# Knowledge Based on Rough Approximations and Ideals

Authors: Rodyna A. Hosny; Tareq M. Al-shami; A. A. Azzam; Ashraf S. Nawar
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Topology is a beneficial structure to study the approximation operators in the rough set theory. In this work, we first introduce six new types of neighborhoods with respect to finite binary relations. We study their main properties and show under what conditions they are equivalent. Then we applied these types of neighborhoods to initiate some topological spaces that are utilized to define new types of rough set models. We compare these models and prove that the best accuracy measures are obtained in the cases ofi and i. Also, we illustrate that our approaches are better than those defined under one arbitrary relation. To improve rough sets’ accuracy, we define some topological spaces using the idea of ideals. With the help of examples, we demonstrate that our methods are better than some methods studied in some published literature. Finally, we give a real-life application showing the merits of the approaches followed in this manuscript.


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## Body

## 1. Introduction and Preliminaries

Information science is an area that is mostly interested in analysis, combination, sorting, storage, retrieval, and protection of information. On occasion, knowledge in information science is constructed in an imperfect or vague manner and has a level of granularity.As one of the treatment methods of these issues of the knowledge system, rough set theory (briefly, RS theory), created by Pawlak [1], has been developed. The main purpose of RS methods is upgrading the approximation issues, which were established to reduce the boundaries and raise the degree of accuracy. RS theory has involved several topological notions; approximation operators have many characterizations of interior and closure operators that are topological operators derived from equivalence relations defined on the universe. Thus, the study of RS by using the concept of topology is helpful to analyze many issues in reality (see [2, 3]).However, equivalence relations in the classical RS theory have some restrictions in generating neighborhood bases. To overcome these restrictions, the authors interested in uncertainty have studied the classical RS theory with respect to different types of relations [4], covering rough sets [5], and constructed different topological structures [6, 7]. Kondo [8] showed that a reflexive relation produces a topology. The authors of [9, 10] investigated the features of rough sets under different kinds of binary relations. Salama [11] explored some topological properties of rough sets and showed that rough sets and topological rough sets constitute a consistency base for data mining. Al-shami [12] improved the accuracy values of subsets using the concept of somewhere dense sets.As yet, concerted studies of the rough set theory and topology are demanded. The idea of neighborhood systems is extracted from the geometrical concept of “near,” and it is inherent in the theory of topological spaces. By considerable researches in the discussion of RS, neighborhood systems have been applied. Yao [13] examined the RS using neighborhood systems for explicating granules. Abd El-Monsef et al. [14] employed the notion of “j-neighborhood space” (j-NS) to generalize the classical RS theory by utilizing various general topologies deduced from binary relations. Mareay [15] defined the concepts of core neighborhoods and applied them to initiate topological approximations. Al-shami et al. [16] initiated different rough set models using Nj-neighborhoods.Also, as a generalization of RS methods, Abu-Donia [17] exploited rκ-neighborhoods to characterize new kinds of approximations of any set with respect to a finite family of binary relations ζκ:κ=1,2,…,n. Then he [18] improved these approximations using rκ-neighborhoods. Recently, Al-shami et al. [19] have defined a new system of neighborhoods called Ej-neighborhoods and applied it to establish new rough and topological approximations. They compared them and showed that the accuracy measure obtained from rough approximations is better than their counterparts obtained from topological approximations. Also, Al-shami [20] has established a new family of neighborhood systems called Cj-neighborhoods. He applied to protect medical staff from new coronavirus (COVID-19). Recently, Al-shami and Ciucci [21] have defined a new class of neighborhood systems namely “subset neighborhood.” They have explored their main properties and investigated their applications to handle some diseases.The interaction between rough set theory and topology began by Skowron [22] and Wiweger [23]. Ideal structure, which was initiated by Kuratowski [24], is defined as a nonempty collection ℐ of subsets of a universe that is closed under finite union and subsets. The concept of an ideal topological space consists of topology and ideal defined on the same universe, and they are independent of each other. Using ideal topological spaces as a new technique to generate new rough approximations was introduced by [25]. After that, some studies were conducted on ideal rough approximations; see, for example, [26, 27]. Recently, Hosny et al. [28] have applied Ej-neighborhoods and ideals to initiate new types of rough set models.By using ideals notion and topologies generated byj-neighborhoods with respect to the finite family of binary relations on a universe, our suggested approach will deal with the gap between topologies concepts and their applications in various fields of real life. We successfully applied our approaches to reduce the boundary region and increase the accuracy value, which is the essential target of RS theory.In the rest of this section, we recall some concepts that help understand this manuscript well.Definition 1.
(see [20]) A binary relation ζ on a nonempty set ℧ is said to be as follows:(i)
Reflexive ifyζy for each y∈℧(ii)
Symmetric ifzζy⇔yζz(iii)
Antisymmetric ifz=y whenever zζy and yζz(iv)
Transitive ifxζz whenever xζy and yζz(v)
Preorder (or quasiorder) if it is reflexive and transitive(vi)
Equivalence (resp. partial order) if it is symmetric (resp. antisymmetric), reflexive and transitive(vii)
Diagonal ifζ=y,y:y∈℧(viii)
Comparable ifzζy or yζz for each y,z∈℧Definition 2.
(see [20]). Let ζ be a binary relation on a nonempty finite set (universe) ℧ and j∈r,l,r,l,i,u,i,u. The j-neighborhoods of z∈℧ (symbolized by jNz are defined as follows:(i)
rNζz=y∈℧:zζy(ii)
lNζz=y∈℧:yζz(iii)
rNζz=∩z∈rNζyrNζy,there existsrNζycontainingz,∅,Otherwise.(iv)
lNζz=∩z∈lNζylNζy,there existslNζycontainingz,∅,Otherwise.(v)
iNζz=rNζz∩lNζz(vi)
uNζz=rNζz∪lNζz(vii)
iNζz=rNζz∩lNζz(viii)
uNζz=rNζz∪lNζzDefinition 3.
A class⊤ of subsets of ℧ is closed under finite intersection, and the arbitrary union is called a topology. We call an ordered pair ℧,⊤ a topological space.Definition 4.
For a setS in ℧,⊤, the interior of S (denoted by int⊤S) is the union of all open subsets that are contained in S; the closure of S (denoted by cl⊤S) is the intersection of all closed subsets containing S.


## 2.j-Neighborhoods Based on a Finite Number of Binary Relations and the Topologies Inferred from Them

In this part, we introduce and explore thej-neighborhood space jNκS that depends on a finite number of arbitrary binary relations. Then we apply them to generate eight diverse topologies and investigate the relationships between them with help of examples. Also, we exploit these topologies to initiate lower and upper (rough) approximations. Comparisons between the accuracy of these types are attained, as well as comparisons that show our methods better in terms of accuracy measure than those given in [14] are presented.Henceforth, for all the following results, we will deal with all the values ofj, j∈r,l,i,u,r,l,i,u, unless otherwise noted.Definition 5.
Letζκ:κ=1,2,…,n be a finite family of binary relations on ℧. The j-neighborhoods of z∈℧ with respect to ζκ (briefly, jNκz) are defined as follows:(1)
rκ-neighborhood: rNκz=y∈℧:zζκyfor eachκ. Equivalently, rNκz=∩κ=1nrNζκz.(2)
lκ-neighborhood: lNκz=y∈℧:yζκzfor eachκ. Equivalently, lNκz=∩κ=1nlNζκz.(3)
iκ-neighborhood: iNκz=rNκz∩lNκz.(4)
uκ-neighborhood: uNκz=rNκz∪lNκz.(5)
rκ-neighborhood: rNκz=∩z∈rNκyrNκy,there existsrNκycontainingz,∅,Otherwise.(6)
lκ-neighborhood: lNκz=∩z∈lNκylNκy,there existslNκycontainingz,∅,Otherwise.(7)
iκ-neighborhood: iNκz=rNκz∩lNκz.(8)
uκ-neighborhood: uNκz=rNκz∪lNκz.Remark 1.
The concepts ofrκ-neighborhood and rκ-neighborhood were introduced and studied in [17, 18], respectively.Definition 6.
Letζκ:κ=1,2,3,…,n be a finite family of binary relations on a universe ℧ and ψj: ℧⟶P℧ be a mapping that assigns for each z in ℧ its jNκz in P℧. Then ℧,ζκ,ψj is called a j-neighborhood space (briefly, ), κ∈1,2,3,…,n.
Now, we investigate the main properties ofjNκS under some types of binary relations.Proposition 1.
Let℧,ζκ,ψj be a jNκS such that ζκ is reflexive for each κ. Then jNκz⊆jNκz for each z∈℧.Proof.
Ifζκ is reflexive for each κ, then rNκz=∩z∈rNκyrNκy⊆rNκz and lNκz=∩z∈lNκylNκy⊆lNκz. Hence, iNκz⊆iNκz and uNκz⊆uNκz.Proposition 2.
Let be ajNκS such that ζκ is preorder for each κ. Then jNκz=jNκz for each z∈℧.Proof.
It follows from the above proposition thatjNκz⊆jNκz for each z∈℧. Conversely, let x∉rNκz. Then there exists y∈℧ such that z∈rNκy and x∉rNκy. Suppose that x∈rNκz. Since ζκ is transitive for each κ, then x∈rNκy; this is a contradiction. Therefore, x∉rNκz, and consequently, rNκz⊆rNκz. Thus, rNκz=rNκz. Similarly, one can prove that lNκz=lNκz. Hence, iNκz=iNκz and uNκz=uNκz.Proposition 3.
Let℧,ζκ,ψj be a jNκS such that ζκ is reflexive and antisymmetric for each κ. Then(i)
Ify∈rNκz, then z∉rNκy and y∉lNκz(ii)
iNκy=iNκy=y for each y∈℧(iii)
jNκy≠jNκz for all y≠z∈℧ in the case of j∈r,l,r,lProof.
(i)
Lety∈rNκz. Then zζκy for each κ. Since ζκ is antisymmetric for each κ, then yζκz does not hold for each κ. Therefore, z∉rNκy and y∉lNκz.(ii)
Sinceζκ is reflexive for each κ, then y∈iNκy. Now, consider z∈iNκy. Then z∈rNκy and z∈lNκy. Therefore, yζκz and zζκy for each κ. Since ζκ is antisymmetric for each κ, we obtain y=z. Thus, iNκy=y. Since is reflexive for each κ, y∈iNκy and iNκy=iNκy, as required.(iii)
Forj=r. Let rNκy=rNκz. By hypothesis, ζκ is reflexive for each κ, we obtain y∈rNκy and z∈rNκz. By assumption, yζκz and zζκy. It follows from the antisymmetric of ζκ for each κ that y=z.
j=r. Let rNκy=rNκz. It follows from the reflexivity of ζκ for each κ that y∈rNκy and z∈rNκz. By assumption, z∈rNκy and y∈rNκz. Consequently, z∈rNκy and y∈rNκz. Thus, yζκz and zζκy. Since ζκ is antisymmetric for each κ, we obtain y=z.Proposition 4.
Let℧,ζκ,ψj be a jNκS. Then the following results hold:(i)
The comparable ofζκ for each κ implies that uNκy=℧ or uNκy=℧∖y for all y∈℧(ii)
The symmetric ofζκ for each κ implies that rNκy=lNκy=iNκy=uNκy and rNκy=lNκy=iNκy=uNκyProof.
Straightforward.Proposition 5.
Let℧,ζκ,ψj be a jNκS such that ζκ is transitive and symmetric for all κ. If the intersection of jNκy and jNκz is nonempty, then jNκy and jNκz are equal for all j.Proof.
We prove the result forj=r. One can prove the other cases in a similar way.
SincerNκy∩rNκz≠∅, there is a∈℧ such that yζκa and zζκa. Since ζκ is symmetric for each κ, aζκz, and since ζκ is transitive for each κ, yζκz. Now, let c∈rNκz. Then zζκc. Since yζκz, c∈rNκy. Hence, rNκz⊆rNκy. Following the similar argument, we find rNκy⊆rNκz.Proposition 6.
Consider℧,ζκ,ψj as a jNκS, where ζκ is reflexive and antisymmetric. Then, for j∈i,i, a class G=jNκy:y∈℧ represents a partition for ℧.Proof.
We suffice by proving case ofj=i.
The reflexivity ofζκ for each κ implies that ℧=∪y∈℧iNκy. Consider y≠z. Suppose that iNκy∩iNκz≠∅. Then there is a∈℧ satisfying a∈iNκy and a∈iNκz. This leads to aζκyζκa and aζκzζκa. Since ζκ is antisymmetric for each κ, a=z and a=y. This is a contradiction with our assumption. Hence, iNκy∩iNκz=∅ for every y≠z∈℧.Proposition 7.
Let℧,ζκ,ψj be a jNκS, where a relation ζκ is a partial order for each κ. The next results hold for the smallest element y∈℧.(i)
rNκy=rNκy=uNκy=uNκy=℧(ii)
lNκy=lNκy=iNκy=iNκy=yProof.
(i)
Lety be the smallest element. Then yζκz for each z∈℧. This implies that rNκy=℧. Consequently, uNκy=℧. To show that rNκy=rNκy, let y∈rNκz. By hypothesis, we find z=y. That is, rNκy represents the only right neighborhood of y. Thus, rNκy=rNκy. As a direct result, we have uNκy=uNκy=℧.(ii)
Similar to (i).Proposition 8.
Let℧,ζκ,ψj be a jNκS, where a relation ζκ is partial order for each . The next results hold for the largest element y∈℧.(i)
lNκy=lNκy=uNκy=uNκy=℧(ii)
rNκy=rNκy=iNκy=iNκy=yProof.
The proofs (i) and (ii) are similar to that of Proposition7.
In the following result, we initiate a topology fromjNκ-neighborhoods.Theorem 1.
Let℧,ζκ,ψj be a j−NκS, κ∈1,2,3,…,n. Then the collection n−⊤j=S⊆℧:∀z∈S,jNκz⊆S is a topology on ℧.Proof.
Obviously,∅ and ℧ belong to n−⊤j. Let Sα∈n−⊤j for each α. Let z∈∪α∈ΛSα. Then there exists β∈Λ such that z∈Sβ. Therefore, jNκz⊆Sβ⊆∪α∈ΛSα. Thus, ∪α∈ΛSα belongs to n−⊤j, as required. Now, let S1,S2∈℧. Then, for each z∈S1∩S2, we have jNκz⊆S1 and jNκz⊆S2. Consequently, jNκz⊆S1∩S2. Hence, we obtain the desired result.
Now, one may study the relations between the topologiesn−⊤j, j∈r,l,i,u,r,l,i,u, which generated from ζκ:κ=1,2,3,…,n.Proposition 9.
Let℧,ζκ,ψj be a j−NκS, κ∈1,2,3,…,n. Then the next properties hold true:(1)
n−⊤u⊆n−⊤r∩n−⊤l(2)
n−⊤r∪n−⊤l⊆n−⊤i(3)
n−⊤u⊆n−⊤r∩n−⊤l(4)
n−⊤r∪n−⊤l⊆n−⊤iProof.
To prove 1, letS∈n−⊤u. Then uNκz⊆S, ∀z∈S. Therefore, rNz∪lNz⊆S, . Thus, rNz⊆S and lNz⊆S, ∀z∈S. Hence, S∈n−⊤r∩n−⊤l, as required. To prove 2, let S∈n−⊤r∩n−⊤l. Then rNκz⊆S, ∀z∈S, or lNκz⊆S, ∀z∈S. Since iNκz⊆rNκz⊆S or iNκz⊆lNκz⊆S, ∀z∈S. Hence, S∈n−⊤i.One can prove the other cases similarly.Example 1.
On℧=δ1,δ2,δ3,δ4,δ5, we have information systems ζ1,ζ2 specified as follows:
ζ1=▲∪δ1,δ2,δ1,δ4,δ2,δ1,δ2,δ4,δ2,δ5,δ3,δ1,δ3,δ2,δ3,δ4,δ3,δ5,δ4,δ2,δ5,δ2,δ5,δ3,δ5,δ4,
ζ2=▲∪δ1,δ2,δ1,δ3,δ1,δ4,δ1,δ5,δ2,δ1,δ2,δ4,δ3,δ4,δ3,δ5,δ4,δ1,δ4,δ2,δ4,δ3,δ4,δ5,δ5,δ3,δ5,δ4 such that ▲ is the identity relation on ℧.
Therefore, we obtain the following:
2−⊤r=∅,℧,δ1,δ2,δ4, 2−⊤l=∅,℧,δ3,δ5, 2−⊤i=∅,℧,δ3,δ5,δ1,δ2,δ4, 2−⊤u=∅,℧, 2−⊤r=∅,℧,δ4,δ2,δ4,δ4,δ5,δ1,δ2,δ4,δ2,δ4,δ5,δ3,δ4,δ5,δ1,δ2,δ4,δ5,δ2,δ3,δ4,δ52−⊤l=∅,℧,δ1,δ2,δ3,δ1,δ2,δ1,δ3,δ1,δ4,δ2,δ3,δ3,δ5,δ1,δ2,δ3,δ1,δ2,δ4,δ1,δ3,δ4,δ1,δ3,δ5,δ2,δ3,δ5,δ1,δ2,δ3,δ4,δ1,δ2,δ3,δ5,δ1,δ3,δ4,δ52−⊤i=P℧. 2−⊤u=∅,℧,δ1,δ2,δ4.Remark 2.
It can be seen from Example1 that the eight topologies [2]–⊤j are different from each other.Example 2.
On℧=δ1,δ2,δ3,δ4, we have information systems ζ1,ζ2 specified as follows:ζ1=δ1,δ1,δ1,δ3,δ2,δ2,δ2,δ4,δ3,δ1,δ3,δ3,δ4,δ3,ζ2=δ1,δ1,δ1,δ2,δ2,δ1,δ2,δ2,δ3,δ3,δ3,δ4,δ4,δ2. Then n−⊤l=P℧, andn−⊤l=∅,℧,δ1,δ2,δ3,δ1,δ2,δ1,δ3,δ1,δ4,δ2,δ3,δ1,δ2,δ3,δ1,δ2,δ4,δ1,δ3,δ4.Remark 3.
n−⊤j, n−⊤j are not necessarily comparable, for any j∈r,l,i,u,r,l,i,u. Suppose j=l, then(1)
Example1 illustrates n−⊤l⊈n−⊤l(2)
Example2 shows that n−⊤l⊈n−⊤lProposition 10.
Let℧,ζκ,ψj be a jNκS, κ∈1,2,3,…,n. Then the topologies n−⊤r and n−⊤l are dual.Proof.
To prove the duality betweenn−⊤r, n−⊤l, we shall prove that S∈n−⊤r if Sc∈n−⊤l, that is,
rNκp⊆S, ∀p∈S⇔lNκq⊆Sc, ∀q∈Sc.
SupposerNκp⊆S, ∀p∈S. If q∈Sc, then lNκq=y∈℧|yζκqfor eachκ. Let lNκq∩S≠∅ for each κ∈1,2,3,…,n. Then there exists x∈S and x∈lNκq, that is, x∈S and xζκq, for each κ∈1,2,3,…,n. So q∈rNκx. x∈S implies that rNκx⊆S. Hence, q∈S. It is a contradiction, so lNκq⊆Sc. By the same manner, we can prove the other side.Remark 4.
In view of Example1, the topologies n−⊤r, n−⊤l are not dual.Definition 7.
Let℧,ζκ,ψj be a jNκS, κ∈1,2,3,…,n. A set S⊆℧ is called a j open set if S∈n−⊤j, and the complement of a j open set is called a j closed set. The family n−Υj of all j closed sets of a jNκS is defined by n−Υj=F⊆U:Fc∈n−⊤j, where Fc is the complement of F.Definition 8.
Let be ajNκS, κ∈1,2,3,…,n. If S⊆℧, then the j lower, j upper approximations, j boundary regions, and j accuracy of S are defined, respectively, as follows:(1)
LjnS=∪O∈n−⊤j:O⊆S=intjnS, where &ImaginaryI;nntjS represents the interior points of S with respect to n−⊤j(2)
UjnS=∩F∈n−⊥j:S⊆F=cljnS, where cljnS represents the closure points of with respect to n−⊤j(3)
nBjS=nUjS−nLjS(4)
, whereUjnS≠0Definition 9.
Let℧,ζκ,ψj be a jNκS, κ∈1,2,3,…,n. A set S⊆℧ is called j definable set if nBjS=∅. Otherwise, is called j rough set.
In the following example, we make a comparison between thej approximations and j accuracy measure of any subset of ℧ for each j∈r,l,i,u,r,l,i,u and . The following example shows that we obtain the best j approximations and j accuracy measure in the cases of j=i and j=i; see Tables 1 and 2.Table 1
Thej approximations and j accuracy for each j∈r,l,i,u.
S⊆℧LrnsLlnsLinsLunsUrnsUlnsUinsUunsθrnsθlnsθinsθunsδ1∅δ1δ1∅δ1δ1,δ2δ1δ1,δ201/210δ2δ2∅δ2∅δ1,δ2δ2δ2δ1,δ21/2010δ3∅∅∅∅δ3,δ4δ3,δ4δ3,δ40000δ4∅∅∅∅δ3,δ4δ3,δ4δ3,δ4δ3,δ40000δ1,δ2δ1,δ2δ1,δ2δ1,δ2δ1,δ2δ1,δ2δ1,δ2δ1,δ2δ1,δ21111δ1,δ3∅δ1δ1∅δ1,δ3,δ4℧δ1,δ3,δ4℧01/41/30δ1,δ4∅δ1δ1∅δ1,δ3,δ4℧δ1,δ3,δ4℧01/41/30δ2,δ3δ2∅δ2∅℧δ2,δ3,δ4δ2,δ3,δ4℧1/401/30δ2,δ4δ2∅δ2∅℧δ2,δ3,δ4δ2,δ3,δ4℧1/401/30δ3,δ4δ3,δ4δ3,δ4δ3,δ4δ3,δ4δ3,δ4δ3,δ4δ3,δ4δ3,δ41111δ1,δ2,δ3δ1,δ2δ1,δ2δ1,δ2℧℧℧℧1/21/21/21/2δ1,δ2,δ4δ1,δ2δ1,δ2δ1,δ2δ1,δ2℧℧℧℧1/21/21/21/2δ1,δ3,δ4δ3,δ4δ1,δ3,δ4δ1,δ3,δ4δ3,δ4δ1,δ3,δ4℧℧2/33/411/2δ2,δ3,δ4δ2,δ3,δ4δ3,δ4δ2,δ3,δ4δ3,δ4℧δ2,δ3,δ4δ2,δ3,δ4℧3/42/311/2℧℧℧℧℧℧℧℧℧1111Table 2
Thej approximations and j accuracy for each j∈r,l,i,u.
S⊆℧LrnsLlnsLinsLunsUrnsUlnsUinsUunsθrnsθlnsθinsθunsδ1δ1δ1δ1δ1δ1δ1δ1δ11111δ2δ2δ2δ2δ2δ2δ2δ2δ21111δ3∅δ3δ3∅δ3,δ4δ3,δ4δ3,δ4δ3,δ401/21/20δ4∅∅∅δ3,δ4δ4δ40000δ1,δ2δ1,δ2δ1,δ2δ1,δ2δ1,δ2δ1,δ2δ1,δ2δ1,δ2δ1,δ21111δ1,δ3δ1δ1,δ3δ1,δ3δ1δ1,δ3,δ4δ1,δ3,δ4δ1,δ3,δ4δ1,δ3,δ41/32/32/31/3δ1,δ4δ1δ1δ1δ1δ1,δ3,δ4δ1,δ4δ1,δ3,δ41/31/21/21/3δ2,δ3δ2δ2,δ3δ2,δ3δ2δ2,δ3,δ4δ2,δ3,δ4δ2,δ3,δ41/32/32/31/3δ2,δ4δ2δ2δ2δ2δ2,δ3,δ4δ2,δ4δ2,δ4δ2,δ3,δ41/31/21/21/3δ3,δ4δ3,δ4δ3,δ4δ3,δ4δ3,δ4δ3,δ4δ3,δ4δ3,δ4δ3,δ41111δ1,δ2,δ3δ1,δ2δ1,δ2,δ3δ1,δ2,δ3δ1,δ2℧℧℧℧1/23/43/41/2δ1,δ2,δ3δ1,δ2δ1,δ2δ1,δ2δ1,δ2℧δ1,δ2,δ3δ1,δ2,δ3℧1/22/32/31/2δ1,δ3,δ4δ1,δ3,δ4δ1,δ3,δ4δ1,δ3,δ4δ1,δ3,δ4δ1,δ3,δ4δ1,δ3,δ4δ1,δ3,δ4δ1,δ3,δ41111δ2,δ3,δ4δ2,δ3,δ4δ2,δ3,δ4δ2,δ3,δ4δ2,δ3,δ4δ2,δ3,δ4δ2,δ3,δ4δ2,δ3,δ4δ2,δ3,δ41111℧℧℧℧℧℧℧℧℧1111Example 3.
On℧=δ1,δ2,δ3,δ4, we have information systems ζ1,ζ2 specified as follows:
ζ1=▲∪δ1,δ2,δ3,δ1,δ3,δ4,δ4,δ3, ζ2=▲∪δ1,δ2,δ2,δ3,δ2,δ4,δ3,δ2,δ3,δ4,δ4,δ3 such that ▲ is an identity relation on ℧, 2−⊤r=∅,℧,δ2,δ1,δ2,δ3,δ4,δ2,δ3,δ4, 2−⊤l=∅,℧,δ1,δ1,δ2,δ3,δ4,δ1,δ3,δ4, 2−⊤i=∅,℧,δ1,δ2,δ1,δ2,δ3,δ4,δ1,δ3,δ4,δ2,δ3,δ4, 2−⊤u=∅,℧,δ1,δ2,δ3,δ4, , 2−⊤l=2−⊤i=∅,℧,δ1,δ2,δ3,δ1,δ2,δ1,δ3,δ2,δ3,δ3,δ4,δ1,δ2,δ3,δ1,δ3,δ4,δ2,δ3,δ4.Remark 5.
In view of Table2 and according to Example 3, one may notice the following:(1)
Forj∈r,l,i,u, θins is the preferable j accuracy, and the best method for building the approximations of sets is specified by utilizing the topology [2]–⊤i(2)
The accuracy measure of any setS, θjns, coincides with the j accuracy measure due to [17] when ζκ, κ∈1,2 is a reflexive relation and j=rRemark 6.
In view of Table2 and according to Example 3, one may notice the following:(1)
LrnS=LunS, LlnS=LinS(2)
UrnS=UunS, UlnS=UinS(3)
θrnS=θunS, θlnS=θinS(4)
θinS is the preferable j accuracy, and the best method for building the approximations of sets is specified by utilizing the topology [2]–⊤i(5)
The accuracy measure of any setS, θjnS, coincides with the j accuracy measure due to [18] when ζκ, κ∈1,2 is a reflexive relation and j=r.Remark 7.
According to Example3, δ1,δ2 and δ3,δ4 are j definable sets, and δ4 is j rough set for all j∈r,l,i,u,r,l,i,u.
One of the obtained merits of using more than one relations is the improvement of the accuracy measures of a subset. To demonstrate this matter, we compare thej accuracy measure obtained using two arbitrary binary relations and j accuracy measure obtained using one arbitrary binary relation in the case of . In fact, this proves that our approach is better than that given in [14]; see the next example.Example 4.
On℧=δ1,δ2,δ3,δ4, we have information systems ζ1,ζ2 specified as follows:
ζ1=δ1,δ1,δ1,δ2,δ1,δ3,δ1,δ4,δ2,δ3,δ2,δ4,δ3,δ2,δ3,δ4,δ4,δ1,δ4,δ2,δ4,δ3,δ4,δ4, ζ2=δ1,δ1,δ1,δ4,δ2,δ2,δ3,δ1,δ3,δ2,δ3,δ4,δ4,δ1,δ4,δ4. Therefore, ⊤ζ1i=∅,℧,δ4,δ1,δ4,δ2,δ4,δ3,δ4,δ1,δ2,δ4,δ1,δ3,δ4,δ2,δ3,δ4, ⊤ζ2i=∅,℧,δ2,δ3,δ1,δ4,δ2,δ3,δ1,δ2,δ4,δ1,δ3,δ4, 2−⊤i=∅,℧,δ2,δ3,δ4,δ1,δ4,δ2,δ3,δ2,δ4,δ3,δ4,δ1,δ2,δ4,δ1,δ3,δ4,δ2,δ3,δ4.
It can be seen from Table3 that the rows with green color show that θ2iS>θζ1iS and θ2iS>θζ2iS.Table 3
Comparison between our approaches and those given in [1] in the case of j=i.
S⊆℧LrnSLlnsLinSLunSUrnSUlnSUinSUunSθrnSδ1∅δ10∅δ1,δ40∅δ10δ2∅δ20δ2δ21δ2δ21δ3∅δ30δ3δ31δ3δ31δ4δ4℧1/4∅δ1,δ40δ4δ1,δ41/2δ1,δ2δ1,δ20δ2δ1,δ2,δ41/3δ2δ1,δ21/2δ1,δ3∅δ1,δ30δ3δ1,δ3,δ41/3δ3δ1,δ31/2δ1,δ4δ1,δ4℧1/2δ1,δ4δ1,δ41δ1,δ4δ1,δ41δ2,δ3∅δ2,δ30δ2,δ3δ2,δ31δ2,δ3δ2,δ31δ2,δ4δ2,δ4℧1/2δ2δ1,δ2,δ41/3δ2,δ4δ1,δ2,δ42/3δ3,δ4δ3,δ4℧1/2δ2δ1,δ3,δ41/3δ3,δ4δ1,δ3,δ42/3δ1,δ2,δ3∅δ1,δ2,δ30℧1/2δ2,δ3δ1,δ2,δ32/3δ1,δ2,δ4δ1,δ2,δ4℧3/4δ1,δ2,δ4δ1,δ2,δ41δ1,δ2,δ4δ1,δ2,δ41δ1,δ3,δ4δ1,δ3,δ4℧3/4δ1,δ3,δ4δ1,δ3,δ41δ1,δ3,δ4δ1,δ3,δ41δ2,δ3,δ4δ2,δ3,δ4℧δ2,δ3℧1/2δ2,δ3,δ4℧3/4℧℧℧1℧℧1℧℧1


## 3. ℐj-Lower and ℐj Upper Approximations

This part aims to generalize the topologies generated byjNκS, j∈r,l,i,u,r,l,i,u utilizing the idea of ideals. After we show the relationships between the new topologies, we apply them to define new rough approximations. We prove that our new techniques produce higher accuracy measures than those given in [17, 18, 26]. Some practical examples are provided.Theorem 2.
Let℧,ζκ,ℐ,ψj be an ℐjNκS, κ∈1,2,3,…,n. If S⊆℧, then the collection n−⊤jℐ=S⊆℧:∀z∈S,jNκz−S∈ℐ is a topology on ℧.Proof.
SupposeSα∈n−⊤jℐ, α∈Δ∀j∈r,l,i,u,r,l,i,u. Let z∈∪α∈ΔSα, then there exists α0∈Δ such that z∈Sα0. Hence, jNκz−Sα0∈ℐ. Since −∪α∈ΔSα⊆−Sα0, then jNκz−∪α∈ΔSα∈ℐ, that is, ∪α∈ΔSα∈n−⊤jℐ.
SupposeS1,S2∈n−⊤jℐ, j∈r,l,i,u,r,l,i,u. Let z∈S1∩S2, then jNκz−S1∈ℐ and jNκz−Sα0∈ℐ. According to properties of ℐ, jNκz−S1∪jNκz−S2∈ℐ. Hence, jNκz−S1∩S2∈ℐ. It follows that S1∩S2∈n−⊤jℐ.
Easily,∅,U∈n−⊤jℐ, j∈r,l,i,u,r,l,i,u. Consequently, n−⊤jℐ is a topology on ℧.Lemma 1.
Letℐ,J be two ideals on a jNκS℧,ζκ,ψj, κ∈1,2,3,…,n. If ℐ⊆J, then n−⊤jℐ⊆n−⊤jJ.Proof.
Straightforward.Proposition 11.
Let℧,ζκ,ℐ,ψj be an ℐjNκS, κ∈1,2,3,…,n. Then the following statements hold:(1)
n−⊤uℐ⊆n−⊤rℐ∩n−⊤lℐ(2)
n−⊤rℐ∪n−⊤lℐ⊆n−⊤iℐ(3)
n−⊤uℐ⊆n−⊤rℐ∩n−⊤lℐ(4)
n−⊤rℐ∪n−⊤lℐ⊆n−⊤iℐProof.
We prove only (1), and the other cases can be proved similarly.
LetS∈n−⊤uℐ. Then uNκz−S∈ℐ, ∀z∈S. Hence, rNz∪∩κ=1nlNz−S∈ℐ, ∀z∈S. So rNz−S∈ℐ and lNz−S∈ℐ, ∀z∈S. Consequently, S∈n−⊤rℐ∩n−⊤lℐ.Example 5.
In Example3, suppose that ℐ=∅,δ4. Then2−⊤rℐ=∅,℧,δ2,δ3,δ1,δ2,δ2,δ3,δ3,δ4,δ1,δ2,δ3,δ2,δ3,δ4,2−⊤lℐ=∅,℧,δ1,δ3,δ1,δ2,δ1,δ3,δ3,δ4,δ1,δ2,δ3,δ1,δ3,δ4,2−⊤iℐ=∅,℧,δ1,δ2,δ3,δ1,δ2,δ1,δ3,δ2,δ3,δ3,δ4,δ1,δ2,δ3,δ1,δ3,δ4δ2,δ3,δ4,2−⊤uℐ=∅,℧,δ3,δ1,δ2,δ3,δ4,δ1,δ2,δ3.
We note that the topologiesn−⊤rℐ, n−⊤lℐ are not dual. Also, the four topologies n−⊤jℐj=r,l,i,u are different from each other.Example 6.
On℧=δ1,δ2,δ3,δ4, suppose that ℐ=∅,δ2,δ4,δ2,δ4 is an ideal and ζ1,ζ2 are two binary relations on ℧ defined as follows:ζ1=δ1,δ1,δ1,δ2,δ1,δ3,δ2,δ3,δ2,δ4,δ3,δ1,δ3,δ3,δ4,δ1,δ4,δ3,δ4,δ4,ζ2=δ1,δ1,δ1,δ2,δ1,δ3,δ1,δ4,δ2,δ2,δ2,δ3,δ2,δ4,δ3,δ1,δ3,δ3,δ3,δ4,δ4,δ1,δ4,δ3.Hence, 2−⊤rℐ=∅,℧,δ3,δ1,δ3,δ2,δ3,δ3,δ4,δ1,δ2,δ3,δ1,δ3,δ4,δ2,δ3,δ4,2−⊤lℐ=∅,℧,δ1,δ2,δ4,δ1,δ2,δ1,δ3,δ1,δ4,δ2,δ4,δ1,δ2,δ3,δ1,δ2,δ4,δ1,δ3,δ4,2−⊤iℐ=P℧,2−⊤uℐ=∅,℧,δ1,δ3,δ1,δ2,δ3,δ1,δ3,δ4,δ2,δ3,δ4.Remark 8.
In view of Example 1.5,(1)
n−⊤rℐ≠n−⊤iℐ and n−⊤lℐ≠n−⊤iℐ(2)
n−⊤rℐ≠n−⊤uℐ and n−⊤lℐ≠n−⊤uℐ(3)
n−⊤iℐ≠n−⊤uℐRemark 9.
n−⊤jℐ and n−⊤jℐ are not necessarily comparable, for any j∈r,l,i,u,r,l,i,u. Suppose j=l, then(1)
Example6 illustrates n−⊤lℐ⊈n−⊤lℐ(2)
In Example2, if ℐ=∅,δ2, then n−⊤lℐ⊈n−⊤lℐTheorem 3.
Let℧,ζκ,ℐ,ψj be an ℐjNκS, κ∈1,2,3,…,n. If S⊆℧, then n−⊤j⊆n−⊤jℐ.Proof.
LetS∈n−⊤j. Then ∩κ=1njNκz⊆S, ∀z∈S. So ∩κ=1njNκz−S=∅∈ℐ, ∀z∈S. Hence, S∈n−⊤jℐ.Using the chemical application mentioned in article [29], we make a comparison among the n−⊤rℐ and n−⊤r with respect to reflexive relations.Example 7.
Let℧=δ1,δ2,δ3,δ4,δ5 be five amino acids (AAs). The (AAs) are characterized in terms of five attributes:A1 = PIE, A2 = surface area, A3 = molecular refractivity, A4 = the side-chain polarity, and A5 = molecular volume. See Table 2 of [29], which displays all quantitative attributes of five AAs.The five reflexive relations on ℧ defined as: ζκ=δμ,δν∈℧×℧|δμAκ−δνAκ〈σκ/2,κ,μ,ν=1,2,3,4,5, where σκ symbolizes the standard deviation of the quantitative attributes Aκ, κ=1,2,3,4,5. The r-neighborhoods for all elements of ℧ with respect to the relations ζκ, κ=1,2,3,4,5 are shown in Table 3 of [29].So one may notice the following:rNκδ1=δ1,δ4, rNκδ2=δ2,δ5, rNκδ3=δ2,δ3,δ4,δ5, rNκδ4=δ4, rNκδ5=δ5.Hence, [5]–⊤r=℧,∅,δ4,δ5,δ1,δ4,δ2,δ5,δ4,δ5,δ1,δ4,δ5,δ2,δ4,δ5,δ1,δ2,δ4,δ5,δ2,δ3,δ4,δ5. If ℐ=∅,δ4, then [5]–⊤rℐ=℧,∅,δ1,δ4,δ5,δ1,δ4,δ1,δ5,δ2,δ5,δ4,δ5,δ1,δ2,δ5,δ1,δ4,δ5,δ2,δ3,δ5,δ2,δ4,δ5,δ1,δ2,δ3,δ5,δ1,δ2,δ4,δ5,δ2,δ3,δ4,δ5. Hence, [5]–⊤r⊆5−⊤rℐ.Remark 10.
In Example6, for any arbitrary binary relations ζ1,ζ2, then [2]–⊤u=∅,℧, and 2−⊤uℐ=∅,℧,δ1,δ3,δ1,δ2,δ3,δ1,δ3,δ4. Hence, 2−⊤u⊆2−⊤uℐ.Definition 10.
Let℧,ζκ,ℐ,ψj be an ℐjNκS, κ∈1,2,3,…,n. A set S of ℧ is called ℐj open set, if S∈n−⊤jℐ, and its complement is called ℐj closed set. The family of all ℐj closed sets of a j neighborhood space is defined by n−Υjℐ=F⊆U:Fc∈n−⊤jℐBy using ideals, we present eight methods for approximating rough sets via interiorintjℐnS, closure cljℐnS, of the topologies n−⊤jℐ, j∈r,l,i,u,r,l,i,u, which will be contributory in decision-making.Definition 11.
Let℧,ζκ,ℐ,ψj be an ℐjNκS, κ∈1,2,3,…,n. If S⊆℧, then the ℐj lower, ℐj upper approximations, ℐj boundary regions, and ℐj accuracy of a set S are defined, respectively, as follows:(1)
LjℐnS=∪O∈n−⊤jℐ:O⊆S=intjℐnS(2)
UjℐnS=∩F∈n−Υjℐ:S⊆F=cljℐnS,(3)
nBjℐS=UjℐnS−LjℐnS(4)
nθjℐS=LjℐnS/UjℐnS, where UjℐnS≠0Example 8.
In Example4, suppose that ℐ=∅,δ1,δ4,δ1,δ4. Then2−⊤rℐ=℧,∅,δ1,δ2,δ4,δ1,δ2,δ1,δ4,δ2,δ3,δ2,δ4,δ1,δ2,δ3,δ1,δ2,δ4,δ2,δ3,δ4,2−⊤lℐ=℧,∅,δ1,δ3,δ1,δ3,δ2,δ3,δ3,δ4,δ1,δ2,δ3,δ1,δ3,δ4,δ2,δ3,δ4,2−⊤uℐ=℧,∅,δ2,δ3,δ1,δ2,δ3,δ2,δ3,δ4,2−⊤uℐ=℧,∅,δ2,δ3,δ4,δ1,δ3,δ2,δ3,δ2,δ4,δ3,δ4,δ1,δ2,δ3,δ1,δ3,δ4,δ2,δ3,δ4,2−⊤iℐ=2−⊤rℐ=2−⊤lℐ=2−⊤iℐ=P℧.Remark 11.
In view of Example8 and Table 4, for any arbitrary binary relations ζ1,ζ2, the ℐj accuracy, j=r, r depending on Definition 11 is better than the j accuracy measure due to [17, 18], j=r, r, respectively, in Example 4 and Table 3.Table 4
ℐj approximations and ℐj accuracy for j∈r,r depending on Definition 11.
S⊆℧Lrℐ2SUrℐ2Sθrℐ2SLrℐ2SUrℐ2Sθrℐ2Sδ1δ1δ11δ1δ11δ2δ2δ2,δ31/2δ2δ21δ3∅δ30δ3δ31δ4δ4δ41δ4δ41δ1,δ2δ1,δ2δ1,δ2,δ32/3δ1,δ2δ1,δ21δ1,δ3δ1δ1,δ3δ1,δ3δ1,δ31δ1,δ4δ1,δ4δ1,δ41δ1,δ4δ1,δ41δ2,δ3δ2,δ3δ2,δ31δ2,δ3δ2,δ31δ2,δ4δ2,δ4b,δ3,δ42/3δ2,δ4δ2,δ41δ3,δ4δ4δ3,δ41/2δ3,δ4δ3,δ41δ1,δ2,δ3δ1,δ2,δ3δ1,δ2,δ31δ1,δ2,δ3δ1,δ2,δ31δ1,δ2,δ4δ1,δ2,δ4℧3/4δ1,δ2,δ4δ1,δ2,δ41δ1,δ3,δ4δ1,δ4δ1,δ3,δ42/3δ1,δ3,δ4δ1,δ3,δ41δ2,δ3,δ4δ2,δ3,δ4δ2,δ3,δ41δ2,δ3,δ4δ2,δ3,δ41℧℧℧1℧℧1Several fundamental properties of theLjℐn, Ujℐn operators in the next proposition are listed. Properties Li, Uij = 1, 2, …, 10, display that approximate operators , Ujℐn are dual to each other. By using intjℐnS and cljℐnS behaviors, the proof of the next proposition is understandable.Proposition 12.
Let℧,ζκ,ℐ,ψj be an ℐjNκS, κ∈1,2,3,…,n. If S,S`⊆℧, then the following properties hold:(1)
L1LjℐnS=UjℐnScc(2)
L2Ljℐn℧=℧(3)
L3 If S⊆S`, then LjℐnS⊆LjℐnS`(4)
L4LjℐnS∩S`⊆LjℐnS∩LjℐnS`(5)
L5LjℐnS∪S`⊇LjℐnS∪LjℐnS`(6)
L6Ljℐn∅=∅(7)
L7LjℐnS⊆S(8)
L8LjℐnS=LjℐnLjℐnS(9)
U1UjℐnS=LjℐnScc(10)
U2Ujℐn∅ = ∅(11)
U3 If S⊆S`, then UjℐnS⊆UjℐnS`(12)
U4UjℐnS∪S`⊇UjℐnS∪UjℐnS`(13)
U5UjℐnS∩S`⊆UjℐnS∩UjℐnS`(14)
U6Ujℐn℧=℧(15)
U7S⊆nUjℐS(16)
U8UjℐnS=UjℐnUjℐSnRemark 12.
Example6 shows that the converse of L3 does not hold; if S=δ1,δ2,δ4 and S`=δ1,δ2,δ3, then LrℐnS⊆LrℐnS`, but S⊈S`.If S=δ1,δ2,δ3 and S`=δ2,δ3,δ4, then LuℐnS∩S`=∅≠δ2,δ3=LuℐnS∩LuℐnS`. Consequently, the converse of L4 is not true.If S=δ1 and S`=δ3, then LuℐnS∪S`=δ1,δ3≠∅=LuℐnS∪LuℐnS`. Consequently, the converse of L5 is not true.If S=δ1 and j=u, then the converse of L7 is not true.According to Theorem3, the proof of the following theorem is evident.Theorem 4.
Let℧,ζκ,ℐ,ψj be an ℐjNκS, κ∈1,2,3,…,n. If S⊆℧, then ∀j∈r,l,i,u,r,l,i,u; the following statements hold:(1)
LunS⊆LuℐnS(2)
LunS⊇LuℐnSLemma 2.
Letℐ,J be two ideals on a jNκS℧,ζκ,ψj, κ∈1,2,3,…,n. If ℐ⊆J, then the following statements hold:(1)
LjℐnS⊇LjJnS(2)
UjℐnS⊇UjJnS(3)
θjℐnS⊇θjJnSProof.
Straightforward.Example9 offers the comparison between Definition 9 and Definition 6, if each ζi, κ∈1,2,3,…,n is a reflexive relation on ℧.Example 9.
In Example3, if ℐ=∅,δ1,δ4,δ1,δ4, then2−⊤rℐ=2−⊤uℐ=∅,℧,δ2,δ3,δ1,δ2,δ2,δ3,δ3,δ4,δ1,δ2,δ3,δ2,δ3,δ4,2−⊤lℐ=2−⊤iℐ=∅,℧,δ1,δ2,δ3,δ1,δ2,δ1,δ3,δ2,δ3,δ3,δ4,δ1,δ2,δ3,δ1,δ3,δ4,δ2,δ3,δ4,2−⊤rℐ=2−⊤lℐ=2−⊤iℐ=2−⊤uℐ=∅,℧,δ1,δ2,δ3,δ1,δ2,δ1,δ3,δ2,δ3,δ3,δ4,δ1,δ3,δ4,δ2,δ3,δ4,δ1,δ2,δ3.Remark 13.
It is clear that from Example3, n−⊤j⊆n−⊤jℐ and ∀j∈r,l,i,u,r,l,i,u.Remark 14.
In Example9 and Table 5:(1)
ℒrℐnS=ℒrℐnS and LlℐnS=LiℐnS=LrℐnS=LlℐnS=LiℐnS=LuℐnS(2)
UuℐnS=UuℐnS and UlℐnS=UiℐnS=UrℐnS=UlℐnS=UiℐnS=UuℐnS(3)
θrℐnS=θuℐnS andTable 5
Theℐj approximations and ℐj accuracy for each j∈r,l with respect to Example 9.
S⊆℧LrnSLlnSLinSLunSUrnSUlnSδ1∅δ1δ1δ101δ2δ2δ2δ1,δ2δ21/21δ3δ3δ3δ3,δ4δ3,δ41/21/2δ4∅∅δ4δ400δ1,δ2δ1,δ2δ1,δ2δ1,δ2δ1,δ211δ1,δ3δ3δ1,δ3δ1,δ3,δ4δ1,δ3,δ41/32/3δ1,δ4∅δ1δ1,δ4δ1,δ401/2δ2,δ3δ2,δ3δ2,δ3℧δ2,δ3,δ41/22/3δ2,δ4δ2δ2δ1,δ2,δ4δ2,δ41/31/2δ3,δ4δ3,δ4δ3,δ4δ3,δ4δ3,δ411δ1,δ2,δ3δ1,δ2,δ3δ1,δ2,δ3℧℧3/43/4δ1,δ2,δ4δ1,δ2δ1,δ2δ1,δ2,δ4δ1,δ2,δ42/32/3δ1,δ3,δ4δ3,δ4δ1,δ3,δ4δ1,δ3,δ4δ1,δ3,δ42/31δ2,δ3,δ4δ2,δ3,δ4δ2,δ3,δ4℧δ2,δ3,δ43/41℧℧℧℧℧11The following propositions are obvious, and the proof is omitted.Proposition 13.
Let℧,ζκ,ℐ,ψj be an ℐjNκS, κ∈1,2,3,…,n. If S⊆℧, then ∀j∈r,l,i,u,r,l,i,u; the following statements hold:(1)
LnuℐS⊆nLrℐS⊆nLiℐS(2)
LnuℐS⊆nLlℐS⊆nLiℐS(3)
LnuℐS⊆nLrℐS⊆nLiℐS(4)
LnuℐS⊆nLlℐS⊆nLiℐSProposition 14.
Let℧,ζκ,ℐ,ψj be an ℐjNκS, κ∈1,2,3,…,n. If S⊆℧, then ∀j∈r,l,i,u,r,l,i,u; the following statements hold:(1)
UiℐnS⊆UrℐnS⊆UuℐnS(2)
UiℐnS⊆UlℐnS⊆UuℐnS(3)
UiℐnS⊆UrℐnS⊆UuℐnS(4)
UrℐnS⊆UlℐnS⊆UiℐnSCorollary 1.
Let℧,ζκ,ℐ,ψj be an ℐjNκS, κ∈1,2,3,…,n. Then ∀j∈r,l,i,u,r,l,i,u and θjnS≤θjℐnS.Definition 12.
Let℧,ζκ,ℐ,ψj be an ℐjNκS, κ∈1,2,3,…,n, ∀j∈r,l,i,u,r,l,i,u. A subset of ℧ is called as follows:(1)
Totally,ℐj definable, if LjℐnS=S=UjℐnS(2)
Internally,ℐj definable, if LjℐnS=S and UjℐnS≠S(3)
Externally,ℐj definable, if LjℐnS≠S and UjℐnS=S(4)
ℐj rough set, if LjℐnS≠S andCorollary 2.
Let℧,ζκ,ℐ,ψj be an ℐjNκS, κ∈1,2,3,…,n. Then ∀j∈r,l,i,u,r,l,i,u; the following statements hold:(1)
Everyj definable subset in ℧ is ℐj definable(2)
Everyℐj rough subset in ℧ is j roughRemark 15.
Example9 shows that the converse of parts of Corollary 2 is not necessarily true.(1)
Ifj=l, then δ2 is ℐj definable, but it is not j definable (see Tables 2 and 5)(2)
Ifj=u, then δ2 is rough, but it is not ℐj rough (see Tables 2 and 5)Remark 16.
In view of Example9 and Table 5, we have the following:(1)
δ1,δ2 is a totally ℐr definable set(2)
δ2,δ4 is an ℐr rough set(3)
δ3 is an internally ℐl definable set(4)
δ4 is an externally ℐl definable setCorollary 3.
Let℧,ζκ,ℐ,ψj be an ℐjNκS, κ∈1,2,3,…,n. Then ∀j∈r,l,i,u,r,l,i,u; the following statements hold:(1)
θjℐnS≤θrℐnS≤θjℐnS(2)
θuℐnS≤θlℐnS≤θiℐnS(3)
θuℐnS≤θrℐnS≤θiℐnS(4)
θuℐnS≤θlℐnS≤θiℐnSIn Example10, we show that the current method in Definition 11 is more accurate in comparison to Hosny’s method [26], for any j=i.Example 10.
Continued from Example4. Table 6 shows the comparison between the current method in Definition 11 and Hosny’s method [26], for any j=i.If ℐ=∅,δ1, then⊤ζ1iℐ=∅,℧,δ4,δ1,δ4,δ2,δ4,δ3,δ4,δ1,δ2,δ4,δ1,δ3,δ4,δ2,δ3,δ4,⊤ζ2iℐ=∅,℧,δ2,δ3,δ4,δ1,δ4,δ2,δ3,δ2,δ4,δ3,δ4,δ1,δ2,δ4,δ1,δ3,δ4,δ2,δ3,δ4,2−⊤iℐ=P℧.Table 6
Comparison between our approaches and those given in the case ofj=i.
S⊆℧Lζ1iSUζ1iSθζ1iSLζ2iSUζ2iSθζ2iSLζ2i2SUi2Sθi2Sδ1∅δ10∅δ10δ1δ11δ2∅δ20δ2δ21δ2δ21δ3∅δ30δ3δ31δ3δ31δ4δ4℧1/4δ4δ1,δ41/2δ4δ41δ1,δ2∅δ1,δ20δ2δ1,δ21/2δ1,δ2δ1,δ21δ1,δ3∅δ1,δ30δ3δ1,δ31/2δ1,δ3δ1,δ31δ1,δ4δ1,δ4℧1/2δ1,δ4δ1,δ41δ1,δ4δ1,δ41δ2,δ3∅δ2,δ30δ2,δ3δ2,δ31δ2,δ3δ2,δ31δ2,δ4δ2,δ4℧1/2δ2,δ4δ1,δ2,δ42/3δ2,δ4δ2,δ41δ3,δ4δ3,δ4℧1/2δ3,δ4δ1,δ3,δ42/3δ3,δ4δ3,δ41δ1,δ2,δ3∅δ1,δ2,δ30δ2,δ3δ1,δ2,δ32/3δ1,δ2,δ3δ1,δ2,δ31δ1,δ2,δ4δ1,δ2,δ4℧3/4δ1,δ2,δ4δ1,δ2,δ41δ1,δ2,δ41δ1,δ3,δ4δ1,δ3,δ4℧3/4δ1,δ3,δ4δ1,δ3,δ41δ1,δ3,δ4δ1,δ3,δ41δ2,δ3,δ4δ2,δ3,δ4℧3/4δ2,δ3,δ4℧3/4δ2,δ3,δ4δ2,δ3,δ41℧℧℧1℧℧1℧℧1


## 4. Network Devices Application

Network devices or networking hardware (network interface cards, hubs, switches, repeaters, bridges, routers, and gateways) are physical devices that are required for communication and interaction between hardware on a computer network. This section is devoted to introducing an applied example (see [30]) of the new method on the network topologies. This application presents a comparison between approaches with respect to n−⊤j and n−⊤jℐ, j∈r,l,i,u,r,l,i,u.Example 11.
Let℧=δ1,δ2,δ3,δ4 be a set of four network topologies and N=N1,N2,N3 be the attributes of network topologies (see Table 7), where δ1 is a bus topology, δ2 is a ring topology, δ3 is a star topology, and δ4 is a mesh topology.N1 = The method of transfer data = broadcast,multicast,unicast = a1,b1,c1, N2 = Cable type = twistedpaircable,thincoaxialcable,thickcoaxialcable,fiberopticcable = a2,b2,c2,d2, andN3 = Bandwidth capacity = 10Mbit/s,10−100Mbit/s,10Mbit/s−40Gbit/s = a3,b3,c3.Consider Table 7: let ζκ, κ∈1,2,3 be arbitrary binary relations as follows:
δmζκδnNδm⊆Nδn, δm,δn∈℧, κ∈1,2,3.
Then we compute the approximations of attributesN1,N2, and N3. ζ1 = ▲∪δ1,δ2,δ1,δ3,δ2,δ1,δ2,δ3,δ4,δ3, ζ2 = ▲∪δ1,δ3,δ1,δ4,δ2,δ3,δ2,δ4,δ3,δ4,δ4,δ3, ζ3 = ▲∪δ2,δ3,δ2,δ4,δ3,δ2,δ3,δ4,δ4,δ2,δ4,δ3. Now, uNκδ1=δ1, uNκδ2=δ2,δ3, uNκδ3=δ2,δ3,δ4, uNκδ4=δ3,δ4.Therefore, [3]–⊤u=∅,℧,δ1,δ2,δ3,δ4.If ℐ = ∅,δ2, then [3]–⊤uℐ=∅,℧,δ1,δ3,δ4,δ1,δ3,δ4,δ2,δ3,δ4.If J = ∅,δ2,δ3,δ2,δ3, then [3]–⊤uJ=∅,℧,δ1,δ2,δ4,δ1,δ2,δ1,δ4,δ2,d,δ3,δ4,δ1,δ2,δ4,δ1,δ3,δ4,δ2,δ3,δ4.
From Table8, we notice the following:(1)
The accuracy of the approximations induced fromn−⊤uℐ for any ideal ℐ is more accurate than the accuracy of approximations induced from n−⊤u(2)
Ifℐ⊆J, then the accuracy of the approximations induced from n−⊤jJ is more accurate than the accuracy of the approximations induced from n−⊤jℐ, ∀j∈r,l,i,u,r,l,i,u(3)
Everyj definable set with respect to n−⊤j is j definable with respect to n−⊤jℐ for any ideal ℐ, but the converse is not true(4)
The approximations induced from for any idealℐ will help extract and discover the hidden information in data that were collected from real-life applications, which are very useful in decision-makingTable 7
Attributes of network topologies.
℧∖SN1N2N3δ1a1b2,c2a3δ2a1c2,d2c3δ3a1,b1,c1a2,b2,c2,d2c3δ4c1a2,b2,c2,d2c3Table 8
Comparisons between the accuracy of some subsets with respect ton−⊤j, n−⊤jℐ, and n−⊤jJ at j=u.
S⊆℧LunSUunSLuℐnSUuℐnSLuℐnSUuJnSθunSθuℐnSθuJnSδ1δ1δ1δ1δ1δ1δ1111δ2∅δ2,δ3,δ4∅δ2δ2δ2001δ3∅δ2,δ3,δ4∅δ2,δ3,δ4∅δ3000δ4∅δ2,δ3,δ4∅δ2,δ3,δ4δ4δ3,δ4001/2δ1,δ2δ1℧δ1δ1,δ2δ1,δ2δ1,δ21/41/21δ1,δ3δ1℧δ1℧δ1δ1,δ31/41/41/2δ1,δ4δ1℧δ1℧δ1,δ4δ1,δ3,δ41/41/42/3δ2,δ3∅δ2,δ3,δ4∅δ2,δ3,δ4δ2δ2,δ3001/2δ2,δ4∅δ2,δ3,δ4∅δ2,δ3,δ4δ2,δ4δ2,δ3,δ4002/3δ3,δ4∅δ2,δ3,δ4δ3,δ4δ2,δ3,δ4δ3,δ4δ3,δ402/31δ1,δ2,δ3δ1℧δ1℧δ1,δ2δ1,δ2,δ31/41/42/3δ1,δ2,δ4δ1℧δ1℧δ1,δ2,δ4℧1/41/43/4δ1,δ3,δ4δ1℧δ1,δ3,δ4℧δ1,δ3,δ4δ1,δ3,δ41/43/41δ2,δ3,δ4δ2,δ3,δ4δ2,δ3,δ4δ2,δ3,δ4δ2,δ3,δ4δ2,δ3,δ4δ2,δ3,δ4111℧℧℧℧℧℧℧111


## 5. Conclusion and Future Work

In this manuscript, we have presented novel types of rough neighborhoods generated from a finite number of binary relations. We have scrutinized their main features and determined under what conditions some of these neighborhoods are equivalent. Then we have established topological spaces from these types of neighborhoods that we used to initiate novel rough set models. The best approximations and accuracy measures have been given in the cases ofi and i as the provided examples illustrated. Thereafter, we have applied the notion of ideals to initiate topological spaces finer than those induced from rough neighborhoods given in Section 2. Some comparisons have been given to show the importance of our approaches compared with some methods in the literature such as [14, 17, 18, 26]. Finally, we have displayed a practical application showing the merits of the approaches followed in this manuscript.In upcoming works, we plan to benefit from the rough neighborhoods given in [12, 20] and the ideals to improve the approximations given herein and increase their accuracy values.



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