# New Topological Approaches to Rough Sets via Subset Neighborhoods

Authors: Esra Dalan Yildirim
Journal: Journal of Mathematics
 (2022)

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## 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, Sj-near open and θβSj-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 θβSj-open sets are more accurate than those based on Sj-open and Sj-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.


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## Body

## 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 j-neighborhood space to generalize the classical rough set theory via different topologies induced by an arbitrary relation. Then, new types of neighborhoods such as j-adhesion neighborhoods [7], E-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 j-near approximations as mathematical instruments modifying and generalizing the j-approximations and studied their applications. Hosny [14] and Nawar et al. [15] presented new kinds of approximations by using δβj-open sets, ∧βj-sets, and θβj-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 usingSj-near open and θβSj-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 j-subset approximations and highest j-subset accuracy measures are obtained for j∈i,i. Moreover, these approximations, boundary regions, and accuracy measures are compared with the previous ones in [6, 11]. In Section 4, the concepts of Sj-near open sets based on the topologies generated by subset neighborhoods are presented and the connections among them are studied. Then, θβSj-open sets are defined as the generalization of Sj-near open sets, and the fundamental properties of them are investigated. In Section 5, j-subset approximations are generalized to Sj-near approximations based on Sj-near open sets. The main properties of these approximations are studied. Then, θβSj-approximations are defined and these approximations are compared with Sj-near approximations, j-subset approximations, and j-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 aboutj-neighborhood and subset neighborhood cited in [6, 11].Definition 1 (see [6]).
Letϱ be an arbitrary binary relation on a nonempty finite set Q. The j-neighborhood of q∈Q (briefly, Njq) for each j∈Y=r,l,r,l,u,i,u,i is defined as follows:(1)
r-neighborhood: Nrq=p∈Q:qϱp(2)
l-neighborhood: Nlq=p∈Q:pϱq(3)
r-neighborhood: Nrq=∩q∈NrpNrp(4)
l-neighborhood: Nlq=∩q∈NlpNlp(5)
u-neighborhood: Nuq=Nrq∪Nlq(6)
i-neighborhood: Niq=Nrq∩Nlq(7)
u-neighborhood: Nuq=Nrq∪Nlq(8)
i-neighborhood: Niq=Nrq∩NlqDefinition 2 (see [6]).
Letφj:Q⟶℘Q be a mapping which assigns for each q∈Q its j-neighborhood in ℘Q. The triple Q,ϱ,φj is called j-neighborhood space (j-NS).Theorem 1 (see [6]).
LetQ,ϱ,φj be a j-NS. Then, ℑj = {P⊆Q:Njp⊆P for each p∈P} is a topology on Q for every j∈Y. The elements of ℑj are called j-open sets and the complement of j-open sets are j-closed sets.Definition 3 (see [6]).
LetQ,ϱ,φj be a j-NS, P⊆Q and j∈Y. Then, the j-lower and j-upper approximations, j-boundary regions, and j-accuracy measures of P are defined as follows, respectively: ϱ¯jP=∪H∈ℑj:H⊆P and ϱ¯jP=∩Kc∈ℑj:P⊆K, BjP=ϱ¯jP−ϱ¯jP and σjP=ϱ¯jP/ϱ¯jP where ϱ¯jP≠0.
Ifϱ¯jP=ϱ¯jP then P is called a j-exact set. Otherwise, it is called a j-rough set.Definition 4 (see [11]).
Letϱ be an arbitrary binary relation on a nonempty finite set Q. The subset neighborhood of q∈Q (briefly, Sjq) for each j∈Y is defined as follows:(1)
Srq=p∈Q:Nrq⊆Nrp(2)
Slq=p∈Q:Nlq⊆Nlp(3)
Suq=Srq∪Slq(4)
Siq=Srq∩Slq(5)
Srq=p∈Q:Nrq⊆Nrp(6)
Slq=p∈Q:Nlq⊆Nlp(7)
Suq=Srq∪Slq(8)
Siq=Srq∩SlqProposition 1 (see [11]).
LetQ,ϱ,φj be a j-NS and q∈Q. Then,(1)
q∈Sjq for each j∈Y(2)
Ifp∈Sjq, then Sjp⊆Sjq for every j∈r,l,i,r,l,iLemma 1 (see [11]).
LetQ,ϱ,φj be a j-NS and ϱ be a similarity relation on Q. Then, Sjq⊆Njq for each q∈Q and for j∈r,l,i,u.Definition 5 (see [11]).
LetQ,ϱ,φj be a j-NS, P⊆Q and j∈Y. Then, the lower TSjP and upper TSjP approximations, Sj-boundary regions, and Sj-accuracy measures of P are defined as follows, respectively: TSjP=q∈Q:Sjq⊆P, TSjP=q∈Q:Sjq∩P≠∅BSjP=TSjP−TSjP, and ASjP=TSjP/TSjP where TSjP≠0.


## 3. Topologies Based onSj-Neighborhoods

This section aims to introduce eight different topologies based onSj-neighborhoods for j∈Y. The relationships among them are given and comparisons between these topologies and ℑj are examined. Then, j-subset approximations, j-subset boundary regions, and j-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 j∈i,i. Also, the proposed approximations in Definition 7 are compared with the previous ones in Definitions 3 and 5.Theorem 2.
LetQ,ϱ,φj be a j-NS. Then, ℑSj = {P⊆Q:Sjp⊆P for each p∈P} is a topology on Q generated by Sj-neighborhood for every j∈Y.Proof.
(1)
∅,Q∈ℑSj(2)
Suppose thatPλ∈ℑSj for each λ∈Δ and p∈∪λ∈ΔPλ. Then, there exists λ0∈Δ such that p∈Pλ0. From here, Sjp⊆Pλ0. This implies Sjp⊆∪λ∈ΔPλ. Hence ∪λ∈ΔPλ∈ℑSj(3)
LetP1,P2∈ℑSj and p∈P1∩P2. Since p∈P1 and p∈P2, we get Sjp⊆P1 and Sjp⊆P2. Thus, Sjp⊆P1∩P2. So, we have P1∩P2∈ℑSjProposition 2.
LetQ,ϱ,φj be a j-NS. Then,(1)
ℑSu⊆ℑSr⊆ℑSi(2)
ℑSu⊆ℑSl⊆ℑSi(3)
ℑSu⊆ℑSr⊆ℑSi(4)
ℑSu⊆ℑSl⊆ℑSiProof.
(1) and (2) LetP∈ℑSu. Then, Sup⊆P for each p∈P. Thus, Srp∪Slp⊆P which implies Srp⊆P and Slp⊆P. Hence, P∈ℑSr and P∈ℑSl. Moreover Srp∩Slp⊆P, that is Sip⊆P, and so P∈ℑSi.
(3) and (4) The proofs are done similarly to the proofs of (1) and (2).Remark 1.
ℑSr and ℑSl, ℑSr and ℑSl, ℑSj and ℑSj need not be comparable.
The following example shows thatℑSr and ℑSl are not comparable and are not dual topologies. Also, it illustrates that ℑSr and ℑSl, ℑSj and ℑSj are incomparable.Example 1.
LetQ=ς1,ς2,ς3,ς4 and ϱ=ς1,ς1,ς1,ς2,ς1,ς3,ς2,ς4,ς2,ς3. Then, we have(i)
ℑSr=∅,Q,ς1,ς2,ς1,ς2 and ℑSl=∅,Q,ς3,ς3,ς4,ς1,ς2,ς3(ii)
ℑSr=∅,Q,ς4,ς1,ς2,ς1,ς2,ς4 and ℑSl=∅,Q,ς1,ς2,ς1,ς2,(iii)
ℑSl=∅,Q,ς3,ς3,ς4,ς1,ς2,ς3 and ℑSl=∅,Q,ς1,ς2,ς1,ς2. Similarly, suitable examples can be found for all other j∈YProposition 3.
LetQ,ϱ,φj be a j-NS and ϱ be a similarity relation on Q. Then, ℑj⊆ℑSj for j∈r,l,i,u.Proof.
It is clear from Lemma1.
The following example shows thatℑj and ℑSj are not comparable for j∈r,l,u,i.Example 2.
LetQ=ς1,ς2,ς3,ς4 and ϱ=ς1,ς1,ς2,ς2,ς3,ς3,ς4,ς4,ς1,ς2,ς2,ς1,ς2,ς4,ς4,ς2,ς3,ς4,ς4,ς3. Then, we have ℑSr = {∅,Q,ς1,ς3,ς1,ς2, ς1,ς3, ς3,ς4, ς1,ς2,ς3, ς1,ς3,ς4} and ℑr = {∅,Q,ς2, ς4, ς1,ς2, ς2,ς4, ς3,ς4, ς1,ς2,ς4, ς2,ς3,ς4}.
The following example shows thatℑj and ℑSj are not comparable if the relation is not similarity relation.Example 3.
Consider Example1 and let j=r. Then, we obtain ℑSr=∅,Q,ς1,ς2,ς1,ς2 and ℑr=∅,Q,ς3,ς4,ς3,ς4,ς2,ς3,ς4.Definition 6.
LetQ,ϱ,φj be a j-NS. H⊆Q is called Sj-open set if H∈ℑSj, the complement of Sj-open set is said to be Sj-closed set. The family ΩSj of all Sj-closed sets is defined by ΩSj=K⊆Q:Kc∈ℑSj.Definition 7.
LetQ,ϱ,φj be a j-NS, P⊆Q and j∈Y. Then, the j-subset lower and j-subset upper approximations are defined as follows, respectively: ϱ¯SjP=∪H∈ℑSj:H⊆P=intSjPϱ¯SjP=∩K∈ΩSj:P⊆K=clSjP where intSjP and clSjP are Sj-interior and Sj-closure of P according to ℑSj, respectively.
Ifϱ¯SjP=ϱ¯SjP, then P is called an Sj-exact set. Otherwise, it is an Sj-rough set.Remark 2.
ϱ¯Sj and ϱ¯Sj provide all the properties of Pawlak approximations.Proposition 4.
LetQ,ϱ,φj be a j-NS and P⊆Q. Then,(1)
ϱ¯SuP⊆ϱ¯SrP⊆ϱ¯SiP(2)
ϱ¯SuP⊆ϱ¯SlP⊆ϱ¯SiP(3)
ϱ¯SuP⊆ϱ¯SrP⊆ϱ¯SiP(4)
ϱ¯SuP⊆ϱ¯SlP⊆ϱ¯SiPProof.
The proofs are clear from Proposition2.Proposition 5.
LetQ,ϱ,φj be a j-NS and P⊆Q. Then,(1)
ϱ¯SiP⊆ϱ¯SrP⊆ϱ¯SuP(2)
ϱ¯SiP⊆ϱ¯SlP⊆ϱ¯SuP(3)
ϱ¯SiP⊆ϱ¯SrP⊆ϱ¯SuP(4)
ϱ¯SiP⊆ϱ¯SlP⊆ϱ¯SuPProof.
The proofs are clear from Proposition2.Theorem 3.
LetQ,ϱ,φj be a j-NS, ϱ be a similarity relation on Q and P⊆Q. For j∈r,l,i,u, we have ϱ¯jP⊆ϱ¯SjP⊆P⊆ϱ¯SjP⊆ϱ¯jP.Proof.
It is clear from Proposition3.
The following example shows that Theorem3 does not hold for j∈r,l,u,i.Example 4.
Consider Example2. For P=ς1,ς3 and R=ς2,ς4, we have ϱ¯SrP=P, ϱ¯SrR=∅, ϱ¯rP=∅, ϱ¯rR=R. Thus, they are not comparable. Besides, we obtain ϱ¯SrP=Q, ϱ¯SrR=R, ϱ¯rP=P and ϱ¯rR=Q. That is, they are also not comparable.Proposition 6.
LetQ,ϱ,φj be a j-NS and P⊆Q. For each j∈r,l,i,r,l,i, we have(1)
ϱ¯SjP=TSjP(2)
ϱ¯SjP=TSjPProof.
(1)
Letq∈ϱ¯SjP. Then, there exists λ0∈Δ such that q∈Gλ0⊆P for Gλ0∈ℑSj. Thus, Sjq⊆Gλ0⊆P. This implies that q∈TSjP. Conversely, let q∈TSjP. Then, Sjq⊆P. Since Sjq∈ℑSj from Proposition 1, we get q∈ϱ¯SjP.(2)
It can be proved similarly.Definition 8.
LetQ,ϱ,φj be a j-NS, P⊆Q and j∈Y. Then, the j-subset boundary regions and j-subset accuracy measures of P are defined as follows, respectively: bSjP=ϱ¯SjP−ϱ¯SjP, σSjP=ϱ¯SjP/ϱ¯SjP, where ϱ¯SjP≠0.
The following two corollaries follow from Propositions4 and 5.Corollary 1.
LetQ,ϱ,φj be a j-NS and P⊆Q. Then,(1)
bSiP⊆bSrP⊆bSuP(2)
bSiP⊆bSlP⊆bSuP(3)
bSiP⊆bSrP⊆bSuP(4)
bSiP⊆bSlP⊆bSuPCorollary 2.
LetQ,ϱ,φj be a j-NS and P⊆Q. Then,(1)
σSuP≤σSrP≤σSiP(2)
σSuP≤σSlP≤σSiP(3)
σSuP≤σSrP≤σSiP(4)
σSuP≤σSlP≤σSiPIn Tables1 and 2, the j-subset boundary regions and j-subset accuracy measures of all subsets of Q are calculated for each j∈Y according to Example 1. Thus, it is seen that the j-subset accuracy measures for j∈i,i are more precise than those for j∈r,l,u,r,l,u.In Table3, the algorithm is given to calculate the accuracy measures induced from the topologies generated by subset neighborhoods.The following two corollaries follow from Theorem3.Table 1
Thej-subset boundary regions and the j-subset accuracy measures for j∈r,l,u,i.
PForj=rForj=lForj=uForj=ibSrPσSrPbSlPσSlPbSuPσSuPbSiPσSiPς1ς3,ς41/3ς1,ς20Q0∅1ς2ς3,ς41/3ς1,ς20Q0∅1ς3ς3,ς40ς1,ς2,ς41/4Q0ς41/2ς4ς3,ς40ς40Q0ς40ς1,ς2ς3,ς41/2ς1,ς20Q0∅1ς1,ς3ς3,ς41/3ς1,ς2,ς41/4Q0ς42/3ς1,ς4ς3,ς41/3ς1,ς2,ς40Q0ς41/2ς2,ς3ς3,ς41/3ς1,ς2,ς41/4Q0ς42/3ς2,ς4ς3,ς41/3ς1,ς2,ς40Q0ς41/2ς3,ς4ς3,ς40ς1,ς21/2Q0∅1ς1,ς2,ς3ς3,ς41/2ς43/4Q0ς43/4ς1,ς2,ς4ς3,ς41/2ς1,ς2,ς40Q0ς42/3ς1,ς3,ς4ς3,ς41/3ς1,ς21/2Q0∅1ς2,ς3,ς4ς3,ς41/3ς1,ς21/2Q0∅1Q∅1∅1∅1∅1Table 2
Thej-subset boundary regions and the j-subset accuracy measures for j∈r,l,u,i.
PForj=rForj=lForj=uForj=ibSrPσSrPbSlPσSlPbSuPσSuPbSiPσSiPς1ς1,ς2,ς30ς3,ς41/3Q0ς31/2ς2ς1,ς2,ς30ς3,ς41/3Q0ς31/2ς3ς30ς3,ς40ς3,ς40ς30ς4ς31/2ς3,ς40ς3,ς40ς31/2ς1,ς2ς32/3ς3,ς41/2ς3,ς41/2ς32/3ς1,ς3ς1,ς2,ς30ς3,ς41/3Q0ς31/2ς1,ς4ς1,ς2,ς31/4ς3,ς41/3Q0ς32/3ς2,ς3ς1,ς2,ς30ς3,ς41/3Q0ς31/2ς2,ς4ς1,ς2,ς31/4ς3,ς41/3Q0ς32/3ς3,ς4ς31/2ς3,ς40ς3,ς40ς31/2ς1,ς2,ς3ς32/3ς3,ς41/2ς3,ς41/2ς32/3ς1,ς2,ς4ς33/4ς3,ς41/2ς3,ς41/2ς33/4ς1,ς3,ς4ς1,ς2,ς31/4ς3,ς41/3Q0ς32/3ς2,ς3,ς4ς1,ς2,ς31/4ς3,ς41/3Q0ς32/3Q∅1∅1∅1∅1Table 3
Algorithm forj-subset accuracy measures.
Input: a binary relation ϱ on a nonempty finite set Q.Output: an accuracy measure σSj of each nonempty subset P of Q.(1)  Input the binary relationϱ on a nonempty finite set Q;(2)forfor eachj∈Υdo(3)   ComputeSj-neighborhoods induced from ϱ;(4)   Generate a topologyℑSj using Theorem 2(5)ifP∈ℑSjthen(6)    Putϱ¯SjP=P;(7)     ComputeclSjP;(8)     Putϱ¯SjP=clSjP;(9)     CalculateσSjP=ϱ¯SjP/ϱ¯SjP.(10)else(11)    ComputeintSjP;(12)ifintSjP=∅then(13)σSjP=0.(14)else(15)     Putϱ¯SjP=intSjP;(16)     ComputeclSjP;(17)     Putϱ¯SjP=clSjP;(18)     CalculateσSjP=ϱ¯SjP/ϱ¯SjP.(19)end(20)end(21)endCorollary 3.
LetQ,ϱ,φj be a j-NS, ϱ be a similarity relation on Q and P⊆Q. For j∈r,l,i,u, we have(1)
bSjP⊆BjP(2)
σjP≤σSjPCorollary 4.
LetQ,ϱ,φj be a j-NS, ϱ be a similarity relation on Q and P⊆Q. If P is j-exact, then it is Sj-exact for j∈r,l,i,u.The converse implication of Corollary4 need not be true.Example 5.
LetQ=ς1,ς2,ς3,ς4 and ϱ=ς1,ς1,ς2,ς2,ς3,ς3,ς4,ς4,ς1,ς3,ς3,ς1,ς1,ς2,ς2,ς1,ς3,ς4,ς4,ς3. Then, ς1,ς2 is Sr-exact but not r-exact.
The following corollary follows from Proposition6.Corollary 5.
LetQ,ϱ,φj be a j-NS and P⊆Q. For each j∈r,l,i,r,l,i, we have(1)
bSjP=BSjP(2)
σSjP=ASjP


## 4.Sj-Near Open Sets and θβSj-Open Sets

Section4 aims to present the concepts of Sj-near open set using the topologies ℑSj. The relationships among these concepts are investigated. From this, it is seen that the concepts of PSj-open and SSj-open sets cannot be compared. Then, the concept of θβSj-open set is introduced as a generalization of the Sj-near open sets, and their main properties are studied.Definition 9.
LetQ,ϱ,φj be a j-NS, P⊆Q and j∈Y. Then, P is called(1)
αSj-open if P⊆intSjclSjintSjP(2)
PSj-open if P⊆intSjclSjP(3)
SSj-open if P⊆clSjintSjP(4)
βSj-open if P⊆clSjintSjclSjP
These sets are calledSj-near open sets, the complement of the Sj-near open sets are Sj-near closed sets. The families of Sj-near open sets (Sj-near closed sets) of Q denoted by ΨSjOQΨSjCQ for each Ψ∈α,P,S,β.Proposition 7.
LetQ,ϱ,φj be a j-NS and j∈Y. Then, the following implications hold.(1)
EverySj-open sets are αSj-open(2)
EveryαSj-open sets are SSj-open(3)
EveryαSj-open sets are PSj-open(4)
EveryPSj-open sets are βSj-open(5)
EverySSj-open sets are βSj-openProof.
Straightforward by Definitions6 and 9.
The converse implications of Proposition7 need not be true.Example 6.
LetQ=ς1,ς2,ς3,ς4 and ϱ=ς1,ς1,ς2,ς1,ς2,ς4,ς3,ς1,ς3,ς3,ς4,ς1,ς4,ς3. Then,(1)
P=ς1,ς2 is αSl-open but not Sl-open(2)
P=ς1,ς2 is SSr-open but not αSr-open(3)
P=ς1,ς2,ς3 is PSr-open but not αSr-open(4)
P=ς1,ς2 is βSr-open but not PSr-open(5)
P=ς1,ς2,ς3 is βSr-open but not SSr-openRemark 3.
The concepts ofSSj-open set and PSj-open set are not comparable for j∈Y. In Example 6, P1=ς1,ς2 is SSr-open but not PSr-open and P2=ς1,ς2,ς3 is PSr-open but not SSr-open.Definition 10.
LetQ,ϱ,φj be a j-NS, P⊆Q and j∈Y. The θSj-closure of P is defined by clSjθP = {q∈Q:P∩clSjH≠∅ for each H∈ℑSj and q∈H}.Definition 11.
LetQ,ϱ,φj be a j-NS, P⊆Q and j∈Y. P is called θβSj-open if P⊆clSjintSjclSjθP. The complement of θβSj-open sets are θβSj-closed sets and the family of θβSj-open (θβSj-closed) sets of Q are denoted by θβSjOQθβSjCQ.Proposition 8.
LetQ,ϱ,φj be a j-NS and j∈Y. Then, every βSj-open sets are θβSj-open.Proof.
Straightforward by definitions ofSj-closure and θSj-closure.
The converse implication of Proposition8 need not be true.Example 7.
Consider Example5. Then, P=ς1,ς2,ς4 is θβSr-open but not βSr-open. In that case, it is not SSr-open, PSr-open, αSr-open and Sr-open.Remark 4.
LetΨ∈P,S,β,θβ and j∈Y.(1)
Proposition2 is not satisfied in general for ΨSjOQ(2)
Although the union of twoΨSj-open sets is ΨSj-open, their intersection need not be ΨSj-openExample 8.
LetΨ=β.(1)
Consider the Example1. Then,(i)
βSrOQ=∅,Q,ς1,ς2,ς1,ς2,ς1,ς3,ς1,ς4,ς2,ς3,ς2,ς4,ς1,ς2,ς3,ς1,ς2,ς4,ς1,ς3,ς4,ς2,ς3,ς4.(ii)
βSlOQ=∅,Q,ς3,ς1,ς3,ς2,ς3,ς3,ς4,ς1,ς2,ς3,ς1,ς3,ς4,ς2,ς3,ς4.(iii)
βSiOQ=∅,Q,ς1,ς2,ς3,ς1,ς2,ς1,ς3,ς2,ς3,ς3,ς4,ς1,ς2,ς3,ς1,ς3,ς4,ς2,ς3,ς4.(iv)
βSuOQ=℘Q.Thus,βSuOQ⊈βSrOQ, βSuOQ⊈βSlOQ, βSuOQ⊈βSiOQ and βSrOQ⊈βSiOQ. Also, ς1,ς3 and ς2,ς3 are βSr-open sets but the intersection of these two sets ς3 is not βSr-open.(2)
LetQ=ς1,ς2,ς3 and ϱ=ς1,ς1,ς2,ς1,ς2,ς2,ς2,ς3,ς3,ς1. Then,(i)
βSrOQ=℘Q−ς1(ii)
βSlOQ=℘Q−ς2(iii)
βSuOQ=℘Q(iv)
βSiOQ=∅,Q,ς3,ς1,ς3,ς2,ς3Thus,βSuOQ⊈βSrOQ⊈βSiOQ and βSuOQ⊈βSlOQ⊈βSiOQ
Other examples can be given forΨ∈P,S,θβ that confirms the correctness of Remark 4.


## 5.Sj-Near Approximations and θβSj-Approximations

Section5 aims to introduce the concepts of Sj-near approximations as a generalization of j-subset approximations by using Sj-near open sets. The fundamental properties of them are given. Also, the concepts of θβSj-approximations are presented and connections between θβSj-approximations, Sj-near approximations, j-subset approximations, and j-approximations are studied.Definition 12.
LetQ,ϱ,φj be a j-NS, P⊆Q, j∈Y and Ψ∈α,P,S,β. The Sj-near lower and Sj-near upper approximations, Sj-near boundary regions, and Sj-near accuracy measures of P are defined as follows, respectively: ϱ¯SjΨP=∪H∈ΨSjOQ:H⊆P = Sj-near interior of P, ϱ¯SjΨP=∩K∈ΨSjCQ:P⊆K = Sj-near closure of P,bSjΨP=ϱ¯SjΨP−ϱ¯SjΨP, σSjΨP=ϱ¯SjΨP/ϱ¯SjΨP, where ϱ¯SjΨP≠0.
Ifϱ¯SjΨP=ϱ¯SjΨP, then P is called an Sj-near exact set. Otherwise, it is an Sj-near rough set.Proposition 9.
LetQ,ϱ,φj be a j-NS, P⊆Q, j∈Y and Ψ∈α,P,S,β. Then,(1)
ϱ¯SjΨP⊆P⊆ϱ¯SjΨP(2)
IfP⊆R then ϱ¯SjΨP⊆ϱ¯SjΨR and ϱ¯SjΨP⊆ϱ¯SjΨR(3)
ϱ¯SjΨP∩R⊆ϱ¯SjΨP∩ϱ¯SjΨR and ϱ¯SjΨP∩R⊆ϱ¯SjΨP∩ϱ¯SjΨR(4)
ϱ¯SjΨP∪R⊇ϱ¯SjΨP∪ϱ¯SjΨR and ϱ¯SjΨP∪R⊇ϱ¯SjΨP∪ϱ¯SjΨR(5)
ϱ¯SjΨP=ϱ¯SjΨPcc and ϱ¯SjΨP=ϱ¯SjΨPcc(6)
ϱ¯SjΨϱ¯SjΨP=ϱ¯SjΨP and ϱ¯SjΨϱ¯SjΨP=ϱ¯SjΨPProof.
The proofs are clear from the definition ofSj-near approximations.
The converse implications and inclusions of Proposition9 are not necessarily true.Example 9.
Consider Example8 (1).(1)
LetP=ς1,ς2 and R=ς2,ς3. Then, we get ϱ¯SlβP=ς1,ς2 and ϱ¯SlβR=Q. Thus, ϱ¯SlβP∩ϱ¯SlβR⊈ϱ¯SlβP∩R=ς2. Also, we have ϱ¯SlβP=∅ and ϱ¯SlβR=ς2,ς3. Hence, ϱ¯SlβP∪R=ς1,ς2,ς3⊈ϱ¯SlβP∪ϱ¯SlβR. Moreover, ϱ¯SlβP⊂ϱ¯SlβR and ϱ¯SlβP⊂ϱ¯SlβR but P⊈R.(2)
LetP=ς1,ς3 and R=ς2,ς3. Then, we get ϱ¯SrβP=ς1,ς3 and ϱ¯SrβR=ς2,ς3. Thus, ϱ¯SrβP∩ϱ¯SrβR⊈ϱ¯SrβP∩R=∅. Also, we obtain ϱ¯SrβP=ς1,ς3 and ϱ¯SrβR=ς2,ς3. From here, ϱ¯SrβP∪R=Q⊈ϱ¯SrβP∪ϱ¯SrβR.Theorem 4.
LetQ,ϱ,φj be a j-NS, P⊆Q, j∈Y and Ψ∈α,P,S,β. Then, ϱ¯SjP⊆ϱ¯SjΨP⊆P⊆ϱ¯SjΨP⊆ϱ¯SjP.Proof.
By Propositions7 and 9 (1), the proof is obvious.
The following corollary follows from Theorem4.Corollary 6.
LetQ,ϱ,φj be a j-NS, P⊆Q, j∈Y and Ψ∈α,P,S,β. Then,(1)
bSjΨP⊆bSjP(2)
σSjP≤σSjΨP
The following two corollaries follow from Theorems3 and 4.Corollary 7.
LetQ,ϱ,φj be a j-NS, ϱ be a similarity relation on Q, P⊆Q, j∈r,l,u,i and Ψ∈α,P,S,β. Then, ϱ¯jP⊆ϱ¯SjP⊆ϱ¯SjΨP⊆P⊆ϱ¯SjΨP⊆ϱ¯SjP⊆ϱ¯jP.Corollary 8.
LetQ,ϱ,φj be a j-NS, ϱ be a similarity relation on Q, P⊆Q, j∈r,l,u,i and Ψ∈α,P,S,β. Then,(1)
bSjΨP⊆bSjP⊆BjP(2)
σjP≤σSjP≤σSjΨPDefinition 13.
LetQ,ϱ,φj be a j-NS, P⊆Q and j∈Y. The θβSj-lower and θβSj-upper approximations, θβSj-boundary regions, and θβSj-accuracy measures of P are defined as follows, respectively: ϱ¯SjθβP=∪H∈θβSjOQ:H⊆P = θβSj-interior of P,ϱ¯SjθβP=∩K∈θβSjCQ:P⊆K = θβSj-closure of P,bSjθβP=ϱ¯SjθβP−ϱ¯SjθβP, σSjθβP=ϱ¯SjθβP/ϱ¯SjθβP, where ϱ¯SjθβP≠0.
Ifϱ¯SjθβP=ϱ¯SjθβP, then P is called a θβSj-exact set. Otherwise, it is a θβSj-rough set.Proposition 10.
LetQ,ϱ,φj be a j-NS, P⊆Q and j∈Y. Then,(1)
ϱ¯SjθβP⊆P⊆ϱ¯SjθβP(2)
IfP⊆R then ϱ¯SjθβP⊆ϱ¯SjθβR and ϱ¯SjθβP⊆ϱ¯SjθβR(3)
ϱ¯SjθβP∩R⊆ϱ¯SjθβP∩ϱ¯SjθβR and ϱ¯SjθβP∩R⊆ϱ¯SjθβP∩ϱ¯SjθβR(4)
ϱ¯SjθβP∪R⊇ϱ¯SjθβP∪ϱ¯SjθβR and ϱ¯SjθβP∪R⊇ϱ¯SjθβP∪ϱ¯SjθβR(5)
ϱ¯SjθβP=ϱ¯SjθβPcc and ϱ¯SjθβP=ϱ¯SjθβPcc(6)
ϱ¯Sjθβϱ¯SjθβP=ϱ¯SjθβP and ϱ¯Sjθβϱ¯SjθβP=ϱ¯SjθβPProof.
The proofs are clear from the definition ofθβSj-approximations.Proposition 11.
LetQ,ϱ,φj be a j-NS, P⊆Q and j∈Y. Then,(1)
ϱ¯SjαP⊆ϱ¯SjPP⊆ϱ¯SjβP⊆ϱ¯SjθβP(2)
ϱ¯SjαP⊆ϱ¯SjSP⊆ϱ¯SjβP⊆ϱ¯SjθβP(3)
ϱ¯SjθβP⊆ϱ¯SjβP⊆ϱ¯SjPP⊆ϱ¯SjαP(4)
ϱ¯SjθβP⊆ϱ¯SjβP⊆ϱ¯SjSP⊆ϱ¯SjαPProof.
(1) By Propositions7 and 8, ϱ¯SjαP=∪H∈αSjOQ:H⊆P⊆∪H∈PSjOQ:H⊆P = ϱ¯SjPP⊆∪H∈βSjOQ:H⊆P = ϱ¯SjβP⊆∪H∈θβSjOQ:H⊆P=ϱ¯SjθβP.
The other proofs can be proved by the same propositions in a similar way.
The following corollary follows from Proposition11.Corollary 9.
LetQ,ϱ,φj be a j-NS, P⊆Q and j∈Y. Then,(1)
bSjθβP⊆bSjβP⊆bSjPP⊆bSjαP(2)
bSjθβP⊆bSjβP⊆bSjSP⊆bSjαP(3)
σSjαP≤σSjPP≤σSjβP≤σSjθβP(4)
σSjαP≤σSjSP≤σSjβP≤σSjθβPTheorem 5.
LetQ,ϱ,φj be a j-NS, P⊆Q, j∈Y and Ψ∈α,P,S,β. Then, ϱ¯SjP⊆ϱ¯SjΨP⊆ϱ¯SjθβP⊆P⊆ϱ¯SjθβP⊆ϱ¯SjΨP⊆ϱ¯SjP.Proof.
By Theorem4, Propositions 10 (1) and 11, the proof is obvious.
The following corollary follows from Theorem5.Corollary 10.
LetQ,ϱ,φj be a j-NS, P⊆Q, j∈Y and Ψ∈α,P,S,β. Then,(1)
bSjθβP⊆bSjΨP⊆bSjP(2)
σSjP≤σSjΨP≤σSjθβPIn Table4, the boundary regions and accuracy measures of all subsets of Q are calculated for Ψ∈α,P,S,β,θβ and j=r according to Example 6. Thus, it is seen that the θβSr-accuracy measure is higher than r-subset accuracy measure and Sr-near accuracy measures for α,P,S,β.The following corollaries follow from Theorem5 and Corollary 7.Table 4
Comparison between the boundary regions and accuracy measures based onSj-open, Sj-near open, and θβSj-open sets for j=r.
PℑSrαSrOQPSrOQSSrOQβSrOQθβSrOQbSrPσSrPbSrαPσSrαPbSrPPσSrPPbSrSPσSrSPbSrβPσSrβPbSrθβPσSrθβPς1ς10ς10ς10ς10ς10∅1ς2ς11/2ς11/2ς11/2∅1∅1∅1ς3ς1,ς3,ς40ς1,ς3,ς40∅1ς3,ς40∅1∅1ς4ς1,ς3,ς40ς1,ς3,ς40∅1ς3,ς40∅1∅1ς1,ς2ς11/2ς11/2ς11/2∅1∅1∅1ς1,ς3ς1,ς3,ς40ς1,ς3,ς40ς11/2ς1,ς3,ς40∅1∅1ς1,ς4ς1,ς3,ς40ς1,ς3,ς40ς11/2ς1,ς3,ς40∅1∅1ς2,ς3ς1,ς3,ς41/4ς1,ς3,ς41/4ς12/3ς1,ς3,ς41/4∅1∅1ς2,ς4ς1,ς3,ς41/4ς1,ς3,ς41/4ς12/3ς1,ς3,ς41/4∅1∅1ς3,ς4ς12/3ς12/3ς12/3∅1∅1∅1ς1,ς2,ς3ς1,ς3,ς41/4ς1,ς3,ς41/4∅1ς3,ς41/2∅1∅1ς1,ς2,ς4ς1,ς3,ς41/4ς1,ς3,ς41/4∅1ς3,ς41/2∅1∅1ς1,ς3,ς4ς12/3ς12/3ς12/3∅1∅1∅1ς2,ς3,ς4ς13/4ς13/4ς13/4ς13/4ς13/4∅1Q∅1∅1∅1∅1∅1∅1Corollary 11.
LetQ,ϱ,φj be a j-NS,, ϱ be a similarity relation on Q, P⊆Q, j∈r,l,u,i and Ψ∈α,P,S,β. Then, ϱ¯jP⊆ϱ¯SjP⊆ϱ¯SjΨP⊆ϱ¯SjθβP⊆P⊆ϱ¯SjθβP⊆ϱ¯SjΨP⊆ϱ¯SjP⊆ϱ¯jP.Corollary 12.
LetQ,ϱ,φj be a j-NS, ϱ be a similarity relation on Q, P⊆Q, j∈r,l,u,i and Ψ∈α,P,S,β. Then,(1)
bSjθβP⊆bSjΨP⊆bSjP⊆BjP(2)
σjP≤σSjP≤σSjΨP≤σSjθβP


## 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 Table5 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 Q=ς1,ς2,ς3,ς4,ς5,ς6,ς7,ς8,ς9,ς10 is the set of 10 patients, the set of condition attributes Λ=Λ1=DB,Λ2=CP,Λ3=H,Λ4=DC,Λ5=HT,Λ6=LTS is the set of most common symptoms of COVID-19 and D=yes=y,no=n is the set of decision attribute. In Table 6, similarities between symptomatic patients are given using the degree of similarity ωp,q which is the ratio of equal attributes between any of the two patients p,q where ωp,q=∑i=1nΛip=Λiq/t and t is the number of condition attributes. Let us take the similarity relation pϱq⇔ωp,q>k, with k∈0,1 and assume k=0.6.
Consider the two sets of yes and no decisions, thatY=ς1,ς2,ς6,ς7,ς9,ς10 and N=ς3,ς4,ς5,ς8. From Table 6, the approximations and accuracy measures of Y and N 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 θβSj-open sets gives the best approximations and highest accuracy.Table 5
COVID-19 information system.
PatientsDBCPHDCHTLTSDecisionς1++−+++yς2++++++yς3++−+−+nς4++−−−−nς5++−+−−nς6+−+++−yς7−−−++−yς8−−−++−nς9−−−−−+yς10−−+++−yTable 6
Similarities between symptomatic patients.
ς1ς2ς3ς4ς5ς6ς7ς8ς9ς10ς110.840.840.50.670.50.50.50.340.34ς20.8410.670.340.50.670.340.340.170.5ς30.840.6710.670.840.340.340.340.50.17ς40.50.340.6710.840.340.340.340.50.17ς50.670.50.840.8410.50.50.50.340.34ς60.50.670.340.340.510.670.670.170.84ς70.50.340.340.340.50.67110.50.84ς80.50.340.340.340.50.67110.50.84ς90.340.170.50.50.340.170.50.510.34ς100.340.50.170.170.340.840.840.840.341Table 7
Comparisons between Abd El-Monsef et al.’s method in [6] and two proposed methods for j=r.
Abd El-Monsef et al.’s method in [6]The method in Definition7The method in Definition13ϱ¯rϱ¯rσrϱ¯Srϱ¯SrσSrϱ¯Srθβϱ¯SrθβσSrθβYς9Q0.1ς2,ς6,ς9ς1,ς2,ς6,ς7,ς8,ς9,ς100.43YY1N∅Q−ς90ς3,ς4,ς5ς1,ς3,ς4,ς5,ς7,ς8,ς100.43NN1


## 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,j-subset approximations were defined by using these topologies and it was shown that the best approximations were obtained for j∈i,i. In addition, Sj-near open and θβSj-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 θβSj-approximations are more precise than j-subset approximations and Sj-near approximations for any arbitrary relation and than j-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 θβSj-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.



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