A New Single-Valued Neutrosophic Rough Sets and Related Topology

Jin, Qiu; Hu, Kai; Bo, Chunxin; Li, Lingqiang

doi:https://doi.org/10.1155/2021/5522021

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 5522021 | https://doi.org/10.1155/2021/5522021

A New Single-Valued Neutrosophic Rough Sets and Related Topology

Qiu Jin,¹Kai Hu,¹Chunxin Bo,¹and Lingqiang Li¹

Academic Editor: Yongqiang Fu

Received01 Mar 2021

Revised19 Jun 2021

Accepted05 Jul 2021

Published17 Jul 2021

Abstract

(Fuzzy) rough sets are closely related to (fuzzy) topologies. Neutrosophic rough sets and neutrosophic topologies are extensions of (fuzzy) rough sets and (fuzzy) topologies, respectively. In this paper, a new type of neutrosophic rough sets is presented, and the basic properties and the relationships to neutrosophic topology are discussed. The main results include the following: (1) For a single-valued neutrosophic approximation space , a pair of approximation operators called the upper and lower ordinary single-valued neutrosophic approximation operators are defined and their properties are discussed. Then the further properties of the proposed approximation operators corresponding to reflexive (transitive) single-valued neutrosophic approximation space are explored. (2) It is verified that the single-valued neutrosophic approximation spaces and the ordinary single-valued neutrosophic topological spaces can be interrelated to each other through our defined lower approximation operator. Particularly, there is a one-to-one correspondence between reflexive, transitive single-valued neutrosophic approximation spaces and quasidiscrete ordinary single-valued neutrosophic topological spaces.

1. Introduction

The original notion of neutrosophic set was proposed by Smarandache [1]. For the convenience of application, Wang et al. [2] investigated the single-valued neutrosophic set (Svns). In Svns, three independent membership functions (truth, indeterminacy, and falsity) are considered; hence, it can be regarded as extensions of fuzzy set [3] and intuitionistic fuzzy set [4]. There are many works on the theory and application of Svns (see Abdel-Basset [5], Ye [6, 7], Samant [8], Yang [9, 10], Zhang [11–13], Zavadskas [14], and Xu [15] as well as Peng’s review paper [16]).

The fusion of neutrosophic sets with rough sets theory [17] is an important research direction. According to Li’s review paper [18], there exists two fundamental combinations of rough sets and neutrosophic sets: Broumi’s rough neutrosophic sets [19] and Sweety’s neutrosophic rough sets [20]. Many other models can be regarded as their extensions [12, 21–24].(i)Broumi’s rough neutrosophic sets [19]: let be an equivalent relation (can be easily extended for an arbitrary binary relation) on . Then, for each neutrosophic set on , a pair of neutrosophic sets and on are defined as the lower and upper approximations of w.r.t. .(ii)Sweety’s neutrosophic rough sets [20]: let be a neutrosophic relation on . Then, for each neutrosophic set on , a pair of neutrosophic sets and on are defined as the lower and upper approximations of w.r.t. . Yang [10] defined a similar model by considering the single-valued neutrosophic relation and single-valued neutrosophic set on .

In this paper, we shall introduce a new model of rough sets fusion with neutrosophic sets under the framework of single-valued neutrosophic approximation space (i.e., a nonempty set together with a single-valued neutrosophic relation on ). For each ordinary subset of , we shall define a pair of single-valued neutrosophic sets and on as the lower and upper approximations of with respect to . Obviously, our model is different from Broumi–Sweety–Yang’s models, since, in our model, the original sets are ordinary subsets of and their approximations are single-valued neutrosophic sets, but, in Broumi–Sweety–Yang’s models, the original sets and their approximations are all (single-valued) neutrosophic sets. Hence, our rough sets will be called ordinary single-valued neutrosophic rough sets.

(Fuzzy) rough sets are closely related to (fuzzy) topology [25–42]. The well-known result may be that there is a one-to-one correspondence between reflexive and transitive (fuzzy) approximation spaces and quasidiscrete (fuzzy) topological spaces [26, 37, 38]. Under the framework of single-valued neutrosophic sets, two kinds of neutrosophic topological spaces are discussed (for more general neutrosophic topology, refer to Al-Omeri [43] and Lupianez [44]).(i)Yang’s single-valued neutrosophic topological spaces [45]: for a nonempty set , Yang defined the single-valued neutrosophic topology on as a subset of (the set of all single-valued neutrosophic sets on ) with some conditions. Yang’s space can be regarded as an extension of Lowen’s fuzzy topological space [46]. Yang also proved that there is a one-to-one correspondence between reflexive and transitive single-valued neutrosophic approximation spaces and his single-valued neutrosophic rough topological spaces.(ii)Kim’s ordinary single-valued neutrosophic topological spaces [47]: for a nonempty set , Kim defined the ordinary single-valued neutrosophic topology on as a neutrosophic set on (the power set of ) with some conditions. Kim’s space can be regarded as an extension of S̆ostak’s fuzzy topology [48] (or Ying’s fuzzifying topology [49]).

In this paper, we shall prove that there are close relationships between our ordinary single-valued neutrosophic rough sets and Kim’s ordinary single-valued neutrosophic topological spaces. The close relationships exhibit that it is meaningful to investigate the new rough sets model.

The method of this paper and the comparison with related literature can be summarized in Table 1.

The remainder of this paper is organized as follows. In Section 2, we will recall some knowledge about neutrosophic sets and rough sets. In Section 3, we shall give the notion of ordinary single-valued neutrosophic upper and lower approximation operators and discuss their properties. Then we will explore the further properties of the proposed approximations corresponding to reflexive (transitive) single-valued neutrosophic approximation space. In Section 4, we will prove that each single-valued neutrosophic approximation space induces an ordinary single valued neutrosophic topological space via our defined lower approximation. In Section 5, we shall verify that each ordinary single-valued neutrosophic topological space induces a single-valued neutrosophic approximation space. In Section 6, we will show that there is a one-to-one correspondence between reflexive and transitive single-valued neutrosophic approximation spaces and quasidiscrete ordinary single-valued neutrosophic topological spaces.

2. Preliminaries

In this section, we recall some knowledge about neutrosophic rough sets and neutrosophic topologies used in this paper.

Unless otherwise stated, we always assume that is a nonempty infinite set. We denote as the power set of and define for .

Definition 1 (see [2]). An Svns on is defined as three membership functions , which are interpreted as truth-membership function, indeterminacy-membership function, and falsity-membership function, respectively. All Svnss are denoted by .
Each is called a single-valued neutrosophic number, and its complement is defined as . We denote the single-valued neutrosophic numbers and . Obviously, and .

Remark 1. Pythagorean fuzzy set [50] is also an important extension of intuitionistic fuzzy set. We can observe that when restricting and , an Svns becomes a Pythagorean fuzzy set.
For , we define as follows: if and if .

Definition 2 (see [2, 6, 10]). Let .(1)We denote if, for any , , , and . By , we mean and (2)We define as (3)We define by ,

Definition 3 (see [10]). An Svns on is referred to a single-valued neutrosophic relation (Svnr) on . Then the pair is said to be a single-valued neutrosophic approximation space (Svnas). Furthermore, is called(i)Reflexive if , , i.e., (ii)Transitive if are all transitive fuzzy relations, that is,

Definition 4 (see Definition 3.1 in [10]). Let be an Svnas. For , the upper and lower approximations of , denoted by , are defined as follows: ,The pair is referred to the single-valued neutrosophic rough sets of . and are said to be the single-valued neutrosophic upper and lower approximation operators, respectively.

Definition 5 (see Definition 8 in [47]). An Svns on , that is, with , is referred to an ordinary single-valued neutrosophic topology (OSvnt) on if fulfills the following conditions:The pair is said to be an ordinary single-valued neutrosophic topological space (OSvnts).
For examples and more results about OSvnts, refer to [47].
The following lemma can be easily observed. We will use it without mentioning again.

Lemma 1. Let . Then the following conditions are equivalent:(1)(2)For all , (3)For all , (4)For all , (5)For all ,

3. Ordinary Single-Valued Neutrosophic Rough Sets for Svnas

In this section, we present the notions and properties of ordinary single-valued neutrosophic upper and lower approximation operators.

Definition 6. Let be an Svnas. For , the upper and lower approximations of , denoted by , are defined as follows: ,The pair is referred to the ordinary single-valued neutrosophic rough sets of . and are said to be the ordinary single-valued neutrosophic upper and lower approximation operators, respectively.

Remark 2. (1)The definition of is an interpretation of the fact that “the join of and is not empty,” and the definition of is an interpretation of the fact that “ is contained in (or equivalent, is contained in ).”(2)For a fuzzy relation on , it is easily observed that induces an Svnr on defined as follows: , . For , we have , where are the fuzzy approximations of ordinary subset w.r.t. fuzzy relation in the work of Yao [51]. Therefore, the single-valued neutrosophic approximations in this paper are a generalization of Yao’s fuzzy approximations.(3)Obviously, the single-valued neutrosophic approximation operators in this paper are different from the single-valued neutrosophic approximation operators in the work of Yang [10], since our operators are defined from to and Yang’s operators are defined from to .

Example 1. Let be an Svnas with and let be defined as in Table 2.
Taking , we haveHence, we obtain and as in Table 3.

Theorem 1. Let be an Svnas. Then we have the following:(1); (2)If , then and (3)For all , and (4)For , and

Proof. For (1)–(3), we prove only the results for lower approximation. The proofs for upper approximation are similar and hence are omitted.(1)For any , we have , . Hence, .(2)For any and , we obtain Hence, .(3)For any , Hence, .(4)For any , Hence, . That is, can be proved similarly.The following theorem gives a characterization on the approximation operators generated by reflexive Svnas.

Theorem 2. Let be an Svnas. Then the following three are equivalent:(1) is reflexive(2) for each (3) for each

Proof. (1) (2). If , thenIf , thenHence, .
(2) (1). For any , by (2), we haveHence, is reflexive.
(2) (3). It can be concluded from Theorem 1 (4).
The following theorem presents a characterization on the approximation operators generated by transitive Svnas.

Theorem 3. Let be an Svnas. Then the following three are equivalent:(1) is transitive.(2)For each and ,(3)For each and ,

Proof. (1) (2). Let and .(i)For any , we have and so Conversely, let ; then for any . Take ; then . It follows that Note that, for any , we have . Since is transitive, we have , which means that . So, and then Hence,(ii) For any , we have and so Conversely, let ; then for any . Take ; then . It follows that Note that, for any , we have . Since is transitive, we have , which means that . So, and then Hence,(iii) For any , we have and soConversely, let ; then for any . Take ; then . It follows thatNote that, for any , we have . Since is transitive, we have , which means that . So,and thenHence,(2) (1). Let .(i)Note that Take any ; then or . Case 1: if , then Case 2: if , then and so By a combination of Cases 1 and 2, we obtain that is, , as desired.(ii)Note that Similar to (i), we can prove that ; that is, , as desired.(iii)Note that Take any ; then or . Case 1: if , then Case 2: if , then and so By a combination of Cases 1 and 2, we obtain that is, , as desired. From (i)–(iii), we know that is transitive. (2) (3). It can be concluded from Theorem 1 (4).

4. Ordinary Single-Valued Neutrosophic Topological Space Induced by Single-Valued Neutrosophic Approximation Space

In this section, we shall consider the OSvnt induced by Svnas through the ordinary single-valued neutrosophic lower approximation operator.

At first, we fix a subclass of ordinary single-valued neutrosophic topological spaces.

Definition 7. An OSvnts is said to be quasidiscrete if it fulfills the following:It is not difficult to see that quasidiscrete OSvnts is an extension of quasidiscrete topological space [10].

Theorem 4. Let be an Svnas. Then the Svns on is defined as follows: for any ,is a quasidiscrete OSvnt on .

Proof. OSvnt1: it follows that OSvnt2s: let . Then Similarly, we can prove that . OSvnt3: let . Then it follows by Theorem 1 (2) that Similarly, we can prove that .

Remark 3. The definition of is an interpretation of the fact that “ is contained in its lower approximation.”

5. Single-Valued Neutrosophic Approximation Space Induced by Ordinary Single-Valued Neutrosophic Topological Space

In this section, we shall consider the Svnas induced by OSvnt.

Theorem 5. Let be an OSvnts. Then the Svnr on is defined as follows: for any ,is reflexive and transitive.

Proof. Reflexivity: it follows that Transitivity: let .(i)Note that Take any with ; then or . Case 1: if , then . So, Case 2: if , then . So, By a combination of Cases 1 and 2, we obtain that(ii)Note that Take any with ; then or . Case 1: if , then . So, Case 2: if , then . So, By a combination of Cases 1 and 2, we obtain that(iii)Similar to (ii), one can prove that .

Remark 4. Note that neither of the topological conditions (OSvnt1)-(OSvnt3) is used in the above theorem. Hence, it can be extended to any single-valued neutrosophic relation on .

6. One-to-One Correspondence between Reflexive and Transitive Single-Valued Neutrosophic Approximation Spaces and Quasidiscrete Ordinary Single-Valued Neutrosophic Topological Spaces

In this section, we prove that there is a one-to-one correspondence between reflexive and transitive Svnas and quasidiscrete OSvnts.

Theorem 6. Let be an Svnas. Then , and if is reflexive and transitive.

Proof. (1)For , Hence, .(2)Let be reflexive and transitive and .(i)Note that We assume that there is an such that but . Putting , by reflexivity of , we have , so , and by we have . This means that . From we know that there exists such that ; that is, and . It follows by the transitivity that a contradiction! Therefore, always implies that . Hence, .(ii)Note that We assume that there is an such that but . Putting , by reflexivity of , we have , so , and by we have . This means that . From we know that there exists such that ; that is, and . It follows by the transitivity that a contradiction! Therefore, always implies that . Hence, .(iii)Similar to (ii), we can prove that . (i)–(iii) show that , and so by (1).(3)If , then it follows by Theorems 4 and 5 that is reflexive and transitive.

Theorem 7. Let be an OSvnts. Then , and if is quasidiscrete.

Proof. (1)Let . Then Similarly, we can prove that .(2)Let .(i)Note that We assume that Then, for any , there is such that and . Putting , by (OSvnt3), we have Note that (indeed, if , then, for any , and so ; hence, ; if , then, for any , we have , and then so , which means that ; hence, ); then it follows by OSvnt2s that Therefore, .(ii)Note that We assume that Then, for any , there is such that and . Putting , by OSvnt3, we have Note that ; then it follows by OSvnt2s that Therefore, .(iii)Similar to (ii), we can prove that . (i)–(iii) show that , and so by (1).(3)If , then it follows by Theorems 4 and 5 that is quasidiscrete.From Theorems 6 and 7, we obtain the following corollary.

Corollary 1. There is a one-to-one correspondence between reflexive and transitive Svnas and quasidiscrete OSvnts with the same underlying set.

Remark 5. We can give a similar discussion on Svnas and ordinary single-valued neutrosophic cotopology in [47] via the ordinary single-valued neutrosophic upper approximation operator.

7. Conclusions

In this paper, we presented a new model of neutrosophic rough sets. The difference between this model and the existing models is that, in our model, the original sets are ordinary subsets of and their approximations are single-valued neutrosophic sets; however, in the existing models, the original sets and their approximations are all (single-valued) neutrosophic sets. We also discussed the basic properties of the proposed rough sets and gave their relationships with Kim’s ordinary single-valued neutrosophic topology. Particularly, we proved by our lower approximation operator that there is a one-to-one correspondence between reflexive and transitive single-valued neutrosophic approximation spaces and quasidiscrete ordinary single-valued neutrosophic topological spaces. In the future work, we shall present a more general single-valued neutrosophic topology such that it can be regarded as an extension of bifuzzy topology in [49]. We will also consider the corresponding single-valued neutrosophic rough sets related to the new single-valued neutrosophic topology. Furthermore, from Remark 1, we know that when restricting single-valued neutrosophic sets to Pythagorean fuzzy sets, we can define a model of Pythagorean fuzzy rough sets. It is well known that Pythagorean fuzzy sets and (fuzzy) rough sets have been applied in many fields, particularly in multiple attribute decision-making [9, 16, 52–55]. Therefore, in the future, we will also consider the potential application of Pythagorean fuzzy rough sets.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no conflicts of interest.

Authors’ Contributions

Qiu Jin and Kai Hu contributed the central idea, and all authors contributed to the writing and revisions.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 11801248 and 11501278) and Natural Science Foundation of Shandong Province (no. ZR2020MA042), and the KeYan Foundation of Liaocheng University (318012030).

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Copyright © 2021 Qiu Jin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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