table3, we notice that(1)
Our method in Definition16 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.,,a,, b,b,c,a,b,c, a,b,d).


## 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.Figure 1
Perseverance of coronaviruses on different kinds of stationary sides [42].Example 6.
Here, we use the notion of nano topology, as defined in Definition10, 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 U=p1,p2,p3,…,p6 and the set of attributes AT=a1,a2,a3 where a1= handled surfaces =Metal,Wood,Paper, a2= protection tools =Muzzle,Glove, and a3= person status =Stayathome,Goouthome, and decision set of the infected persons with COVID-19 is D=Yes,No.
Now, suppose the reflexive relation ispiRpj⇔Vpi⊆Vpj. This relation can be applied on Table 4 to get =p1,p1,p1,p3,p2,p2,p3,p1,p3,p3,p4,p4,p5,p5,p6,p1,p6,p3,p6,p6. Thus, the after set of each element in U of this relation is p1R=p3R=p1,p3, p2R=p2, p4R=p4, p5R=p5, and p6R=p1,p3,p6. Accordingly, the r-adhesion neighbourhoods are Arp1=Arp3=p1,p3, Arp2=p2, Arp4=p4, Arp5=p5, and Arp6=p6.
Now, from Table4, there are two types of patients according to infections of COVID-19.Table 4
Multi-information table of causing factors for COVID-19 infection.
a1a2a3Dp1M,WM,GGYesp2WGSNop3M,WM,GGYesp3PMSNop3PMGYesp6MGGNoCase 1.
(Infection). Using the set X=p1,p3,p5, we get R¯rX=R¯rX=X and BrX=∅. Hence, by Definition 8, the nano topology of X is τNr=U,∅,p1,p3,p5 with a base βNr=U,∅,p1,p3,p5. The reduction proceeds as follows:Step 1.
a1 is removed. Then, r-adhesion neighbourhoods are Ar−a1p1=Ar−a1p3=p1,p3, Ar−a1p2=p2, Ar−a1p4=p4, Ar−a1p5=p5, and Ar−a1p6=p6. Accordingly, R¯r−a1X=R¯r−a1X=X and Br−a1X=∅. Therefore, by Definition 10, τNr−a1=U,∅,p1,p3,p5=τNr. Also, βNr−a1=U,∅,p1,p3,p5=βNr.Step 2.
a2 is removed. So, r-adhesion neighbourhoods are Ar−a2p1=Ar−a2p3=p1,p3, Ar−a2p2=p2, Ar−a2p4=p4, Ar−a2p5=p5, and Ar−a2p6=p6. Accordingly, R¯r−a2X=R¯r−a2X=X and Br−a2X=∅. Therefore, by Definition 10, τNr−a2=U,∅,p1,p3,p5=τNr. Also, βNr−a2=U,∅,p1,p3,p5=βNr.Step 3.
a3 is removed. Then, r-adhesion neighbourhoods are Ar−a3p1=Ar−a3p3=p1,p3, Ar−a3p2=p2, Ar−a3p4=Ar−a3p5=p4,p5, and Ar−a3p6=p6. Accordingly, R¯r−a3X=p1,p3,R¯r−a3X=p1,p3,p4,p5 and Br−a3X=p4,p5. Therefore, by Definition 10, τNr−a3=U,∅,p1,p3,p4,p5,p1,p3,p5≠τNr. In this case, βNr−a3=U,p1,p3,p4,p5≠βNr. So, the CORE is the attribute a3. This means that “Person status” is the effective factor for COVID-19 infection.Case 2.
(Noninfection). Using the set Y=p2,p6, we get R¯rY=R¯rY=Y and BrY=∅. Hence, by Definition 8, τNr=U,∅,p2,p6 with a base βNr=U,∅,p2,p6. Here, the reduction proceeds as follows:Step 4.
a1 is removed. Hence, like procedures of Case 1, τNr−a1=U,∅,p2,p6=τNr with a base βNr−a1=U,∅,p2,p6=βNr.Step 5.
a2 is removed. Hence, like procedures of Case 1, τNr−a2=U,∅,p2,p6=τNr with a base βNr−a2=U,∅,p2,p6=βNr.Step 6.
a3 is removed. Then, like procedures of Case 1, τNr−a3=U,∅,p2,p6,p4,p5,p2,p4,p5,p6≠τNr with a base βNr−a3=U,p2,p6,p4,p5≠βNr. Therefore, the CORE is a3. 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.Algorithm 1: A decision making via j-rough approximations.
Step 1: Write the finite setU and the set of attributes represent the data as an information table, with rows labelled by attributes (C), columns labelled by objects, and table entries representing attribute values.Step 2: Compute thej-upper approximation, j-lower approximation, and j-boundary for each X⊆U. As j=r, say R¯rX, R¯rY, and BrY, respectively, as defined in Definition 11.Step 3: EstablishτNr on U with a base βNr by Definition 10.Step 4: Removeai from (C). Accordingly, find the r-upper approximation, r-lower approximation, and r-boundary X⊆U on C−(ai).Step 5: Generate the nano topologyτNr−ai on U and its base βNr−ai by Definition 10.Step 6: Repeat Steps4 and 5, ∀ai in C.Step 7:ai which satisfies βNr−ai≠βNr forms the COREU.


## 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.



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*Source: 2989236-2022-06-24.xml*