Abstract
Rough set has an important role to deal with uncertainty objects. The aim of this article is to introduce some kinds of generalization for rough sets through minimal neighborhoods using special kinds of binary relations. Moreover, four different types of dual approximation operators will be constructed in terms of minimal neighborhoods. The comparison between these types of approximation operators is discussed. Some new kinds of topological structures induced by minimal neighborhoods are established and some of their properties are studied. Finally, we give a comparison between these topologies that help for determining the major components of COVID-19 infections. In this application, the components of infections help the expert in decision making in medicine.
1. Introduction and Preliminaries
Topology is an area of mathematics that is very important, whose definitions and concepts exist inside branches of mathematics and many applications. Topological spaces and their generalizations [1] are considered as basic definitions in system analysis. Recently, topological structures have been used to study graphs [2]. Also, many researchers suggested topological models in analysis [3], chemistry [4], medicine [5], and physics [6] and for determining COVID-19 [7]. Furthermore, Lashin et al. [8] have generated other topologies using a general binary relation. Some researchers have used a topology to represent structures such as fractals [2] in terms of binary relations.
Rough set theory (RST, for short) is initiated in 1982 by Pawlak [9]. RST is a mathematical method to study the uncertainty data, but not qualitatively. Also, Pawlak studied the relation between topology and its generalization and RST. Minimal structure in topology and rough sets are studied in [10], and many applications are discussed in [11]. The fundamental idea of RST is the common lower and upper approximations, which have been constructed using many types of neighborhoods, such as right and left neighborhoods [12], minimal right [13], and minimal left [14] neighborhoods. New kinds of neighborhoods called -neighborhoods are defined by Al-Shami et al. [15]. Al-Shami has defined new kinds of neighborhoods called -neighborhoods and studied them in medical applications [16]. Also, Al-Shami has defined maximal neighborhoods and studied their characteristics and applications in the medical field [17]. The range of rough sets applications today is far broader than it was in the past, and it has been used in many science and engineering domains, including computer network [18], solution of missing attribute values [19], and medical application [16].
In this paper, some ideas of minimal neighborhoods in topology are integrated. Other types of minimal structures will be investigated from a view point of minimal neighborhoods. We think that topological space will serve as a crucial foundation for modifying information extraction and processing. Several foundational ideas regarding topology and RST are introduced. Finally, the present approximations will be applied to have the best tools for determining the major components of COVID-19 infections. This medical application may be useful for experts in decision making.
Definition 1. (see [1]). If is a family from a nonempty universal set . is a topology on if satisfies the following: (i) and are in ; (ii) if for , then ; and (iii) if , then .
RST emerged out of the necessity to depict subsets of in units of equivalence classes as a partition, which defines a topological structure known as approximation space symbolized by , where is an equivalence relation [20, 21]. To represent the equivalence class containing , we will use . RST has come from the need to represent subsets from the universe set in terms of the equivalence class called approximation space , where is a knowledge about an element of . The equivalence class of is also known as the granules, elementary sets, or blocks.
Definition 2. (see [21]). In with , the lower and upper approximations of are defined by and , respectively.
is divided to three disjoint regions in by using Definition 2; the boundary region (briefly ), the positive region (briefly, ), and the negative region (briefly ) are defined by , , and , respectively.
Definition 3. (see [9]). If is an approximation space with , then the accuracy of is defined by , where and denotes the cardinality.
The definition of approximations uses the concept of knowledge granules. is the set of all granules that exist in , is the set of all granules that have a nonempty intersection with , and is the set of all granules that exist in the upper approximation and do not exist in the lower approximation. These definitions of approximation operators were given by Pawlak [9].
In , if , then each of the following is true [21], where is the complement.
Definition 4. (see [22]). The binary relation is called(i)serial: , s.t , (ii)inverse serial: , s.t , (iii)reflexive: , (iv)symmetric: , if , then
Definition 5. (see [23]). Let be a binary relation and . Then, and are called right neighborhood and left neighborhood of , respectively.
Definition 6. (see [24]). If is a binary relation and , then minimal right neighborhood is .
2. Generalization for Rough Sets Based on Minimal Neighborhood Systems
This section aims to investigate RST based on the minimal neighborhood system. Four types of approximation operators are generated. In addition, a comparison between our study and another approach is investigated.
Definition 7. If is a binary relation, then four kinds of minimal neighborhood systems are defined as follows:(i)Minimal right neighborhood (ii)Minimal left neighborhood (iii)Minimal of union neighborhood (iv)Minimal of intersection neighborhood
Remark 8. It is clear that ,
Definition 9. Let be minimal neighborhood systems with binary relation , where and . Then, is an approximation space (briefly, approximation space).
Remark 10. In Definition 9, is the same for all , when is an equivalence relation.
Lemma 11. If , then , where and .
Proof. Consider . Let and . Then, . Therefore, . By the same manner, the proof is verified for . In the case , if , then and . Thus, and . Hence, . Therefore, .
Lemma 11 does not hold for , in general.
Example 1. If with , then and . Hence, , , , , , , , , , , , , , and . Clearly, , but .
The proof of Lemma 12 is clear, so it is omitted.
Lemma 12. Let be a symmetric relation and . Then, for each .
Lemma 13. If is an inverse serial and a serial relation, then and , respectively.
Proof. Let be a serial relation. Then, , s.t . Hence, for some . Therefore, .
Let be an inverse serial and serial relation. Then, and are not true, in general.
Example 2. If and , then and . Hence, and . Clearly, , is serial and , but . Now, let . Then, and . Hence, , , and . Clearly, , is inverse serial and , but .
Lemma 14. If is a reflexive relation, then and , and .
Proof. Assume that is reflexive and . Then, and . Thus, . Therefore, , .
Proposition 15. If is a reflexive relation and , then(i)(ii)(iii)(iv)(v)
Proof. (i) Let be a reflexive relation. Then, for all . Consider . Then, . Therefore, . By the same manner, proofs of (ii), (iii), (iv), and (v) are verified.
The converse of Proposition 15 and equality are not true, in general.
Example 3. In Example 1, all properties of Proposition 15 are satisfied. However, is not reflexive relation, since . Also, , , , and .
Proposition 16. Let be a reflexive and symmetric relation. Then, and , and .
Proof. Suppose that is reflexive and symmetric relation. Then, , . Hence, the proof is clear by Proposition 15.
Definition 17. In with , the lower and upper approximations of are defined by and , respectively.
Definition 18. For all , if , then the set is called exact. Otherwise, the set is called rough.
Definition 19. For each , the boundary set, the positive set, and the negative set are defined by , , and , respectively.
Theorem 20. Let be a binary relation and . Then, lower approximation and upper approximation have the following characteristics, where is the complement.
Proof. The properties (L1), (L4), (H2), and (H4) are clear. Therefore, the remaining properties can be proved as follows: (L5) = = = . (L6) Let , where . Then, . Hence, . So, . Therefore, . Conversely, let , where . Then, . We want to show that . Let . Then, by Lemma 11, we have . Thus, . Hence, and then . So, and . Therefore, . (H6) Similar to the proof of (L6). (L7) Let . Then, . (H7) Similar to the proof of (L7). (L8) and (H8) can be proved directly by using (L7) and (H7). Properties (L2), (L3), (H1), and (H3) are not true, in general.
Example 4. In Example 1, if and , then , , , , and .
Proposition 21. Let be a reflexive relation. Then,
Proof. It is sufficient to prove (L3) and proofs of (L2), (H1), and (H3) are similar. Assume that is reflexive and for all . Thus, . However, . Hence, . Therefore, .
Properties (L6) and (H6) are not true, for with reflexive relation, in general.
Example 5. Let and . Then, and . Hence, , , , , , , , , , , , , , , , and . However, and .
Proposition 22. If is a symmetric relation and , then
Proof. The proof is obvious using Lemma 12 and Theorem 20.
Equality in the properties (L6) and (H6) are not true, in general.
Example 6. In Example 1, consider , , and . Then, , , , , , , , and .
To satisfy the majority of Pawlak’s approximations qualities from Theorem 20 and Proposition 21, must be a reflexive relation. Therefore, minimal approximations are a generalization for rough approximations. In Table 1, we compare our study with other techniques of rough approximations. The symbol indicates that Pawlak’s property is verified.
Definition 23. Let be a reflexive relation with and . Then, accuracy of approximation of the subset is , where and denotes the cardinality.
Corollary 24. From Definition 23 and Proposition 21, we deduce that with a reflexive relation , (i) ; (ii) if , then the subset is exact. Otherwise, is rough.
3. Relationship between Approximation Operators
In this section, several kinds of approximation operators are compared. Also, boundary and accuracy of approximations are investigated.
Remark 25. From Tables 2 and 3 and Example 5, different kinds of approximations operators, boundary and accuracy, are compared. is the best accuracy.
Theorem 26. If is a binary relation and , then(i)(ii)(iii)(iv)
Proof. (i) If , then . Thus, . Hence, . Therefore, . Now, let . Then, . However, . Hence, . Therefore, . (ii) If , then . However, and thus . Hence, . Therefore, . Now, let . Then, . However, . Thus, . Hence, . Therefore, . By the same manner, the proof is verified for (ii) and (iv).
Equality in Theorem 26 does not hold, in general.
Example 7. In Example 1, , , , , , , , and .
Theorem 27. Let be a reflexive relation and . Then,(i)(ii)
Proof. By using Proposition 21 and Theorem 26, the proof is obvious.
The equality in Theorem 27 is not true, in general.
Example 8. In Example 5 and Tables 2 and 3, , , , and .
In the following implications, several kinds of approximations operators with a reflexive relation and are compared.
Theorem 28. If is a reflexive relation and , then(i)(ii)
Proof. (i) Consider . Thus, and . By using Theorem 30, and . Hence, . Therefore, . Now, let . Then, and . By using Theorem 27, and . Hence, . Therefore, . By the same manner, (ii) is verified.
Corollary 29. Let be a reflexive relation and . Then,(i)(ii)
The proof of Theorem 30 is clear. So, the proof is omitted.
Theorem 30. Let be a reflexive relation and . Then,(i) is exact is exact is exact(ii) is exact is exact is exact
The equality in Theorem 28 and Corollary 29 does not hold, in general.
Example 9. In Example 5 and Tables 2 and 3, , , , and .
The converse in Theorem 30 is not true, in general.
Example 10. In Example 5 and Tables 2 and 3, we have is exact, but is neither exact nor exact. Also, is exact, but is not exact. Furthermore, is exact, but is neither exact nor exact. Finally, is exact, but is not exact.
4. Topological Spaces Induced by Minimal Neighborhoods
In this section, various topologies are created by using the minimal of neighborhoods. The comparison between these new types of topologies is studied.
It is easy to prove the conditions of topology for the class in Theorem 31, so the proof must be omitted.
Theorem 31. If is approximation space and is a binary relation, then the families are topologies on , for all .
Example 11. In Example 1, we have
Theorem 32. If are topologies and is a binary relation, then(i)(ii)
Proof. If , then for all . However, and then for all . Hence, . Therefore, . Now, let , then for all . However, and then for all . Hence, . Therefore, . By the same manner, the proof is true for (ii).
The equality in Theorem 32 is not true, in general.
Example 12. In Example 11, we have .
From Theorem 32, it is easy to prove Theorem 33. So, we omit the proof.
Theorem 33. If are topologies and is a symmetric relation, then .
5. Applications: COVID-19 Infections and Heart Attacks
In this section, some suggested methodologies to practical issues are presented and particularly in the area of patient diagnostics where more precise judgments are required. Therefore, medical applications are examined for the proposed approximations characteristics in terms of minimal neighborhoods. These examples show how the generalization of rough sets using minimal neighborhoods can effectively handle and represent a variety of real-world issues. It is illustrated that the utilization of minimal neighborhoods in the context of RST aids in the elimination of data uncertainty and ambiguity.
Example 13. This example aims to illustrate the significance of present approximations in order to obtain the best tools for determining the major components of COVID-19 infections in humans. The World Health Organization and medical organizations with expertise in COVID-19 gathered the data in Table 4 [27]. Due to the similar properties in the rows (objects), data from 500 patients were reduced to 10 patients. Therefore, the set of objects is .
The attributes (most common symptoms) of COVID-19 are given as follows: {Difficulty breathing = , Chest pain = , High Headache = , Dry cough = , Temperature = , Loss of smell or taste = } and Decision COVID-19, as shown in Table 4.
From Table 4, the symptoms are given as follows: , , , , , , , , and .
The relation is given as follows: . Consequently, , , , , , , , , , , , , , , , , , , , , and .
Then, , , and .
Hence, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and .
Patients with confirmed COVID-19 infections are and then (i) , , , and ; (ii) , , , and ; (iii) , , , and ; and (iv) , , , and .
According to the proposed fourth type, patients , and are unquestionably infected with COVID-19 using the current technique, as shown in Table 4.
Example 14. The presented methodologies are used to make decisions on heart attacks. Table 5 shows the set of objects (patients) as collected from Al-Azhar University’s cardiology department (Hospital of Sayed Glal University, Cairo, Egypt) [11]. It was shortened to since the properties in rows (objects) are same. The study covered patients with a variety of symptoms, and the set of attributes = {Breathlessness = , Orthopnea = , Paroxysmal nocturnal dyspnea = , Reduced exercise tolerance = , Ankle swelling = } and decision of heart attacks is ruled out or confirmed = D as illustrated in Table 5.
From Table 5, the symptoms are given as follows: , , , , , , and .
The relation is given as follows: . Consequently, , , , , , , , , , and .
Then, and .
Hence, , , , , , , , , , , , , , , , , , , , , , , , and .
In Table 5, patients with confirmed heart attacks are . Then,(i), , , and (ii), , , and (iii), , , and (iv), , , and From the proposed first and fourth types, the patients , and have certainly undergone heart attacks, which is consistent with Table 5. In addition, the fourth type is a best accuracy and the topology which constructed by is the best choice for decision making.
6. Conclusion and Future Work
The current paper examines four different kinds of generalization for rough sets which contain four different types of lower and upper approximations that construct by minimal neighborhoods. The properties of these approximations are discussed. Many comparisons have been made between our generalization and other generalizations. The approximation operators pave the way for additional topological advances to RST and applications. There are four topologies established through our study. Medical applications are shown in two examples and used for decision making. In a future work, these results can be studied in bi-neighborhoods. Also, this research will be helpful and will open new doors in the study of topologies which approach rough sets through minimal neighborhoods, as well as applications for graphs [2] and in the study of minimal neighborhoods as applications of these new concepts.
Data Availability
The data used to support the findings of this study are included within the article.
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.