# Covering Rough Sets in Terms of Topological Bi-Neighbourhoods and Their Applications

Authors: Amal T. Abushaaban; Abdelfattah A. El Atik; O. A. Embaby
Journal: Journal of Mathematics
 (2023)

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## Abstract

Neighbourhoods are one of the important topics in topology that relies on two types of neighbours of a point and has many applications in the graph theory and in the sciences and medical sciences. In this work, some types of bi-j-neighbourhoods on generalized bicovering approximation space are introduced. Moreover, some kinds of bicovering rough sets using relations are presented. Some properties of these new types of covering are discussed. Pawlaks’ properties are studied in the case of bicovering approximation space. More properties on different bi-neighbourhoods such as bi-j-neighbourhoods, complementary bi-j-neighbourhoods, and bi-j-adhesions are investigated. A comparison between these new types of bi-neighbourhoods and bi-covering is presented with the help of some counterexamples. Finally, we give an application of our results in the rheumatic fever data information by generated topologies.


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## Body

## 1. Introduction

Covering rough sets has been initiated by Zhu [1] who pointed to the definition of covering of universal discourse U; as any partition of U. The covering of U; is not necessarily a partition, in general. Elementary sets are the members of the cover, and any union of them is called a definable set. Some authors used covering to induce several kinds of rough sets [2–5]. In addition, generalized rough sets were developed by using the covering [6–9]. Abd El-Monsef et al. [10] generalized a covering approximation space, and Zhang and Wang [11] related β-operator with a fuzzy covering approximation space to get a generalization of β-fuzzy covering approximation space.Pawlak [12] was the first one who developed rough set theory to work in information systems about vagueness information. The covering π; of U; with the idea of rough set was initiated by Zakowski [4], and some researchers have developed the subject of covering rough sets [2, 5, 7, 10, 24]. Topological notions and axiomatic systems are discussed relating with rough sets [25–28]. Classical rough sets and neighbourhood were related to covering rough sets [29–32] as an expansion of rough sets.Abo-Elhamayel [17] was one of these researchers who was talking about bi--covering rough sets. The development of the topological neighbourhoods subject has continued between the researchers. Nawar et al. [18] found the reduction by topologies which are inducing from neighbourhood systems. There are many topological models induced by approximation spaces that depend on its relation [19]. If the relation is equivalence, then a clopen space is defined [20, 21]. Moreover, for any arbitrary relation, one of topology, supratopology, and infratopology is constructed [22, 23].Throughout the study, we induce new kinds of topologies using generalizedbi–covering approximation space that can be used to compute the reduction for data in the information system. In addition, the accurate approximations are calculated when two relations are used. An introduction and main concepts and results which used in the article are stated in Sections 1 and 2. New kinds of bi-j-neighbourhoods on generalized bicovering approximation space are presented, and the comparison between them is illustrated in Section 3. In addition, we study different approximations depending on Nbij,Obij;, and dbij; of some types of bicovering rough sets. Furthermore, relationships between different kinds of approximations are constructed in Section 4. Finally, some properties of approximation operators are proved, and their related counterexamples are studied in Section 5. For applications of results, different topologies are generated by Nbij,Obij;, and dbij; and are given in Section 6.


## 2. Preliminaries

Some fundamental definitions and propositions inℛ-left and ℛ-right neighbourhoods, right cover, left cover, right bi-cover, left bi-cover, and some neighbourhoods which are necessary for our study are presented in this section.


### 2.1. Pawlak Approximation Space

LetU≠ϕ; be a finite set, ℛ; be an equivalence relation on U;, and h⊆U; denotes to the equivalence class containing h∈U. An approximation space is denoted as U,ℛ. The following two definitions present the concepts of lower and upper approximations of a subset A; of U; in approximation space U,ℛ.Definition 1 (see [13, 14]).
LetU,ℛ; be an approximation space and A⊆U. The ℛ–lower and ℛ–upper approximations of A; are defined, respectively, by ℛ¯A=h∈U:h⊆A; and ℛ¯A=h∈U:h∩A≠∅. Based on lower and upper approximations of A⊆U,U; divided into three disjoint regions. A boundary region (briefly BNDℛA) is defined by BNDℛA=ℛ¯A−ℛ¯A.Proposition 1 (see [15]).
The following properties are for Pawlaks’ rough sets:L1ℛ¯ℋ⊆ℋL2ℛ¯∅=∅L3ℛ¯U=UL4ℛ¯ℋ∩T=ℛ¯ℋ∩ℛ¯TL5 If ℋ⊆T, then ℛ¯ℋ⊆ℛ¯TL6ℛ¯ℋ∪ℛ¯T⊆ℛ¯ℋ∪TL7ℛ¯ℋc=ℛ¯ℋcL8ℛ¯ℛ¯ℋ=ℛ¯ℋL9ℛ¯ℛ¯ℋc=ℛ¯ℋcH1ℋ⊆ℛ¯ℋH2ℛ¯∅=∅H3ℛ¯U=UH4ℛ¯ℋ∪T=ℛ¯ℋ∪ℛ¯TH5 If ℋ⊆T, then ℛ¯ℋ⊆ℛ¯TH6ℛ¯ℋ∩ℛ¯T⊇ℛ¯ℋ∩TH7ℛ¯ℋc=ℛ¯ℋcH8ℛ¯ℛ¯ℋ=ℛ¯ℋH9ℛ¯ℛ¯ℋc=ℛ¯ℋc


### 2.2. A Generalized Covering Approximation Space

Definition 2 (see [16]).
If∀h∈U∃t∈U; s.t. hℛt, then ℛ; is called a binary serial relation on U.Definition 3 (see [17]).
lℛh=t:t∈U,tℛh; and rℛh=t:t∈U,hℛt; are, respectively, the definitions of ℛ–left and ℛ–right neighbourhoods of an element h; in U.Proposition 2 (see [17]).
U=∪h∈Uℛh; (resp., U=∪h∈Uhℛ); if ℛ; (resp., ℛ−1); is a serial relation on U.Definition 4 (see [10]).
LetU; be a domain of discourse and C=K:K⊆U; be a family of nonempty subsets of U. If U=∪K:K⊆U, then C; is a covering of U. For serial relations ℛ; and ℛ−1; on U, right cover (briefly, r-cover) is Cr=hℛ:∀h∈U, and left cover (briefly, l-cover) is Cl=ℛh:∀h∈U. U,ℛ,Cr,Cl; is called a generalized covering approximation space (for short, GCAS).Definition 5 (see [10]).
LetU,ℛ,Cr,Cl; be a GCAS. j-neighbourhoods Njh,j∈r,l,u,i; are defined by (1) r–neighbourhood: Nrh=∩K∈Cr:h∈K; (2) l–neighbourhood: Nlh=∩K∈Cl:h∈K; (3) u–neighbourhood: Nuh=Nrh∪Nlh; (4) i--neighbourhood: Nih=NrhcapNlh.


### 2.3. Generalized Bicovering Approximation Space

Definition 6 (see [17]).
Letℛ1,ℛ2,ℛ1−1;, and ℛ2−1; be serial relations on U,Cri=hℛi:∀h∈U; and Cli=ℛih:∀h∈U,i∈1,2. Right bicover (briefly, r--bicover) is Cr12=∪i∈1,2Cri, and left bicover (briefly, l--bicover) is Cl12=∪i∈1,2Cli. U,ℛ1,ℛ2,Cl12,Cr12; is called a generalized bicovering approximation space GBCAS.Definition 7 (see [17]).
LetU,ℛ1,ℛ2,Cr12,Cl12; be a GBCAS. The bi-j-neighbourhoods Nbijh,j∈Λ=r12,l12,r12∨l12,r12∧l12; are defined as follows: (1) r12-neighbourhood:Nbir12h=∩K∈Cr12:h∈K (2) l12-neighbourhood: Nbil12h=∩K∈Cl12:h∈K (3) r12∨l12-neighbourhood: Nbir12∨l12h=Nbir12h∪Nbil12h (4) r12∧l12-neighbourhood: Nbir12∧l12h=Nbir12h∩Nbil12hProposition 3 (see [17]).
LetU,ℛ1,ℛ2,Cr12,Cl12; be a GBCAS. ∀h∈U. ∀j∈Λ,h∈Nbijh,∀h∈U.Proposition 4 (see [17]).
LetU,ℛ1,ℛ2,Cr12,Cl12; be a GBCAS. ∀j∈Λ1=r12,l12,r12∧l12, if h∈Nbijt, then Nbijh⊆Nbijt.Remark 1 (see [17]).
LetU,ℛ1,ℛ2,Cr12,Cl12; be a GBCAS. ∀j=r12∨l12, if h∈Nbijt, then Nbijh⊈Nbijt.Definition 8 (see [17]).
LetU,ℛ1,ℛ2,Cr12,Cl12; be a GBCAS. ∀j∈Λ,A⊆U,N¯bijA=h∈A:Nbijh⊆A,, N¯bijA=h∈U:Nbijh∩A≠∅,ℬbijA=N¯bijA−N¯bijA ; and αbijA=N¯bijA/N¯bijA, where N¯bijA≠0; are, respectively, the definitions of: bi-j- lower approximation N¯bijA,bi−j− upper approximation N¯bijA,bi-j- boundary region and bi-j- accuracy of the approximations of A.


## 2.1. Pawlak Approximation Space

LetU≠ϕ; be a finite set, ℛ; be an equivalence relation on U;, and h⊆U; denotes to the equivalence class containing h∈U. An approximation space is denoted as U,ℛ. The following two definitions present the concepts of lower and upper approximations of a subset A; of U; in approximation space U,ℛ.Definition 1 (see [13, 14]).
LetU,ℛ; be an approximation space and A⊆U. The ℛ–lower and ℛ–upper approximations of A; are defined, respectively, by ℛ¯A=h∈U:h⊆A; and ℛ¯A=h∈U:h∩A≠∅. Based on lower and upper approximations of A⊆U,U; divided into three disjoint regions. A boundary region (briefly BNDℛA) is defined by BNDℛA=ℛ¯A−ℛ¯A.Proposition 1 (see [15]).
The following properties are for Pawlaks’ rough sets:L1ℛ¯ℋ⊆ℋL2ℛ¯∅=∅L3ℛ¯U=UL4ℛ¯ℋ∩T=ℛ¯ℋ∩ℛ¯TL5 If ℋ⊆T, then ℛ¯ℋ⊆ℛ¯TL6ℛ¯ℋ∪ℛ¯T⊆ℛ¯ℋ∪TL7ℛ¯ℋc=ℛ¯ℋcL8ℛ¯ℛ¯ℋ=ℛ¯ℋL9ℛ¯ℛ¯ℋc=ℛ¯ℋcH1ℋ⊆ℛ¯ℋH2ℛ¯∅=∅H3ℛ¯U=UH4ℛ¯ℋ∪T=ℛ¯ℋ∪ℛ¯TH5 If ℋ⊆T, then ℛ¯ℋ⊆ℛ¯TH6ℛ¯ℋ∩ℛ¯T⊇ℛ¯ℋ∩TH7ℛ¯ℋc=ℛ¯ℋcH8ℛ¯ℛ¯ℋ=ℛ¯ℋH9ℛ¯ℛ¯ℋc=ℛ¯ℋc


## 2.2. A Generalized Covering Approximation Space

Definition 2 (see [16]).
If∀h∈U∃t∈U; s.t. hℛt, then ℛ; is called a binary serial relation on U.Definition 3 (see [17]).
lℛh=t:t∈U,tℛh; and rℛh=t:t∈U,hℛt; are, respectively, the definitions of ℛ–left and ℛ–right neighbourhoods of an element h; in U.Proposition 2 (see [17]).
U=∪h∈Uℛh; (resp., U=∪h∈Uhℛ); if ℛ; (resp., ℛ−1); is a serial relation on U.Definition 4 (see [10]).
LetU; be a domain of discourse and C=K:K⊆U; be a family of nonempty subsets of U. If U=∪K:K⊆U, then C; is a covering of U. For serial relations ℛ; and ℛ−1; on U, right cover (briefly, r-cover) is Cr=hℛ:∀h∈U, and left cover (briefly, l-cover) is Cl=ℛh:∀h∈U. U,ℛ,Cr,Cl; is called a generalized covering approximation space (for short, GCAS).Definition 5 (see [10]).
LetU,ℛ,Cr,Cl; be a GCAS. j-neighbourhoods Njh,j∈r,l,u,i; are defined by (1) r–neighbourhood: Nrh=∩K∈Cr:h∈K; (2) l–neighbourhood: Nlh=∩K∈Cl:h∈K; (3) u–neighbourhood: Nuh=Nrh∪Nlh; (4) i--neighbourhood: Nih=NrhcapNlh.


## 2.3. Generalized Bicovering Approximation Space

Definition 6 (see [17]).
Letℛ1,ℛ2,ℛ1−1;, and ℛ2−1; be serial relations on U,Cri=hℛi:∀h∈U; and Cli=ℛih:∀h∈U,i∈1,2. Right bicover (briefly, r--bicover) is Cr12=∪i∈1,2Cri, and left bicover (briefly, l--bicover) is Cl12=∪i∈1,2Cli. U,ℛ1,ℛ2,Cl12,Cr12; is called a generalized bicovering approximation space GBCAS.Definition 7 (see [17]).
LetU,ℛ1,ℛ2,Cr12,Cl12; be a GBCAS. The bi-j-neighbourhoods Nbijh,j∈Λ=r12,l12,r12∨l12,r12∧l12; are defined as follows: (1) r12-neighbourhood:Nbir12h=∩K∈Cr12:h∈K (2) l12-neighbourhood: Nbil12h=∩K∈Cl12:h∈K (3) r12∨l12-neighbourhood: Nbir12∨l12h=Nbir12h∪Nbil12h (4) r12∧l12-neighbourhood: Nbir12∧l12h=Nbir12h∩Nbil12hProposition 3 (see [17]).
LetU,ℛ1,ℛ2,Cr12,Cl12; be a GBCAS. ∀h∈U. ∀j∈Λ,h∈Nbijh,∀h∈U.Proposition 4 (see [17]).
LetU,ℛ1,ℛ2,Cr12,Cl12; be a GBCAS. ∀j∈Λ1=r12,l12,r12∧l12, if h∈Nbijt, then Nbijh⊆Nbijt.Remark 1 (see [17]).
LetU,ℛ1,ℛ2,Cr12,Cl12; be a GBCAS. ∀j=r12∨l12, if h∈Nbijt, then Nbijh⊈Nbijt.Definition 8 (see [17]).
LetU,ℛ1,ℛ2,Cr12,Cl12; be a GBCAS. ∀j∈Λ,A⊆U,N¯bijA=h∈A:Nbijh⊆A,, N¯bijA=h∈U:Nbijh∩A≠∅,ℬbijA=N¯bijA−N¯bijA ; and αbijA=N¯bijA/N¯bijA, where N¯bijA≠0; are, respectively, the definitions of: bi-j- lower approximation N¯bijA,bi−j− upper approximation N¯bijA,bi-j- boundary region and bi-j- accuracy of the approximations of A.


## 3. New Kinds ofbi-j-Neighbourhoods on Generalized Bicovering Approximation Space

In this section, new kinds ofbi-j-neighbourhoods are introduced, and their properties are investigated.Definition 9.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS. The complementary bi-j-neighbourhood of h is Obijh=t∈U:h∈Nbijt,∀h∈U, and j∈Λ=r12,l12,r12∧l12,r12∨l12.Example 1.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS, where U=ω,ψ,υ,m,ℛ1=ω,ω,ω,ψ,ω,m,ψ,ω,ψ,m,υ,ω,υ,m,υ,υ,m,ω,m,m, and ℛ2=ω,ω,ω,ψ,ω,m,ψ,ω,ψ,m,υ,ω,υ,m,υ,υ,m,ω, we have Table 1 from Definition 3.
The covers are given byCr1=ω,ψ,m,ω,m,ω,υ,m;Cr2=ω,ψ,m,ω,m,ω,υ,m,ω;Cl1=U,ω,υ;Cl2=U,ω,υ,ω,ψ,υ;Cr12=ω,ψ,m,ω,m,ω,υ,m,ω; andCl12=U,ω,υ,ω,ψ,υ.
j-neighbourhoods, bi-j-neighbourhoods, and complementary bi-j-neighbourhoods are presented, respectively, in Tables 2–4.Table 1
Theℛ-right and ℛ-left neighbourhoods of h∈U.
h∈Uωψυmhℛ1ω,ψ,mω,mω,υ,mω,mℛ1hUωυUhℛ2ω,ψ,mω,mω,υ,m{ω}ℛ2hUωυω,ψ,υTable 2
j-neighbourhoods Njh.
h∈UωψυmNr1hω,mω,ψ,mω,υ,mω,mNr2hωω,ψ,mω,υ,mω,mNl1hωUυUNl2hωω,ψ,υυUTable 3
Bi-j-neighbourhoodsNbijh.
h∈UωψυmNbir12hωω,ψ,mω,υ,mω,mNbir12hωω,ψ,υυUNbir12∨l12hωUω,υ,mUNbir12∧l12hωω,ψυω,mTable 4
Complementary bi-j-neighbourhoods Obijh.
h∈UωψυmObir12hUψυψ,υ,mObir12hω,ψ,mψ,mψ,υ,mmObir12∨l12hUψ,mψ,υ,mψ,υ,mObir12∧l12hω,ψ,mψυmDefinition 10.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS. ∀j∈Λ,bi-j-adhesions of h is defined as follows: dbijh=t∈U:Nbijh=Nbijt.Example 2.
Thebi-j-adhesions for GBCASU,ℛ1,ℛ2,Cr12,Cl12 in Example 1 is given by Table 5.Table 5
Bi-j-adhesions dbijh.
h∈Uωψυmdbir12hωψυmdbir12hωψυmdbir12∨l12hωψ,mυψ,mdbir12∧l12hωψυmLemma 1.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS. If h∈U, then h∈Obijh,∀j∈Λ.Proof.
Since∀h∈U,h∈Nbijh, from Proposition 3, then by Definition 9, h∈Obijh.Lemma 2.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS. Then, ∀j∈Λ and h∈U, and we have dbijh⊆Nbijh and dbijh⊆Obijh.Proof.
Suppose thatt∈dbijh. Then, Nbijh=Nbijt, by Definition 10. If h∈Nbijh,∀h∈U, by Proposition 3, then t∈Nbijt, and so, t∈Nbijh. Hence, dbijh⊆Nbijh. Similarly, dbijh⊆Obijh is proved.
The other inclusion for Lemma2 is not true, in general.Example 3.
From Tables3 and 5, we get Nbijψ⊈dbijψ∀j∈Λ. Also, Obijω⊈dbijω∀j∈Λ.Lemma 3.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS. Then, for each j∈Λ1=r12,l12,r12∧l12 and h∈U, we have dbijh=Nbijh∩Obijh.Proof.
By Lemma2, we get dbijh⊆Nbijh, and dbijh⊆Obijh,∀h∈U. Then, dbijh⊆Nbijh∩Obijh. Let g∈Nbijh∩Obijh. Then, g∈Nbijh and g∈Obijh. So, by Definition 9, g∈Nbijh and h∈Nbijg. By Proposition 4, we have Nbijg⊆Nbijh and Nbijh⊆Nbijg. So, Nbijh=Nbijg. Hence, g∈dbijh. Therefore, Nbijh∩Obijh⊆dbijh,∀j∈Λ1 and h∈U.Remark 2.
It is noted thatdbir12∨l12h≠Nbir12∨l12h∩Obir12∨l12h, for some h∈U.Example 4.
From Examples1 and 2, dbir12∨l12υ=υ≠Nbir12∨l12υ∩Obir12∨l12υ=υ,m.
The proof of Proposition5 and its Corollary 1 is clear from Definition 9. So, it is omitted.Proposition 5.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS. Then, for any h,t∈U,t∈Obijh if and only if h∈Nbijt,∀j∈Λ.Corollary 1.
IfU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS, then Nbijh=t∈U:h∈Obijt,∀h∈U,∀j∈Λ.Example 5.
In Example1, take j=r12. Then:(1)
Obir12ω=U, since ω∈Nbir12ω,ω∈Nbir12ψ,ω∈Nbir12υ and ω∈Nbir12m.(2)
Obir12ψ=ψ, since ψ∈Nbir12ψ.(3)
Obir12υ=υ, since υ∈Nbir12υ.(4)
Obir12m=ψ,υ,m, since m∈Nbir12ψ,m∈Nbir12υ and m∈Nbir12m.Also, (5)Nbir12ω=ω, since ω∈Obir12ω.(6)
Nbir12ψ=ω,ψ,m, since ψ∈Obir12ω,ψ∈Obir12ψ and ψ∈Obir12m.(7)
Nbir12υ=ω,υ,m, since υ∈Obir12ω,υ∈Obir12υ and υ∈Obir12m.(8)
Nbir12m=ω,m, since m∈Obir12ω and m∈Obir12m.
Now, the relationship between complementarybi-j-neighbourhoods is discussed.Proposition 6.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS. Then, for any h,t∈U if t∈Obijh, we have Obijt⊆Obijh,∀j∈r12,l12,r12∧l12.Proof.
Letω∈Obijt=h∈U:t∈Nbijh. Then, t∈Nbijω=h∈U:ω∈Obijh. Therefore, ω∈Obijh.Remark 3.
Proposition6 is not satisfied for j=r12∨l12, in general.Example 6.
From Example1 and Table 4, m∈Obir12∨l12ψ=ψ,m, but Obir12∨l12m=ψ,υ,m⊈Obir12∨l12ψ=ψ,m.


## 4. Some Kinds of Bicovering Rough Sets with Some Relations between Them

In this section, new types ofbi-j-neighbourhoods Nbijh, complementary bi-j-neighbourhoods Obijh, and bi-j-adhesions dbij of bicovering rough sets are presented. Furthermore, the relationship between these kinds of approximations is constructed.Definition 11.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS. The complementary (bi-j-lower approximations, bi-j-upper approximations, bi-j-boundary region and bi-j-accuracy of the approximations) of A is defined, respectively, by O¯bijA=h∈A:Obijh⊆A,O¯bijA=h∈U:Obijh∩A≠∅,BbijA=O¯bijA−O¯bijA, and βbijA=O¯bijA/O¯bijA, where O¯bijA≠0,∀A⊆U, and j∈Λ.Example 7.
In Example1, let A=ω,υ,m. Then, O¯bir12A=υ,O¯bil12A=m,O¯bir12∨l12A=∅,O¯bir12∧l12A=υ,m. O¯bir12A=ω,υ,m,O¯bil12A=U,O¯bir12∨l12A=U,O¯bir12∧l12A=ω,υ,m. Bbir12A=ω,m,Bbil12A=ω,ψ,υ,Bbir12∨l12A=U,Bbir12∧l12A=ω. βbir12A=1/3,βbil12A=1/4,βbir12∨l12A=0,βbir12∧l12A=2/3.Definition 12.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS. The adhesion (bi-j-lower approximations, bi-j-upper approximations, bi-j-boundary region and bi-j-accuracy of the approximations) of A is defined, respectively, by d¯bijA=h∈A:dbijh⊆A, d¯bijA=h∈U:dbijh∩A≠∅, BbijA=d¯bijA−d¯bijA, and γbijA=d¯bijA/d¯bijA, where d¯bijA≠0,∀A⊆U, and j∈Λ.Example 8.
In Example2, let A=ω,υ,m. Then, d¯bir12A=ω,υ,m,d¯bil12A=ω,υ,m,d¯bir12∨l12A=ω,υ,d¯bir12∧l12A=ω,υ,m. d¯bir12A=ω,υ,m,d¯bil12A=ω,υ,m,d¯bir12∨l12A=U,d¯bir12∧l12A=ω,υ,m. Bbir12A=∅,Bbil12A=∅,Bbir12∨l12A=ψ,m,Bbir12∧l12A=∅. γbir12A=1,γbil12A=1,γbir12∨l12A=1/2,γbir12∧l12A=1.Definition 13.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS. The bi-j-lower, bi-j-upper approximations, bi-j-boundary region, and bi-j-accuracy of the approximations of A are defined, respectively, by ℒbijA=h∈A:Nbijh⊆A∨Obijh⊆A, UbijA=h∈U:Nbijh∩A≠∅∧Obijh∩A≠∅, BbijA=UbijA−ℒbijA, and ζbijA=ℒbijA/UbijA where UbijA≠0,∀A⊆U, and j∈Λ.Example 9.
In Example1, let A=ω,υ,m. Then, ℒbir12A=ω,υ,m∪υ=ω,υ,m, ℒbil12A=ω,υ∪m=ω,υ,m, ℒbir12∨l12A=ω,υ∪∅=ω,υ, ℒbir12∧l12A=ω,υ,m∪υ,m=ω,υ,m. Ubir12A=U∩ω,υ,m=ω,υ,m,Ubil12A=U∩U=U,Ubir12∨l12A=U∩U=U and Ubir12∧l12A=Ucapω,υ,m=ω,υ,m. Bbir12A=ω,υ,m−ω,υ,m=∅,Bbil12A=U−ω,υ,m=ψ,Bbir12∨l12A=U−ω,υ=ψ,m and Bbir12∧l12A=ω,υ,m−ω,υ,m=∅. ζbir12A=1,ζbil12A=3/4,ζbir12∨l12A=1/2 and ζbir12∧l12A=1.Definition 14.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS. The bi-j-lower, bi-j-upper approximations, bi-j-boundary region, and bi-j-accuracy of the approximations of A are defined, respectively, by ℜ¯bijA=h∈A:Nbijh⊆A∧Obijh⊆A, ℜ¯bijA=h∈A:Nbijh∩A≠∅∨Obijh∩A≠∅, and BbijA=ℜ¯bijA−ℜ¯bijA and ξbijA=ℜ¯bijA/ℜ¯bijA, where ℜ¯bijA≠0,∀A⊆U, and j∈Λ.Example 10.
In Example1, let A=ω,υ,m. ℜ¯bir12A=ω,υ,m∩υ=υ,ℜ¯bil12A=ω,υ∩m=∅,ℜ¯bir12∨l12A=ω,υ∩∅=∅ and ℜ¯bir12∧l12A=ω,υ,m∩υ,m=υ,m. ℜ¯bir12A=U∪ω,υ,m=U,ℜ¯bil12A=U∪U=U,ℜ¯bir12∨l12A=U∪U=U and ℜ¯bir12∧l12A=U∪ω,υ,m=U. Bbir12A=U−υ=ω,ψ,m,Bbil12A=U−∅=U,Bbir12∨l12A=U−∅=U and Bbir12∧l12A=U−υ,m=ω,ψ. ξbir12A=1/4,ξbil12A=0,ξbir12∨l12A=0, and ξbir12∧l12A=1/2.Theorem 1.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS. Then, for every j∈Λ and A⊆U, we have(1)
d¯bijA⊆UbijA⊆N¯bijA⊆ℜ¯bijA.(2)
d¯bijA⊆UbijA⊆O¯bijA⊆ℜ¯bijA.(3)
ℜ¯bijA⊆O¯bijA⊆ℒbijA⊆d¯bijA.(4)
ℜ¯bijA∪N¯bijA=ℒbijA⊆d¯bijA.(5)
O¯bijA∪N¯bijA=ℜ¯bijA.(6)
O¯bijA∩N¯bijA=UbijA.(7)
O¯bijA∩N¯bijA=L¯bijA.(8)
O¯bijA∩N¯bijA=ℛ¯bijA.Proof.
(1) Leth∈d¯bijA. Then, dbijh∩A≠∅, but dbijA⊆NbijA and dbijA⊆ObijA by Lemma 2. Then, Nbijh∩A≠∅ and Obijh∩A≠∅. Therefore, h∈UbijA. Thus, d¯bijA⊆UbijA. Now, let h∈UbijA. Then, Nbijh∩A≠∅, and so, h∈N¯bijA. Thus, UbijA⊆N¯bijA. Also, let h∈N¯bijA which implies Nbijh∩A≠∅. Then, h∈ℜ¯bijA. Hence, N¯bijA⊆ℜ¯bijA. Proofs of (2)–(4) are similar to (1). (5) h∈O¯bijA∪N¯bijA iff Obijh∩A≠∅∨Nbijh∩A≠∅ iff h∈ℜ¯bijA. Thus, O¯bijA∪N¯bijA=ℜ¯bijA. Proofs of (6)–(8) are similar to (5).Corollary 2.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS and A⊆U. Then, ∀j∈Λ:(1)
ξbijA⩽αbijA⩽ζbijA⩽γbijA.(2)
ξbijA⩽βbijA⩽ζbijA⩽γbijA.Example 11.
From Examples1, 7–10, Table 6 is given, and Corollary 2 will be applied.Table 6
Different kinds of bi-j-lower approximations of A.
j∈ΛξbijAαbijAβbijAζbijAγbijAr121/43/41/311l1201/21/43/41r12∨l1201/201/21/2r12∧l121/23/42/311Remark 4.
(1) By Corollary2, the accurate approximation is d¯bijA,d¯bijA, since γbijA; is the greatest value among the accuracies ξbijA,αbijA,βbijA, and ζbijA; in Example 11.
(2) Tables7–10 emphasize Corollary 2.
(3) In Table10, ζbir12∧l12; is identical to γbir12∧l12; which has the largest accurate approximations comparing with the other accuracies of the approximations.
(4) Table10 has most accurate approximations comparing with Tables 7–9 as αbir12∧l12,βbir12∧l12,ζbir12∧l12,ξbir12∧l12; and γbir12∧l12; have great values in Table 10 comparing with the accuracies of the approximations when j=r12,l12,r12∨l12.Table 7
Comparison between the accuracies forj=r12∨l12.
AProcess of Definition8Process of Definition11Process of Definition13Process of Definition14Process of Definition12N¯bir12∨l12N¯bir12∨l12Bbir12∨l12αbir12∨l12O¯bir12∨l12O¯bir12∨l12Bbir12∨l12βbir12∨l12ℒbir12∨l12Ubir12∨l12Bbir12∨l12ζbir12∨l12ℜ¯bir12∨l12ℜ¯bir12∨l12Bbir12∨l12ξbir12∨l12d¯bir12∨l12d¯bir12∨l12Bbir12∨l12γbir12∨l12ωωUψ,υ,m1/4∅ωω0ωω∅1∅UU0ωω∅1ψ∅ψ,mψ,m0∅UU0∅ψ,mψ,m0∅UU0∅ψ,mψ,m0υ∅ψ,υ,mψ,υ,m0∅ω,υ,mω,υ,m0∅υ,mυ,m0∅UU0υυ∅1m∅ψ,υ,mψ,υ,m0∅UU0∅ψ,υ,mψ,υ,m0∅UU0∅ψ,mψ,m0ω,ψωUψ,υ,m1/4∅UU0ωUψ,υ,m1/4∅UU0ωω,ψ,mψ,m1/3ω,υωUψ,υ,m1/4∅ω,υ,mω,υ,m0ωω,υ,mυ,m1/3∅UU0ω,υω,υ∅1ω,mωUψ,υ,m1/4∅UU0ωUψ,υ,m1/4∅UU0ωω,ψ,mψ,m1/3ψ,υ∅ψ,υ,mψ,υ,m0∅UU0∅ψ,υ,mψ,υ,m0∅UU0υψ,υ,mψ,m1/3ψ,m∅ψ,υ,mψ,υ,m0ψUω,υ,m1/4ψψ,υ,mυ,m1/3∅UU0ψ,mψ,m∅1υ,m∅ψ,υ,mψ,υ,m0∅UU0∅ψ,υ,mψ,υ,m0∅UU0υψ,υ,mψ,m1/3ω,ψ,υωUψ,υ,m1/4∅UU0ωUψ,υ,m1/4∅UU0ω,υUψ,m1/2ω,ψ,mωUψ,υ,m1/2ψUω,υ,m1/4ω,ψUυ,m1/2∅UU0ω,ψ,mω,ψ,m∅1ω,υ,mω,υUψ,m1/2∅UU0ω,υUψ,m1/2∅UU0ω,υUψ,m1/2ψ,υ,m∅ψ,υ,mψ,υ,m0ψ,υ,mUω3/4ψ,υ,mψ,υ,m∅1∅UU0ψ,υ,mψ,υ,m∅1UUU∅1UU∅1UU∅1UU∅1UU∅1Table 8
Comparison between the accuracies forj=r12.
AProcess of Definition8Process of Definition11Process of Definition13Process of Definition14Process of Definition12N¯bir12N¯bir12Bbir12αbir12O¯bir12O¯bir12Bbir12βbir12ℒbir12Ubir12Bbir12ζbir12ℜ¯bir12ℜ¯bir12Bbir12ξbir12d¯bir12d¯bir12Bbir12γbir12ωωUψ,υ,m1/4∅ωω0ωω∅1∅UU0ωω∅1ψ∅ψψ0ψω,ψ,mω,m1/3ψψ∅1∅ω,ψ,mω,ψ,m0ψψ∅1υ∅υυ0υω,υ,mω,m1/3υυ∅1∅ω,υ,mω,υ,m0υυ∅1m∅ψ,υ,mψ,υ,m0∅ω,mω,m0∅mm0∅UU0mm∅1ω,ψωUψ,υ,m1/4ψω,ψ,mω,m1/3ω,ψω,ψ,mm2/3∅UU0ω,ψω,ψ∅1ω,υωUψ,υ,m1/4υω,υ,mω,m1/3ω,υω,υ,mm2/3∅UU0ω,υω,υ∅1ω,mω,mUψ,υ1/2∅ω,mω,m0ω,mω,m∅1∅UU0ω,mω,m∅1ψ,υ∅ψ,υψ,υ0ψ,υUω,m1/2ψ,υψ,υ∅1∅UU0ψ,υψ,υ∅1ψ,m∅ψ,υ,mψ,υ,m0ψω,ψ,mω,m1/3ψψ,mm1/2∅UU0ψ,mψ,m∅1υ,m∅ψ,υ,mψ,υ,m0υω,υ,mω,m1/3υυ,mm1/2∅UU0υ,mυ,m∅1ω,ψ,υωUψ,υ,m1/4ψ,υUω,m1/2ω,ψ,υUm3/4∅UU0ω,ψ,υω,ψ,υ∅1ω,ψ,mω,ψ,mUυ3/4ψω,ψ,mω,m1/3ω,ψ,mω,ψ,m∅1ψUω,υ,m1/4ω,ψ,mω,ψ,m∅1ω,υ,mω,υ,mUψ3/4υω,υ,mω,m1/3ω,υ,mω,υ,m∅1υUω,ψ,m1/4ω,υ,mω,υ,m∅1ψ,υ,m∅ψ,υ,mψ,υ,m0ψ,υ,mUω3/4ψ,υ,mψ,υ,m∅1∅UU0ψ,υ,mψ,υ,m∅1UUU∅1UU∅1UU∅1UU∅1UU∅1Table 9
Comparison between the accuracies forj=l12.
AProcess of Definition8Process of Definition11Process of Definition13Process of Definition14Process of Definition12N¯bil12N¯bil12Bbil12αbil12O¯bil12O¯bil12Bbil12βbil12ℒbil12Ubil12Bbil12ζbil12ℜ¯bil12ℜ¯bil12Bbil12ξbil12d¯bil12d¯bil12Bbil12γbil12ωωω,ψ,mψ,m1/3∅ωω0ωω∅1∅ω,ψ,mω,ψ,m0ωω∅1ψ∅ψ,mψ,m0∅ω,ψ,υω,ψ,υ0∅ψψ0∅UU0ψψ∅1υυψ,υ,mψ,m1/3∅υυ0υυ∅1∅ψ,υ,mψ,υ,m0υυ∅1m∅mm0mUω,ψ,υ1/4mm∅1∅UU0mm∅1ω,ψωω,ψ,mψ,m1/3∅ω,ψ,υω,ψ,υ0ωω,ψψ1/2∅UU0ω,ψω,ψ∅1ω,υω,υUψ,m1/2∅ω,υω,υ0ω,υω,υ∅1∅UU0ω,υω,υ∅1ω,mωω,ψ,mψ,m1/3mUω,ψ,υ1/4ω,mω,ψ,mψ2/3∅UU0ω,mω,m∅1ψ,υυψ,υ,mψ,m1/3∅ω,ψ,υω,ψ,υ0υψ,υψ1/2∅UU0ψ,υψ,υ∅1ψ,m∅ψ,mψ,m0ψ,mUω,υ1/2ψ,mψ,m∅1∅UU0ψ,mψ,m∅1υ,mυψ,υ,mψ,m1/3mUω,ψ,υ1/4υ,mψ,υ,mψ2/3∅UU0υ,mυ,m∅1ω,ψ,υω,ψ,υUm3/4∅ω,ψ,υω,ψ,υ0ω,ψ,υω,ψ,υ∅1∅UU0ω,ψ,υω,ψ,υ∅1ω,ψ,mωω,ψ,mψ,m1/3ω,ψ,mUυ3/4ω,ψ,mω,ψ,m∅1ωUψ,υ,m1/4ω,ψ,mω,ψ,m∅1ω,υ,mω,υUψ,m1/2mUω,ψ,υ1/4ω,υ,mUψ3/4∅UU0ω,υ,mω,υ,m∅1ψ,υ,mυψ,υ,mψ,m1/3ψ,υ,mUω3/4ψ,υ,mψ,υ,m∅1υUω,ψ,m1/4ψ,υ,mψ,υ,m∅1UUU∅1UU∅1UU∅1UUU1UU∅1Table 10
Comparison between the accuracies forj=r12∧l12.
AProcess of Definition8Process of Definition11Process of Definition13Process of Definition14Process of Definition12N¯bir12∧l12N¯bir12∧l12Bbir12∧l12αbir12∧l12O¯bir12∧l12O¯bir12∧l12Bbir12∧l12βbir12∧l12ℒbir12∧l12Ubir12∧l12Bbir12∧l12ζbir12∧l12ℜ¯bir12∧l12ℜ¯bir12∧l12Bbir12∧l12ξbir12∧l12d¯bir12∧l12d¯bir12∧l12Bbir12∧l12γbir12∧l12ωωω,ψ,mψ,m1/3∅ωω0ωω∅1∅ω,ψ,mω,ψ,m0ωω∅1ψ∅ψψ0ψω,ψω1/2ψψ∅1∅ω,ψω,ψ0ψψ∅1υυυ∅1υυ∅1υυ∅1υυ∅1υυ∅1m∅mm0mω,mω1/2mm∅1∅ω,mω,m0mm∅1ω,ψω,ψω,ψ,mm2/3ψω,ψω1/2ω,ψω,ψ∅1ψω,ψ,mω,m1/3ω,ψω,ψ∅1ω,υω,υUψ,m1/2υω,υω1/2ω,υω,υ∅1υUω,ψ,m1/4ω,υω,υ∅1ω,mω,mω,ψ,mψ2/3mω,mω1/2ω,mω,m∅1mω,ψ,mω,ψ1/4ω,mω,m∅1ψ,υυψ,υψ1/2ψ,υω,ψ,υω2/3ψ,υψ,υ∅1υω,ψ,υω,ψ1/3ψ,υψ,υ∅1ψ,m∅ψ,mψ,m0ψ,mω,ψ,mω2/3ψ,mψ,m∅1∅ω,ψ,mω,ψ,m0ψ,mψ,m∅1υ,mυυ,mm1/2υ,mω,υ,mω2/3υ,mυ,m∅1υω,υ,mω,m1/3υ,mυ,m∅1ω,ψ,υω,ψ,υUm3/4ψ,υω,ψ,υω2/3ω,ψ,υω,ψ,υ∅1ψ,υUω,m1/2ω,ψ,υω,ψ,υ∅1ω,ψ,mω,ψ,mω,ψ,m∅1ω,ψ,mω,ψ,m∅1ω,ψ,mω,ψ,m∅1ω,ψ,mω,ψ,m∅1ω,ψ,mω,ψ,m∅1ω,υ,mω,υ,mUψ3/4υ,mω,υ,mω2/3ω,υ,mω,υ,m∅1υ,mUω,ψ1/2ω,υ,mω,υ,m∅1ψ,υ,mυψ,υ,mψ,m1/3ψ,υ,mUω3/4ψ,υ,mψ,υ,m∅1υUω,ψ,m1/4ψ,υ,mψ,υ,m∅1UUU∅1UU∅1UU∅1UU∅1UU∅1


## 5. Approximation Operators in Generalized Bicovering Approximation Space

Here, we prove propertiesL1-H9 in Proposition 1 on different types of approximation operators in generalized bicovering approximation space, and some opposite examples are presented.Proposition 7.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS and ℋ⊆U. Then, properties L1,H1,L2,H2,L3, and H3 in Proposition 1 are satisfied for operators N¯bij,O¯bij,ℒbij,ℜ¯bij,d¯bij,N¯bij,O¯bij,Ubij,ℜ¯bij, and d¯bij,∀j∈Λ.Proof.
The proof follows directly from Definitions8, 11–14.Proposition 8.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS and ℋ,T⊆U. Then, the properties L4 and H4 are satisfied for the operators N¯bij,O¯bij,d¯bij,N¯bij,O¯bij, and d¯bij.Proof.
We proveL4 for the operator N¯bij and remainder proofs are similar. Now, we will prove that N¯bijℋ∩T=N¯bijℋ∩N¯bijT. Let h∈N¯bijℋ∩T. Then, h∈ℋ∩T∋Nbijh⊆ℋ∩T. So, h∈ℋ∧h∈H∋Nbijh⊆ℋ∧Nbijh⊆T. That is h∈ℋ∋Nbijh⊆ℋ∧h∈T∋Nbijh⊆T. Therefore, h∈N¯bijℋ∧h∈N¯bijT. So, h∈N¯bijℋ∩N¯bijT. Hence, N¯bijℋ∩T⊆N¯bijℋ∩N¯bijT. Now, let h∈N¯bijℋ∩N¯bijT. Then, h∈N¯bijℋ∧h∈N¯bijT. So, h∈ℋ∋Nbijh⊆ℋ∧h∈H∋Nbijh⊆T. Therefore, h∈ℋ∧h∈T∋Nbijh⊆ℋ∧Nbijh⊆T. That is h∈ℋ∩T∋NNbijh⊆ℋ∩T. Then, h∈N¯bijℋ∩T. Hence, N¯Nbijℋ∩N¯bijT⊆N¯Nbijℋ∩T.Remark 5.
The propertiesL4 and H4 are not satisfied, in general, for the operators ℒbij and Ubij,∀j∈r12,l12,r12∨l12.Example 12.
From Tables7–9, we get ℒbir12ω,ψ,m∩ψ,υ,m=ℒbir12ψ,m=ψ≠ψ,m=ω,ψ,m∩ψ,υ,m=ℒbir12ω,ψ,m∩ℒbir12ψ,υ,m. ℒbil12ω,ψ,υ∩ω,ψ,m=ℒbil12ω,ψ=ω≠ω,ψ=ω,ψ,m∩ω,ψ,m=ℒbil12ω,ψ,υ∩ℒbil12ω,ψ,m. ℒbir12∨l12ω,υ,m∩ψ,υ,m=ℒbir12∨l12υ,m=∅≠ω,υ=ω,υ∩ψ,υ,m=ℒbir12∨l12ω,υ,m∩ℒbir12∨l12ψ,υ,m. Also, by Tables 7–9, we get Ubir12ψ,υ∪ω=Ubir12ω,ψ,υ=U≠ω,ψ,υ=ω∪ψ,υ=Ubir12ω∪Ubir12ψ,υ. Ubil12m∪ω,υ=Ubil12ω,υ,m=U≠ω,υ,m=m∪ω,υ=Ubil12m∪Ubil12ω,υ. Ubir12∨l12ω∪ψ=Ubir12∨l12ω,ψ=U≠ω,ψ,m=ω∪ψ,m=Ubir12∨l12ω∪Ubir12∨l12ψ.Proposition 9.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS and ℋ,T⊆U. Then, the properties L5, and H5 are satisfied for the operators N¯bij,O¯bij,d¯bij,ℒbij,ℜ¯bij,N¯bij,O¯bij,d¯bij,Ubij, and ℜ¯bij.Proof.
We proveL5 for the operator N¯bij and other proofs are similar. Now, if ℋ⊆T, then N¯bijℋ⊆N¯bijT. Suppose that h∈N¯bijℋ. Then, h∈ℋ⊆T and so Nbijh⊆ℋ⊆T, that is, h∈N¯bijT. Hence, N¯bijℋ⊆N¯bijT.Proposition 10.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS and ℋ,T⊆U. Then, the properties L6 and H6 are satisfied for the operators N¯bij,O¯bij,d¯bij,ℒbij,ℜ¯bij,N¯bij,O¯bij,d¯bij,Ubij and ℜ¯bij.Proof.
We proveL6 for N¯bij and remainder proofs are similar. Now, we prove that N¯bijℋ∪N¯bijT⊆N¯bijℋ∪T. Suppose that h∈N¯bijℋ∪N¯bijT. Then, h∈N¯bijℋ∨h∈N¯bijT. So, h∈ℋ∋Nbijh⊆ℋ∨h∈H∋Nbijh⊆T. Then, h∈ℋ∨h∈T∋Nbijh⊆ℋ∨Nbijh⊆T. That is h⊆ℋ∪T∋Nbijh⊆ℋ∪T. Therefore, h∈N¯bijℋ∪T.Proposition 11.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS and ℋ,T⊆U. Then, the properties L6 and H6 are satisfied for the operators N¯bij,O¯bij,d¯bij,ℒbij,ℜ¯bij,N¯bij,O¯bij,d¯bij,Ubij and ℜ¯bij.Proof.
We proveL7 for the operator N¯bij and remainder proofs are similar. Now, if ℋc is a complement of ℋ in U, then N¯bijℋc=N¯bijℋc. Suppose that h∈N¯bijℋc. Then, h∈h∈ℋc∋Nbijh⊆ℋc. So, h∈U∋Nbijh∩ℋc≠∅. That is h∈h∈U∋Nbijh∩ℋ=∅. Therefore, h∈h∈U∋Nbijh∩ℋ≠∅c. Then, h∈N¯bijℋc. Hence, N¯bijℋc⊆N¯bijℋc. Now, suppose that h∈N¯bijℋc. Then, h∈h∈U∋Nbijh∩ℋ≠∅c. So, h∈h∈U∋Nbijh∩ℋ=∅. That is h∈h∈ℋc∋Nbijh⊆ℋc. Therefore, h∈N¯bijℋc. Hence, N¯bijℋc⊆N¯bijℋc.Proposition 12.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS and ℋ,T⊆U. Then, properties L8 and H8 are satisfied for the operators N¯bij,O¯bij,d¯bij,ℒbij,N¯bij,O¯bij,d¯bij and Ubij.Proof.
We proveL8 for operator N¯bij, and the rest is similar. Now, we will prove that N¯bijN¯bijℋ=N¯bijℋ. Suppose that h∈N¯bijN¯bijℋ=h∈N¯bijℋ:Nbijh⊆N¯bijℋ, =h∈h∈ℋ:Nbijh⊆ℋ:Nbijh⊆h∈ℋ:Nbijh⊆ℋ=h∈ℋ:Nbijh⊆ℋ=N¯bijℋ.Remark 6.
The propertiesL8 and H8 are not satisfied, in general, for operators ℜ¯bij and ℜ¯bij,∀j∈r12,l12,r12∨l12.Example 13.
From Tables7 and 8, we get ℜ¯bir12ℜ¯bir12ω,ψ,m=ℜ¯bir12ψ=∅≠ψ=ℜ¯bir12ω,ψ,m. ℜ¯bil12ℜ¯bil12ψ,υ,m=ℜ¯bil12υ=∅≠υ=ℜ¯bil12ψ,υ,m. ℜ¯bir12ℜ¯bir12ψ=ℜ¯bir12ω,ψ,m=U≠ω,ψ,m=ℜ¯bir12ψ. ℜ¯bil12ℜ¯bil12υ=ℜ¯bil12ψ,υ,m=U≠ψ,υ,m=ℜ¯bil12υ.
Example14 illustrates that L8 and H8 are not satisfied neither for ℜ¯bir12∨l12 nor for ℜ¯bir12∨l12.Example 14.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS, where U=ω,ψ,υ,m and ℛ1=ω,ω,ψ,ψ,υ,υ,υ,ψ,υ,m,m,ω,ℛ2=ω,ω,ψ,ψ,υ,υ,ψ,υ,υ,ψ,υ,m,m,ω. The ℛ− right and ℛ− left neighbourhoods in U for relations ℛ1 and ℛ2 are in Table 11.
The covers according to Definitions4 and 6 are Cr1=ω,ψ,ψ,υ,m; Cr2=ω,ψ,υ,ψ,υ,m; Cl1=ω,m,ψ,υ,υ; Cl2=ω,m,ψ,υ,υ; Cr12=ω,ψ,ψ,υ,m,ψ,υ; and Cl12=ω,m,ψ,υ,υ. The bi-j-neighbourhoods and complementary bi-j-neighbourhoods are presented in Tables 12 and 13, respectively.
Now, from Table14, we haveN¯bir12∨l12ω,ψ,m=ω;O¯bir12∨l12ω,ψ,m=ω,m;N¯bir12∨l12ω=∅;O¯bir12∨l12ω=∅;ℜ¯bir12∨l12ω,ψ,m=ω; andℜ¯bir12∨l12ω=∅.
Also, we haveN¯bir12∨l12υ=ψ,υ,m;O¯bir12∨l12υ=ψ,υ;N¯bir12∨l12ψ,υ,m=U;O¯bir12∨l12ψ,υ,m=U;ℜ¯bir12∨l12υ=ψ,υ,m; andℜ¯bir12∨l12ψ,υ,m=U.
It is noted thatL8ℜ¯bir12∨l12ℜ¯bir12∨l12ω,ψ,m=ℜ¯bir12∨l12ω=∅≠ω=ℜ¯bir12∨l12ω,ψ,m, andH8ℜ¯bir12∨l12ℜ¯bir12∨l12υ=ℜ¯bir12∨l12ψ,υ,m=U≠ψ,υ,m=ℜ¯bir12∨l12υ.Table 11
Theℛ-right and ℛ-left neighbourhoods in U for the two relations.
h∈Uωψυmhℛ1{ω}{ψ}{ψ,υ,m}{ω}ℛ1h{ω,m}{ψ,υ}{υ}{υ}hℛ2{ω}{ψ,υ}{ψ,υ,m}{ω}ℛ2hω,m{ψ,υ}{ψ,υ}υTable 12
The bi-j-neighbourhoods of h∈U.
h∈UωψυmNbir12h{ω}{ψ}{ψ,υ}{ψ,υ,m}Nbil12hω,mψ,υ{υ}ω,mNbir12∨l12h{ω,m}ψ,υψ,υUNbir12∧l12h{ω}ψ{υ}mTable 13
The complementary bi-j-neighbourhoods of h∈U.
h∈UωψυmObir12hω{ψ,υ,m}{υ,m}{m}Obil12hω,mψ{ψ,υ}ω,mObir12∨l12hω,mψ,υ,m{ψ,υ,m}{ω,m}Obir12∧l12h{ω}ψ{υ}mTable 14
Comparison between the Definitions8 and 11 for j=r12∧l12 and j=r12∨l12.
AMethods of Definitions8 and 11N¯bir12∧l12AN¯bir12∧l12AO¯bir12∧l12AO¯bir12∧l12AN¯bir12∨l12AN¯bir12∨l12AO¯bir12∨l12AO¯bir12∨l12Aωωωωω∅ω,m∅ω,mψψψψψ∅ψ,υ,m∅ψ,υυυυυυ∅ψ,υ,m∅ψ,υmmmmm∅ω,m∅Uω,ψω,ψω,ψω,ψω,ψ∅U∅Uω,υω,υω,υω,υω,υ∅U∅Uω,mω,mω,mω,mω,mωω,mω,mUψ,υψ,υψ,υψ,υψ,υψ,υψ,υ,m∅ψ,υψ,mψ,mψ,mψ,mψ,m∅U∅Uυ,mυ,mυ,mυ,mυ,m∅U∅Uω,ψ,υω,ψ,υω,ψ,υω,ψ,υω,ψ,υψ,υU∅Uω,ψ,mω,ψ,mω,ψ,mω,ψ,mω,ψ,mωUω,mUω,υ,mω,υ,mω,υ,mω,υ,mω,υ,mωUω,mUψ,υ,mψ,υ,mψ,υ,mψ,υ,mψ,υ,mψ,υUψ,υUUUUUUUUUUProposition 13.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS and ℋ,T⊆U. Then, the properties L9 and H9 are satisfied for the operators d¯bij and d¯bij.Proof.
Directly from Definition12.Remark 7.
L9is not satisfied for N¯bij,O¯bij,ℒbij and ℜ¯bij∀j∈Λ except for ℒbir12∨l12A. Also, H9 is not satisfied for the operators N¯bij,O¯bij,Ubij and ℜ¯bij∀j∈Λ except for Ubir12∧l12. Counterexamples can be found in Tables 7–10.
In Table15, symbol — means “satisfying,” and × means “not satisfying.”
The neighbourhoodsNj,Oj and dj induce different kinds topologies which were generated by Nawar et al. [18]. Here, we use Nbij,Obij and dbij to generate another different topologies that are suitable for applications.Table 15
Different properties on approximation operators.
N¯bijO¯bijℒbijℜ¯bijd¯bijL1—————L2—————L3—————L4——××—L5—————L6—————L7—————L8———×—L9××××—N¯bijO¯bijUbijℜ¯bijd¯bijH1—————H2—————H3—————H4——××—H5—————H6—————H7—————H8———×—H9××××—Theorem 2.
LetU,ℛ1,ℛ2,Cr12,Cl12 be a GBCAS and A⊆U. Then, for each j∈Γ:(1)
τbi1j=A⊆U:Nbijh⊆A∀h∈A,(2)
τbi2j=A⊆U:Obijh⊆A∀h∈A,(3)
τbi3j=A⊆U:dbijh⊆A∀h∈A,(4)
τbi4j=A⊆U:Nbijh∪Obijh⊆A∀h∈A,(5)
τbi5j=A⊆U:Nbijh∩Obijh⊆A∀h∈A are topologies on U.Proof.
It is clear that∅ and U∈τbi1j. Suppose that As:s∈S is a family of elements in τbi1j and h∈∪sAs. Then, ∃s0∈S s.t. h∈As0. Thus, Nbijh⊆As0 which implies Nbijh⊆∪sAs and so ∪sAs∈τbi1j. Finally, suppose that A,ℬ∈τbi1j and h∈A∩ℬ. Then, h∈A and h∈ℬ which implies Nbijh⊆A and Nbijh⊆ℬ. Thus, Nbijh⊆A∩ℬ, and hence, A∩ℬ∈τbi1j. Proofs of (2)–(5) are similar.


## 6. An Application (Rheumatic Fever (RF))

A topological application of data information reduction and RF information will use in this study. Despite of having the same diagnosisD, however, from one person to another, there are many different symptoms of RF disease. We have five patients of RF. RF usually begins at an early age and remains with the patient until his death. In Table 16 [18], the RF datasets describe the adjectives as follows: sex (S) =M,F=s1,s2, pharyngitis (P) =not suffer, suffers=p1,p2, arthritis (A) =not suffer, started at knee, started in ankle = a1,a2,a3, carditis (C) =not affected, affected=c1,c2, chorea (Ch) =not suffer, suffers=ch1,ch2, ESR (E) =elevated,normal=e1,e2, abdominal pain (AP) =not exist,exists=ap1,ap2, and headache (He) =not exist,exists=he1,he2. The decision adjective is diagnosis D=rheumatic carditisrc,rheumatic arthritisra,rheumatic carditis and arthritisrca=d1,d2,d3. Characterizing the five patients by eight adjectives makes the decision adjective for patients in Table 17. Table 18 contains coding (RF) data, by using the adjectives in Table 17. Table 19 shows the RF data in multivalued information system MIS.Table 16
Rheumatic fever (RF) data.
AdjectiveAdjective valuesAdjective refers to(S)s1Ms2F(P)p1Not sufferp2Suffers(A)a1Not suffera2Started at kneea3Started in ankle(C)c1Not affectedc2Affected(Ch)ch1Not sufferch2Suffers(E)e1Elevatede2Normal(AP)ap1Existsap2Not exist(He)he1Not existhe2ExistsDd1(rc)d2(ra)d3(rca)Table 17
RF information system.
(S)(P)(A)(C)(Ch)(E)(AP)(He)Dl1s1p1a1c2ch1e2ap1he1d1l2s1p2a2c1ch1e2ap2he1d2l3s2p2a3c2ch1e2ap2he1d3l4s2p2a2c2ch2e2ap2he1d3l5s1p2a2c2ch2e1ap2he2d3Table 18
Coding RF data.
Adj. symbolSymptomsAdj. valuesLabelsαS,Chα1S⟶s1α2Ch⟶ch2α3Each ofS,Ch⟶s2,ch1βP,A,Eβ1P⟶p2β2A⟶a2β3A⟶a3β4E⟶e2β5Each ofP,A,E⟶p1,a1,e1δC,AP,Heδ1C⟶c2δ2AP⟶ap2δ3He⟶he2δ4Each ofC,AP,He⟶c1,ap1,he1DDiagnosisd1(rc)d2(ra)d3(rca)d1,d2Table 19
RF data in MIS.
PatientsαβδDl1α1β4δ1d2l2α1β1,β2,β4δ2d1l3α3β1,β3,β4δ1,δ2d3l4α2β1,β2,β4δ1,δ2d3l5α1,α2β1,β2δ1,δ2,δ3d3The following binary relations are defined byℛ1a=l,t:fal⊆fat,∀a∈Ad and ℛ2a=l,t:fal≤fat,∀a∈Ad, where Ad=α,β,δ is the set of condition adjectives. ℛ1ℬ=l,t:fal⊆fat,∀a∈ℬ,ℬ⊆Ad and ℛ2ℬ=l,t:fal≤fat,∀a∈ℬ,ℬ⊆Ad. Clearly, ℛ1a,ℛ2a,ℛ1ℬ and ℛ2ℬ are dominance but not symmetric. We can consider in general that a is an element of the power set of the set of condition adjectives α,β,δ. ℛ1D=l,t:fDl⊆fDt and ℛ2D=l,t:fDl≤fDt are nonequivalence relations, where D is the decision adjective.In Table19, let U=l1,l2,l3,l4,l5 be the set of objects and Θ=α,β,δ,αβ,αδ,βδ,αβδ,D. By constructing the biright coverings and bileft coverings using biright coverings (briefly, bi-r-cover) Cbir12J=Cr1J∪Cr2J,J∈Θ; and bileft coverings (briefly, bi-l-cover) Cbil12J=Cl1J∪Cl2J,J∈Θ.After, we create tables ofNbijJl,ObijJl,dbijJl,NbijJl∪ObijJl and NbijJl∩ObijJl, where j∈Λ. Next, we get topologies for the adjective α and the bi-r12- neighbourhoods Nbir12l as follows τbir12α=U,∅,l3,l5,l3,l5,l4,l5,l1,l2,l5,l3,l4,l5,l1,l2,l3,l5,l1,l2,l4,l5. The another topologies are given by the same manner.By the above discussion, we observe the following:(1)
The set of adjectivesℬ⊆At is a reduction if τbijℬ≤τbijD, for j∈Λ.(2)
ℬ is a minimal, where τbijℬ≤τbijD iff ∀G1∈τbijℬ,∃G2∈τbijD s.t. G1⊆G2,G1,G2≠U.(3)
ForNbir12l, we get τbir12δ≤τbir12D, and for Obil12l, we get τbil12δ≤τbil12D. Therefore, δ is the only reduction of a given information system.(4)
In Nawar’s study [18], the reduction of its system is either δ or α,β, or α,δ, or β,δ.(5)
Comparing between both results, it is noted that a reduction of our study is more better than Nawar’s reduction. In other words, the symptomδ is the best choice for expert to diagnose the rheumatic fever.


## 7. Conclusions and Discussion

Throughout this study, some kinds of bi-j-neighbourhoods as a generalization of bicovering of Abo-Elhamayel are introduced, and hence, we used them to obtain five kinds of approximation operators in generalized bicovering approximation space. The comparisons between the accuracy of the suggested approximations are imposed. Corollary 2 proves that the accuracy of the introduced approximations d¯bijA,d¯bijA is the best comparing with the other constructed approximations in this work and Abo-Elhamayel one. Further, some properties of the given approximation operators are studied. Moreover, using the bi-j-neighbourhoods, five kinds of topologies are generated. Finally, we applied these topologies making reduction of adjectives in information tables concerning (RF) data.



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