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 -neighbourhoods and -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.
1. Introduction
Covering rough sets has been initiated by Zhu [1] who pointed to the definition of covering of universal discourse ; as any partition of . The covering of ; 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 -operator with a fuzzy covering approximation space to get a generalization of -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 ; of ; 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 --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 generalized –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 ;, and ; 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 ;, and ; and are given in Section 6.
2. Preliminaries
Some fundamental definitions and propositions in -left and -right neighbourhoods, right cover, left cover, right -cover, left -cover, and some neighbourhoods which are necessary for our study are presented in this section.
2.1. Pawlak Approximation Space
Let ; be a finite set, ; be an equivalence relation on ;, and ; denotes to the equivalence class containing . An approximation space is denoted as . The following two definitions present the concepts of lower and upper approximations of a subset ; of ; in approximation space .
Definition 1 (see [13, 14]). Let ; be an approximation space and . The –lower and –upper approximations of ; are defined, respectively, by ; and . Based on lower and upper approximations of ; divided into three disjoint regions. A boundary region (briefly ) is defined by .
Proposition 1 (see [15]). The following properties are for Pawlaks’ rough sets: If , then If , then
2.2. A Generalized Covering Approximation Space
Definition 2 (see [16]). If ; s.t. , then ; is called a binary serial relation on .
Definition 3 (see [17]). ; and ; are, respectively, the definitions of –left and –right neighbourhoods of an element ; in .
Proposition 2 (see [17]). ; (resp., ); if ; (resp., ); is a serial relation on .
Definition 4 (see [10]). Let ; be a domain of discourse and ; be a family of nonempty subsets of . If , then ; is a covering of . For serial relations ; and ; on , right cover (briefly, -cover) is , and left cover (briefly, -cover) is . ; is called a generalized covering approximation space (for short, ).
Definition 5 (see [10]). Let ; be a . -neighbourhoods ; are defined by (1) –neighbourhood: ; (2) –neighbourhood: ; (3) –neighbourhood: ; (4) --neighbourhood: .
2.3. Generalized Bicovering Approximation Space
Definition 6 (see [17]). Let ;, and ; be serial relations on ; and . Right bicover (briefly, --bicover) is , and left bicover (briefly, --bicover) is . ; is called a generalized bicovering approximation space .
Definition 7 (see [17]). Let ; be a . The neighbourhoods ; are defined as follows: (1) -neighbourhood: (2) -neighbourhood: (3) -neighbourhood: (4) -neighbourhood:
Proposition 3 (see [17]). Let ; be a . . .
Proposition 4 (see [17]). Let ; be a . , if , then .
Remark 1 (see [17]). Let ; be a . , if , then .
Definition 8 (see [17]). Let ; be a . , ; and , where ; are, respectively, the definitions of: lower approximation upper approximation boundary region and accuracy of the approximations of .
3. New Kinds of Neighbourhoods on Generalized Bicovering Approximation Space
In this section, new kinds of neighbourhoods are introduced, and their properties are investigated.
Definition 9. Let be a . The complementary neighbourhood of is , and .
Example 1. Let be a , where , and , we have Table 1 from Definition 3.
The covers are given by ; ; ; ; ; and .-neighbourhoods, neighbourhoods, and complementary neighbourhoods are presented, respectively, in Tables 2–4.
Definition 10. Let be a . adhesions of is defined as follows: .
Example 2. The adhesions for in Example 1 is given by Table 5.
Lemma 1. Let be a . If , then .
Proof. Since , from Proposition 3, then by Definition 9, .
Lemma 2. Let be a . Then, and , and we have and .
Proof. Suppose that . Then, , by Definition 10. If , by Proposition 3, then , and so, . Hence, . Similarly, is proved.
The other inclusion for Lemma 2 is not true, in general.
Example 3. From Tables 3 and 5, we get . Also, .
Lemma 3. Let be a . Then, for each and , we have .
Proof. By Lemma 2, we get , and . Then, . Let . Then, and . So, by Definition 9, and . By Proposition 4, we have and . So, . Hence, . Therefore, and .
Remark 2. It is noted that , for some .
Example 4. From Examples 1 and 2, .
The proof of Proposition 5 and its Corollary 1 is clear from Definition 9. So, it is omitted.
Proposition 5. Let be a . Then, for any if and only if .
Corollary 1. If be a , then .
Example 5. In Example 1, take . Then:(1), since and .(2), since .(3), since .(4), since and . Also, (5) , since .(6), since and .(7), since and .(8), since and .Now, the relationship between complementary neighbourhoods is discussed.
Proposition 6. Let be a . Then, for any if , we have .
Proof. Let . Then, . Therefore, .
Remark 3. Proposition 6 is not satisfied for , in general.
Example 6. From Example 1 and Table 4, , but .
4. Some Kinds of Bicovering Rough Sets with Some Relations between Them
In this section, new types of neighbourhoods , complementary neighbourhoods , and adhesions of bicovering rough sets are presented. Furthermore, the relationship between these kinds of approximations is constructed.
Definition 11. Let be a . The complementary (lower approximations, upper approximations, boundary region and accuracy of the approximations) of is defined, respectively, by , and , where , and .
Example 7. In Example 1, let . Then, . . . .
Definition 12. Let be a . The adhesion (lower approximations, upper approximations, boundary region and accuracy of the approximations) of is defined, respectively, by , , , and , where , and .
Example 8. In Example 2, let . Then, . . . .
Definition 13. Let be a . The lower, upper approximations, boundary region, and accuracy of the approximations of are defined, respectively, by , , , and where , and .
Example 9. In Example 1, let . Then, , , , . and . and . and .
Definition 14. Let be a . The lower, upper approximations, boundary region, and accuracy of the approximations of are defined, respectively, by , , and and , where , and .
Example 10. In Example 1, let . and . and . and . , and .
Theorem 1. Let be a . Then, for every and , we have(1).(2).(3).(4).(5).(6).(7).(8).
Proof. (1) Let . Then, , but and by Lemma 2. Then, and . Therefore, . Thus, . Now, let . Then, , and so, . Thus, . Also, let which implies . Then, . Hence, . Proofs of (2)–(4) are similar to (1). (5) iff iff . Thus, . Proofs of (6)–(8) are similar to (5).
Corollary 2. Let be a and . Then, :(1).(2).
Example 11. From Examples 1, 7–10, Table 6 is given, and Corollary 2 will be applied.
Remark 4. (1) By Corollary 2, the accurate approximation is , since ; is the greatest value among the accuracies , and ; in Example 11.
(2) Tables 7–10 emphasize Corollary 2.
(3) In Table 10, ; is identical to ; which has the largest accurate approximations comparing with the other accuracies of the approximations.
(4) Table 10 has most accurate approximations comparing with Tables 7–9 as ; and ; have great values in Table 10 comparing with the accuracies of the approximations when .
5. Approximation Operators in Generalized Bicovering Approximation Space
Here, we prove properties - in Proposition 1 on different types of approximation operators in generalized covering approximation space, and some opposite examples are presented.
Proposition 7. Let be a and . Then, properties , and in Proposition 1 are satisfied for operators , and .
Proof. The proof follows directly from Definitions 8, 11–14.
Proposition 8. Let be a and . Then, the properties and are satisfied for the operators , and .
Proof. We prove for the operator and remainder proofs are similar. Now, we will prove that . Let . Then, . So, . That is . Therefore, . So, . Hence, . Now, let . Then, . So, . Therefore, . That is . Then, . Hence, .
Remark 5. The properties and are not satisfied, in general, for the operators and .
Example 12. From Tables 7–9, we get . . . Also, by Tables 7–9, we get . . .
Proposition 9. Let be a and . Then, the properties , and are satisfied for the operators , and .
Proof. We prove for the operator and other proofs are similar. Now, if , then . Suppose that . Then, and so , that is, . Hence, .
Proposition 10. Let be a and . Then, the properties and are satisfied for the operators and .
Proof. We prove for and remainder proofs are similar. Now, we prove that . Suppose that . Then, . So, . Then, . That is . Therefore, .
Proposition 11. Let be a and . Then, the properties and are satisfied for the operators and .
Proof. We prove for the operator and remainder proofs are similar. Now, if is a complement of in , then . Suppose that . Then, . So, . That is . Therefore, . Then, . Hence, . Now, suppose that . Then, . So, . That is . Therefore, . Hence, .
Proposition 12. Let be a and . Then, properties and are satisfied for the operators and .
Proof. We prove for operator , and the rest is similar. Now, we will prove that . Suppose that , .
Remark 6. The properties and are not satisfied, in general, for operators and .
Example 13. From Tables 7 and 8, we get . . . .
Example 14 illustrates that and are not satisfied neither for nor for .
Example 14. Let be a , where and . The right and left neighbourhoods in for relations and are in Table 11.
The covers according to Definitions 4 and 6 are ; ; ; ; ; and . The neighbourhoods and complementary neighbourhoods are presented in Tables 12 and 13, respectively.
Now, from Table 14, we have ; ; ; ; ; and .Also, we have ; ; ; ; ; and .It is noted that , and .
Proposition 13. Let be a and . Then, the properties and are satisfied for the operators and .
Proof. Directly from Definition 12.
Remark 7. is not satisfied for and except for . Also, is not satisfied for the operators and except for . Counterexamples can be found in Tables 7–10.
In Table 15, symbol means “satisfying,” and means “not satisfying.”
The neighbourhoods and induce different kinds topologies which were generated by Nawar et al. [18]. Here, we use and to generate another different topologies that are suitable for applications.
Theorem 2. Let be a and . Then, for each :(1),(2),(3),(4),(5) are topologies on .
Proof. It is clear that and . Suppose that is a family of elements in and . Then, s.t. . Thus, which implies and so . Finally, suppose that and . Then, and which implies and . Thus, , and hence, . 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 diagnosis , 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) , pharyngitis (P) , arthritis (A) = , carditis (C) , chorea (Ch) , ESR (E) , abdominal pain (AP) , and headache (He) . The decision adjective is diagnosis . 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.
The following binary relations are defined by and , where is the set of condition adjectives. and . Clearly, and are dominance but not symmetric. We can consider in general that is an element of the power set of the set of condition adjectives . and are nonequivalence relations, where is the decision adjective.
In Table 19, let be the set of objects and . By constructing the biright coverings and bileft coverings using biright coverings (briefly, bi-r-cover) ; and bileft coverings (briefly, bi-l-cover) .
After, we create tables of and , where . Next, we get topologies for the adjective and the neighbourhoods as follows . The another topologies are given by the same manner.
By the above discussion, we observe the following:(1)The set of adjectives is a reduction if , for .(2) is a minimal, where iff s.t. .(3)For , we get , and for , we get . Therefore, is the only reduction of a given information system.(4)In Nawar’s study [18], the reduction of its system is either or , or , or .(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 symptom 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 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.
Data Availability
The related data are included within the manuscript.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
The authors completed this study and wrote and approved the final version of the manuscript.