# A New Single-Valued Neutrosophic Rough Sets and Related Topology

Authors: Qiu Jin; Kai Hu; Chunxin Bo; Lingqiang Li
Journal: Journal of Mathematics
 (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 spaceU,R, 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.


---

## Body

## 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 R be an equivalent relation (can be easily extended for an arbitrary binary relation) on U. Then, for each neutrosophic set A on U, a pair of neutrosophic sets R¯A and R¯A on U are defined as the lower and upper approximations of A w.r.t. U,R.(ii)
Sweety’s neutrosophic rough sets [20]: let R be a neutrosophic relation on U. Then, for each neutrosophic set A on U, a pair of neutrosophic sets R¯A and R¯A on U are defined as the lower and upper approximations of A w.r.t. U,R. Yang [10] defined a similar model by considering the single-valued neutrosophic relation and single-valued neutrosophic set on U.In this paper, we shall introduce a new model of rough sets fusion with neutrosophic sets under the framework of single-valued neutrosophic approximation spaceU,R (i.e., a nonempty set U together with a single-valued neutrosophic relation R on U). For each ordinary subset A of U, we shall define a pair of single-valued neutrosophic sets R¯A and R¯A on U as the lower and upper approximations of A with respect to U,R. Obviously, our model is different from Broumi–Sweety–Yang’s models, since, in our model, the original sets are ordinary subsets of U 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 U, Yang defined the single-valued neutrosophic topology on U as a subset τ of SvnsU (the set of all single-valued neutrosophic sets on U) 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 U, Kim defined the ordinary single-valued neutrosophic topology on U as a neutrosophic set τ on PU (the power set of U) 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 Table1.Table 1
Method and comparison.
Rough setRAR¯A,R¯ATopologyτRBijectionRough neutrosophic sets [19]−++Yang’s topology [45]×Neutrosophic rough sets [10, 20]+++Yang’s topology [45]√Our neutrosophic rough sets+−+Kim’s topology [47]√Notes: “+” represents that the set is a single-valued neutrosophic set, and “−” represents that it is not; “√” represents that there is a bijection between the considered rough sets and topologies, and “×” represents that there is no bijection.The remainder of this paper is organized as follows. In Section2, 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 thatU is a nonempty infinite set. We denote PU as the power set of U and define Ac=U−A for A∈PU.Definition 1 (see [2]).
AnSvnsA=AT,AI,AF on U is defined as three membership functions AT,AI,AF:U⟶0,1, which are interpreted as truth-membership function, indeterminacy-membership function, and falsity-membership function, respectively. All Svnss are denoted by SvnsU.
Eachα=α1,α2,α3∈0,13 is called a single-valued neutrosophic number, and its complement is defined as αc=α3,1−α2,α1. We denote the single-valued neutrosophic numbers ⊤=1,0,0 and ⊥=0,1,1. Obviously, ⊤c=⊥ and ⊥c=⊤.Remark 1.
Pythagorean fuzzy set [50] is also an important extension of intuitionistic fuzzy set. We can observe that when restricting 0≤ATx2+AFx2≤1 and AIx=1−ATx2−AFx2, an Svns becomes a Pythagorean fuzzy set.
ForA∈PU, we define ⊤A∈SvnsU as follows: ∀x∈U,⊤Ax=⊤ if x∈A and ⊤Ax=⊥ if x∉A.Definition 2 (see [2, 6, 10]).
LetA,B,Ajj∈J∈SvnsU.(1)
We denoteA⊑B if, for any x∈U, ATx≤BTx, AIx≥BIx, and AFx≥BFx. By A=B, we mean A⊑B and B⊑A(2)
We defineAc∈SvnsU as ∀x∈U,Acx=Axc=AFx,1−AIx,ATx(3)
We define⊔j∈JAj,⊓j∈JAj∈SvnsU by ∀x∈U,(1)⊔j∈JAjx=∨j∈JAjTx,∧j∈JAjIx,∧j∈JAjFx,⊓j∈JAjx=∧j∈JAjTx,∨j∈JAjIx,∨j∈JAjFx.Definition 3 (see [10]).
AnSvnsR on U×U is referred to a single-valued neutrosophic relation (Svnr) on U. Then the pair U,R is said to be a single-valued neutrosophic approximation space (Svnas). Furthermore, R is called(i)
Reflexive if∀x∈U, Rx,x=⊤, i.e., RTx,x=1,RIx,x=0,RFx,x=0(ii)
Transitive ifRT,RIc,RFc are all transitive fuzzy relations, that is, ∀x,y,z∈U(2)RTx,y∧RTy,z≤RTx,z,RIx,y∨RIy,z≥RIx,z,RFx,y∨RFy,z≥RFx,z.Definition 4 (see Definition 3.1 in [10]).
LetU,R be an Svnas. For A∈SvnsU, the upper and lower approximations of A, denoted by YR¯A,YR¯A∈SvnsU, are defined as follows: ∀x∈U,(3)YR¯ATx=∨y∈URTx,y∧ATy,YR¯AIx=∧y∈URIx,y∨AIy,YR¯AFx=∧y∈URFx,y∨AFy,YR¯ATx=∧y∈URFx,y∨ATy,YR¯AIx=∨y∈U1−RIx,y∧AIy,YR¯AFx=∨y∈URTx,y∧AFy.
The pairYR¯A,YR¯A is referred to the single-valued neutrosophic rough sets of A. YR¯ and YR¯ are said to be the single-valued neutrosophic upper and lower approximation operators, respectively.Definition 5 (see Definition 8 in [47]).
AnSvnsτ on PU, that is, τ=τT,τI,τF with τT,τI,τF:PU⟶0,1, is referred to an ordinary single-valued neutrosophic topology (OSvnt) on U if τ fulfills the following conditions:(4)OSvnt1τ∅=τU=⊤,OSvnt2For anyA,B∈PU,τTA∩B≥τTA∧τTB,τIA∩B≤τIA∨τIB,τFA∩B≤τFA∨τFB,OSvnt3For anyAjj∈J∈PU,τT∪j∈JAj≥∧j∈JτTAj,τI∪j∈JAj≤∨j∈JτIAj,τF∪j∈JAj≤∨j∈JτFAj.
The pairU,τ is said to be an ordinary single-valued neutrosophic topological space (OSvnts).
For examples and more results aboutOSvnts, refer to [47].
The following lemma can be easily observed. We will use it without mentioning again.Lemma 1.
Letα,β∈0,1. Then the following conditions are equivalent:(1)
α≤β(2)
For allγ∈0,1, γ<α⇒γ<β(3)
For allγ∈0,1, γ<α⇒γ≤β(4)
For allγ∈0,1, γ≤α⇒γ≤β(5)
For allγ∈0,1, γ>β⇒γ≥α


## 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.
LetU,R be an Svnas. For A∈PU, the upper and lower approximations of A, denoted by R¯A,R¯A∈SvnsU, are defined as follows: ∀x∈U,(5)R¯ATx=∨y∈ARTx,y,R¯AIx=∧y∈ARIx,y,R¯AFx=∧y∈ARFx,y,R¯ATx=∧y∉ARFx,y,R¯AIx=∨y∉A1−RIx,y,R¯AFx=∨y∉ARTx,y.
The pairR¯A,R¯A is referred to the ordinary single-valued neutrosophic rough sets of A. R¯ and R¯ are said to be the ordinary single-valued neutrosophic upper and lower approximation operators, respectively.Remark 2.
(1)
The definition ofR¯ATx is an interpretation of the fact that “the join of RTx and A is not empty,” and the definition of R¯ATx is an interpretation of the fact that “RTx is contained in A (or equivalent, Ac is contained in RFx).”(2)
For a fuzzy relationr on U, it is easily observed that r induces an SvnrRr on U defined as follows: ∀x,y∈U×U, RrTx,y=rx,y,RrIx,y=0,RrFx,y=1−rx,y. For A∈PU, we have Rr¯AT=∨y∈Arx,y=r¯A,Rr¯AT=∨y∉A1−rx,y=r¯A, where r¯A,r¯A 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 PU to SvnsU and Yang’s operators are defined from SvnsU to SvnsU.Example 1.
LetU,R be an Svnas with U=x1,x2,x3 and let R be defined as in Table 2.
TakingA=x1,x2, we have(6)R¯ATx1=RFx1,x3=0.4,R¯AIx1=1−RIx1,x3=1,R¯AFx1=RTx1,x3=1,R¯ATx2=RFx2,x3=1,R¯AIx2=1−RIx2,x3=1,R¯AFx2=RTx2,x3=0.6,R¯ATx3=RFx3,x3=0,R¯AIx3=1−RIx3,x3=1,R¯AFx1=RTx3,x3=1,R¯ATx1=RTx1,x1∨RTx1,x2=0∨0.3=0.3,R¯AIx1=RIx1,x1∧RIx1,x2=0∧0.1=0,R¯AFx1=RFx1,x1∧RFx1,x2=1∧0.6=0.6,R¯ATx2=RTx2,x1∨RTx2,x2=0∨0.6=0.6,R¯AIx2=RIx2,x1∧RIx2,x2=0.2∧0.5=0.2,R¯AFx2=RFx2,x1∧RFx2,x2=0.4∧1=0.4,R¯ATx3=RTx3,x1∨RTx3,x2=1∨1=1,R¯AIx3=RIx3,x1∧RIx3,x2=0∧0.5=0,R¯AFx3=RFx3,x1∧RFx3,x2=1∧1=1.
Hence, we obtainR¯A and R¯A as in Table 3.Table 2
Svnas.
Rx1x2x3x1(0, 0, 1)(0.3, 0.1, 0.6)(1, 0, 0.4)x2(0, 0.2, 0.4)(0.6, 0.5, 1)(0.6, 0,1)x3(1, 0, 1)(1, 0.5, 1)(1, 0, 0)Table 3
The upper and lower approximation.
R¯AR¯Ax1(0.4, 1, 1)(0.3, 0, 0.6)x2(1, 1, 0.6)(0.6, 0.2, 0.4)x3(0, 1, 1)(1, 0, 1)Theorem 1.
LetU,R be an Svnas. Then we have the following:(1)
R¯U=⊤U; R¯∅=⊤∅(2)
IfA⊆B, then R¯A⊑R¯B and R¯A⊑R¯B(3)
For allAjj∈J∈PU, R¯∩j∈JAj=⊓j∈JR¯Aj and R¯∪j∈JAj=⊔j∈JR¯Aj(4)
ForA∈PU, R¯A=R¯Acc and R¯A=R¯AccProof.
For (1)–(3), we prove only the results for lower approximation. The proofs for upper approximation are similar and hence are omitted.(1)
For anyx∈U, we have R¯UTx=∧y∉URFx,y=1, R¯UIx=∨y∉U1−RIx,y=0,R¯UFx=∨y∉URTx,y=0. Hence, R¯U=⊤U.(2)
For anyx∈U and A⊆B, we obtain(7)R¯ATx=∧y∉ARFx,y≤∧y∉BRFx,y=R¯BTx,R¯AIx=∨y∉A1−RIx,y≥∨y∉B1−RIx,y=R¯BIx,R¯AFx=∨y∉ARTx,y≥∨y∉BRTx,y=R¯BFx.Hence,R¯A⊑R¯B.(3)
For anyx∈U,(8)⊓j∈JR¯AjTx=∧j∈JR¯AjTx=∧j∈J∧y∉AjRFx,y=∧y∉∩j∈JAjRFx=R¯∩j∈JAjTx,⊓j∈JR¯AjIx=∨j∈JR¯AjIx=∨j∈J∨y∉Aj1−RIx,y=∨y∉∩j∈JAj1−RIx,y=R¯∩j∈JAjIx,⊓j∈JR¯AjFx=∨j∈JR¯AjFx=∨j∈J∨y∉AjRTx,y=∨y∉∩j∈JAjRTx,y=R¯∩j∈JAjFx.Hence,R¯∩j∈JAj=⊓j∈JR¯Aj.(4)
For anyx∈U,(9)R¯AcTcx=R¯AcFx=∨y∈ARTx,y=R¯ATx,R¯AcIcx=1−R¯AcIx=1−∨y∈A1−RIx,y=∧y∈ARIx,y=R¯AIx,R¯AcFcx=R¯AcTx=∧y∈ARFx,y=R¯AFx.Hence,R¯A=R¯Acc. That is, R¯A=R¯Acc can be proved similarly.
The following theorem gives a characterization on the approximation operators generated by reflexiveSvnas.Theorem 2.
LetU,R be an Svnas. Then the following three are equivalent:(1)
R is reflexive(2)
R¯A⊑⊤A for each A∈PU(3)
⊤A⊑R¯A for each A∈PUProof.
(1)⇒ (2). If x∈A, then(10)R¯TAx≤⊤ATx=1,R¯IAx≥⊤AIx=0,R¯FAx≥⊤AFx=0.
Ifx∉A, then(11)R¯TAx=∧y∉ARFx,y≤RFx,x=10=⊤ATx,R¯IAx=∨y∉A1−RIx,y≥1−RIx,x=11−0=1=⊤AIx,R¯FAx=∨y∉ARTx,y≥RTx,x=11=⊤AFx.
Hence,R¯A⊑⊤A.
(2)⇒ (1). For any x∈U, by (2), we have(12)RTx,x=∨y∉U−xRTx,y=R¯U−xFx≥⊤U−xFx=1,1−RIx,x=∨y∉U−x1−RIx,y=R¯U−xIx≥⊤U−xIx=1⇒RIx,x=0,RFx,x=∧y∉U−xRFx,y=R¯U−xTx≤⊤U−xTx=0.
Hence,R is reflexive.
(2)⇔ (3). It can be concluded from Theorem 1 (4).
The following theorem presents a characterization on the approximation operators generated by transitiveSvnas.Theorem 3.
LetU,R be an Svnas. Then the following three are equivalent:(1)
R is transitive.(2)
For eachA∈PU and x∈U,(13)R¯ATx=∨B⊆AR¯BTx∧∧y∈BR¯BTy,R¯AIx=∧B⊆AR¯BIx∨∨y∈BR¯BIy,R¯AFx=∧B⊆AR¯BFx∨∨y∈BR¯BFy.(3)
For eachA∈PU and x∈U,(14)R¯ATx=∧A⊆BR¯BTx∨∨y∉BR¯BTy,R¯AIx=∨A⊆BR¯BIx∧∧y∉BR¯BIy,R¯AFx=∨A⊆BR¯BFx∧∧y∉BR¯BFy.Proof.
(1)⇒ (2). Let A∈PU and x∈U.(i)
For anyB⊆A, we have R¯BTx≤R¯ATx and so(15)∨B⊆AR¯BTx∧∧y∈BR¯BTy≤∨B⊆AR¯BTx=R¯ATx.Conversely, letα=R¯ATx=∨y∉ARFx,y; then RFx,y≥α for any y∉A. Take Bx=z∈U|RFx,z<α; then Bx⊆A. It follows that(16)R¯BxTx=∧y∉BxRFx,y≥α,∧y∈BxR¯BxTy=∧y∈Bx∧z∉BxRFy,z.Note that, for anyy∈Bx,z∉Bx, we have RFx,y<α,RFx,z≥α. Since R is transitive, we have RFx,y∨RFy,z≥RFx,z≥α, which means that RFy,z≥α. So,(17)∧y∈BxR¯BxTy=∧y∈Bx∧z∉BxRFy,z≥α,and then(18)∨B⊆AR¯BTx∧∧y∈BR¯BTy≥R¯BxTx∧∧y∈BxR¯BxTy≥α=R¯ATx.Hence,(19)∨B⊆AR¯BTx∧∧y∈BR¯BTy=R¯ATx.(ii)
For anyB⊆A, we have R¯BIx≥R¯AIx and so(20)∧B⊆AR¯BIx∨∨y∈BR¯BIy≥∧B⊆AR¯BIx=R¯AIx.Conversely, letα=R¯AIx=∨y∉A1−RIx,y; then 1−RIx,y≤α for any y∉A. Take Bx=z∈U|1−RIx,z>α; then Bx⊆A. It follows that(21)R¯BxIx=∨y∉Bx1−RIx,y≤α,∨y∈BxR¯BxIy=∨y∈Bx∨z∉Bx1−RIy,z.Note that, for anyy∈Bx,z∉Bx, we have 1−RIx,y>α,1−RIx,z≤α. Since R is transitive, we have 1−RIx,y∧1−RIy,z≤1−RIx,z≤α, which means that 1−RIy,z≤α. So,(22)∨y∈BxR¯BxIy=∨y∈Bx∨z∉Bx1−RIy,z≤α,and then(23)∧B⊆AR¯BIx∨∨y∈BR¯BIy≤R¯BxIx∨∨y∈BxR¯BxIy≤α=R¯AIx.Hence,(24)∧B⊆AR¯BIx∨∨y∈BR¯BIy=R¯AIx.(iii)
For anyB⊆A, we have R¯BFx≥R¯AFx and so(25)∧B⊆AR¯BFx∨∨y∈BR¯BFy≥∧B⊆AR¯BFx=R¯AFx.
Conversely, letα=R¯AFx=∨y∉ARTx,y; then RTx,y≤α for any y∉A. Take Bx=z∈U|RTx,z>α; then Bx⊆A. It follows that(26)R¯BxFx=∨y∉BxRTx,y≤α,∨y∈BxR¯BxFy=∨y∈Bx∨z∉BxRTy,z.
Note that, for anyy∈Bx,z∉Bx, we have RTx,y>α,RTx,z≤α. Since R is transitive, we have RTx,y∧RTy,z≤RTx,z≤α, which means that RTy,z≤α. So,(27)∨y∈BxR¯BxFy=∨y∈Bx∨z∉BxRTy,z≤α,and then(28)∧B⊆AR¯BFx∨∨y∈BR¯BFy≤R¯BxFx∨∨y∈BxR¯BxFy≤α=R¯AFx.
Hence,(29)∧B⊆AR¯BFx∨∨y∈BR¯BFy=R¯AFx.
(2)⇒ (1). Let x,y,z∈U.(i)
Note that(30)RTx,z=R¯U−zFx=2∧A⊆U−zR¯AFx∨∨u∈AR¯AFu,RTx,y=R¯U−yFx=2∧B⊆U−yR¯BFx∨∨v∈BR¯BFv,RTy,z=R¯U−zFy=2∧C⊆U−zR¯CFy∨∨w∈CR¯CFw.Take anyA⊆U−z; then y∈A or y∉A.Case 1: ify∈A, then(31)R¯AFx∨∨u∈AR¯AFu≥∨u∈AR¯AFu,byy∈A=R¯AFy∨∨u∈AR¯AFu,byA⊆U−z≥∧C⊆U−zR¯CFy∨∨w∈CR¯CFw≥∧B⊆U−yR¯BFx∨∨v∈BR¯BFv∧∧C⊆U−zR¯CFy∨∨w∈CR¯CFw.Case 2: ify∉A, then A⊆U−y and so(32)R¯AFx∨∨u∈AR¯AFu,byA⊆U−y≥∧B⊆U−yR¯BFx∨∨v∈BR¯BFv≥∧B⊆U−yR¯BFx∨∨v∈BR¯BFv∧∧C⊆U−zR¯CFy∨∨w∈CR¯CFw.By a combination of Cases 1 and 2, we obtain(33)∧A⊆U−zR¯AFx∨∨u∈AR¯AFu≥∧B⊆U−yR¯BFx∨∨v∈BR¯BFv∧∧C⊆U−zR¯CFy∨∨w∈CR¯CFw,that is,RTx,z≥RTx,y∧RTy,z, as desired.(ii)
Note that(34)1−RIx,z=R¯U−zIx=2∧A⊆U−zR¯AIx∨∨u∈AR¯AIu,1−RIx,y=R¯U−yIx=2∧B⊆U−yR¯BIx∨∨v∈BR¯BIv,1−RIy,z=R¯U−zIy=2∧C⊆U−zR¯CIy∨∨w∈CR¯CIw.Similar to (i), we can prove that1−RIx,z≥1−RIx,y∧1−RIy,z; that is, RIx,z≤RIx,y∨RIy,z, as desired.(iii)
Note that(35)RFx,z=R¯U−zTx=2∨A⊆U−zR¯ATx∧∧u∈AR¯ATu,RFx,y=R¯U−yTx=2∨B⊆U−yR¯BTx∧∧v∈BR¯BTv,RFy,z=R¯U−zTy=2∨C⊆U−zR¯CTy∧∧w∈CR¯CTw.Take anyA⊆U−z; then y∈A or y∉A.Case 1: ify∈A, then(36)R¯ATx∧∧u∈AR¯ATu≤∧u∈AR¯ATu,byy∈A=R¯ATy∧∧u∈AR¯ATu,byA⊆U−z≤∨C⊆U−zR¯CTy∧∧w∈CR¯CTw≤∨B⊆U−yR¯BTx∧∧v∈BR¯BTv∨∨C⊆U−zR¯CTy∧∧w∈CR¯CTw.Case 2: ify∉A, then A⊆U−y and so(37)R¯ATx∧∧u∈AR¯ATu,byA⊆U−y≤∨B⊆U−yR¯BTx∧∧v∈BR¯BTv≤∨B⊆U−yR¯BTx∧∧v∈BR¯BTv∨∨C⊆U−zR¯CTy∧∧w∈CR¯CTw.By a combination of Cases 1 and 2, we obtain(38)∨A⊆U−zR¯ATx∧∧u∈AR¯ATu≤∨B⊆U−yR¯BTx∧∧v∈BR¯BTv∨∨C⊆U−zR¯CTy∧∧w∈CR¯CTw,that is,RFx,z≤RFx,y∨RFy,z, as desired.From (i)–(iii), we know thatR 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 theOSvnt 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.
AnOSvntsU,τ is said to be quasidiscrete if it fulfills the following:(39)OSvnt2sfor anyAj∈PUj∈J,τT∩j∈JAj≥∧j∈JτTAj,τI∩j∈JAj≤∨j∈JτIAj,τF∩j∈JAj≤∨j∈JτFAj.
It is not difficult to see that quasidiscreteOSvnts is an extension of quasidiscrete topological space [10].Theorem 4.
LetU,R be an Svnas. Then the SvnsτR on PU is defined as follows: for any A∈PU,(40)τRTA=∧x∈AR¯ATx,τRIA=∨x∈AR¯AIx,τRFA=∨x∈AR¯AFx,is a quasidiscrete OSvnt on U.Proof.
OSvnt1: it follows that(41)τRT∅=∧x∈∅R¯∅Tx=1,τRI∅=∨x∈∅R¯∅Ix=0,τRF∅=∨x∈∅R¯∅Fx=0,τRTU=∧x∈UR¯UTx=1,τRIU=∨x∈UR¯UIx=0,τRFU=∨x∈UR¯UFx=0.OSvnt2s: letAj∈PUj∈J. Then(42)∧j∈JτRTAj=∧j∈J∧xj∈AjR¯AjTxj≤∧j∈J∧x∈∩j∈JAjR¯AjTx,by Theorem13=∧x∈∩j∈JAjR¯∩j∈JAjTx=τRT∩j∈JAj,∨j∈JτRIAj=∨j∈J∨xj∈AjR¯AjIxj≥∨j∈J∨x∈∩j∈JAjR¯AjIx,by Theorem13=∨x∈∩j∈JAjR¯∩j∈JAjIx=τRI∩j∈JAj.Similarly, we can prove that∨j∈JτRFAj≥τRF∩j∈JAj.OSvnt3: letAj∈PUj∈J. Then it follows by Theorem 1 (2) that(43)∧j∈JτRTAj=∧j∈J∧xj∈AjR¯AjTxj,≤∧j∈J∧xj∈AjR¯∪j∈JAjTxj=∧x∈∪j∈JAjR¯∪j∈JAjTx=τRT∪j∈JAj,∨j∈JτRIAj=∨j∈J∨xj∈AjR¯AjIxj,≥∨j∈J∨xj∈AjR¯∪j∈JAjIxj=∨x∈∪j∈JAjR¯∪j∈JAjIx=τRI∪j∈JAj.Similarly, we can prove that∨j∈JτRFAj=τRF∪j∈JAj.Remark 3.
The definition ofτRA is an interpretation of the fact that “A 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 theSvnas induced by OSvnt.Theorem 5.
LetU,τ be an OSvnts. Then the SvnrRτ on U is defined as follows: for any x,y∈U×U,(44)RτTx,y=∧x,y∈A×AcτFA,RτIx,y=∨x,y∈A×Ac1−τIA,RτFx,y=∨x,y∈A×AcτTA,is reflexive and transitive.Proof.
Reflexivity: it follows that(45)RτTx,x=∧x,x∈A×AcτFA=1,RτIx,x=∨x,x∈A×Ac1−τIA=0,RτFx,x=∨x,x∈A×AcτTA=0.Transitivity: letx,y,z∈U.(i)
Note that(46)RτTx,y=∧x,y∈A×AcτFA,RτTy,z=∧y,z∈B×BcτFB,RτTx,z=∧x,z∈D×DcτFD.Take anyD∈PU with x,z∈D×Dc; then y∈D or y∈Dc.Case 1: ify∈D, then y,z∈D×Dc. So,(47)RτTx,y∧RτTy,z≤RτTy,z=∧y,z∈B×BcτFB≤τFD.Case 2: ify∈Dc, then x,y∈D×Dc. So,(48)RτTx,y∧RτTy,z≤RτTx,y=∧x,y∈A×AcτFA≤τFD.By a combination of Cases 1 and 2, we obtain that(49)RτTx,y∧RτTy,z≤∧x,z∈D×DcτFD=RτTx,z.(ii)
Note that(50)RτIx,y=∨x,y∈A×Ac1−τIA,RτIy,z=∨y,z∈B×Bc1−τIB,RτIx,z=∨x,z∈D×Dc1−τID.Take anyD∈PU with x,z∈D×Dc; then y∈D or y∈Dc.Case 1: ify∈D, then y,z∈D×Dc. So,(51)RτIx,y∨RτIy,z≥RτIy,z=∨y,z∈B×Bc1−τIB≥1−τID.Case 2: ify∈Dc, then x,y∈D×Dc. So,(52)RτIx,y∨RτIy,z≥RτIx,y=∨x,y∈A×Ac1−τIA≥1−τID.By a combination of Cases 1 and 2, we obtain that(53)RτIx,y∨RτIy,z≥∨x,z∈D×Dc1−τID=RτIx,z.(iii)
Similar to (ii), one can prove thatRτFx,y∨RτFy,z≥RτFx,z.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 onPU.


## 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 transitiveSvnas and quasidiscrete OSvnts.Theorem 6.
LetU,R be an Svnas. Then RτR⊒R, and RτR=R if R is reflexive and transitive.Proof.
(1)
Forx,y∈U,(54)RτRTx,y=∧x,y∈A×AcτRFA=∧x,y∈A×Ac∨z∈AR¯AFz=∧x,y∈A×Ac∨z∈A∨w∉ARTz,w=∧x,y∈A×Ac∨z,w∈A×AcRTz,w,takingz=x,w=y≥∧x,y∈A×AcRTx,y=RTx,y,RτRIx,y=∨x,y∈A×Ac1−τRIA=∨x,y∈A×Ac1−∨z∈AR¯AIz=∨x,y∈A×Ac1−∨z∈A∨w∉A1−RIz,w=∨x,y∈A×Ac∧z,w∈A×AcRIz,w,takingz=x,w=y≤∨x,y∈A×AcRIx,y=RIx,y,RτRFx,y=∨x,y∈A×AcτRTA=∨x,y∈A×Ac∧z∈AR¯ATz=∨x,y∈A×Ac∧z∈A∧w∉ARFz,w=∨x,y∈A×Ac∧z,w∈A×AcRFz,w,takingz=x,w=y≤∨x,y∈A×AcRFx,y=RFx,y.Hence,RτR⊒R.(2)
LetR be reflexive and transitive and x,y∈U.(i)
Note that(55)RτRTx,y≤RTx,y⇔∀α∈0,1,α<RτRTx,yimpliesα<RTx,y.We assume that there is anα0∈0,1 such that α0<RτRTx,y but α0≥RTx,y. Putting A0=z∈U|RTx,z>α0, by reflexivity of R, we have RTx,x=1>α0, so x∈A0, and by α0≥RTx,y we have y∈A0c. This means that x,y∈A0×A0c. From(56)α0<RτRTx,y=∧x,y∈A×Ac∨z,w∈A×AcRTz,w,we know that there existsz,w∈A0×A0c such that RTz,w>α0; that is, RTx,z>α0 and RTx,w≤α0. It follows by the transitivity that(57)α0<RTx,z∧RTz,w≤RTx,w≤α0,a contradiction! Therefore,α<RτRTx,y always implies that α<RTx,y. Hence, RτRTx,y≤RTx,y.(ii)
Note that(58)RτRIx,y≥RIx,y⇔∀α∈0,1,α≤RIx,yimpliesα≤RτRIx,y.We assume that there is anα0∈0,1 such that α0≤RIx,y but α0>RτRIx,y. Putting A0=z∈U|RIx,z<α0, by reflexivity of R, we have RIx,x=0<α0, so x∈A0, and by α0≤RIx,y we have y∈A0c. This means that x,y∈A0×A0c. From(59)α0>RτRIx,y=∨x,y∈A×Ac∧z,w∈A×AcRIz,w,we know that there existsz,w∈A0×A0c such that RIz,w<α0; that is, RIx,z<α0 and RIx,w≥α0. It follows by the transitivity that(60)α0>RIx,z∨RIz,w≥RIx,w≥α0,a contradiction! Therefore,α≤RIx,y always implies that α≤RτRIx,y. Hence, RτRIx,y≥RIx,y.(iii)
Similar to (ii), we can prove thatRτRFx,y≥RFx,y.(i)–(iii) show thatR⊒RτR, and so RτR=R by (1).(3)
IfRτR=R, then it follows by Theorems 4 and 5 that R is reflexive and transitive.Theorem 7.
LetU,R be an OSvnts. Then τRτ⊒τ, and τRτ=τ if τ is quasidiscrete.Proof.
(1)
LetA∈PU. Then(61)τRτTA=∧x∈ARτ¯ATx=∧x∈A∧y∈AcRτFx,y=∧x,y∈A×Ac∨x,y∈B×BcτTB,takingB=A≥∧x,y∈A×AcτTA=τTA,τRτIA=∨x∈ARτ¯AIx=∨x∈A∨y∈Ac1−RτIx,y=∨x,y∈A×Ac1−∨x,y∈B×Bc1−τIB=∨x,y∈A×Ac∧x,y∈B×BcτIB,takingB=A≤∨x,y∈A×AcτIA=τIA.Similarly, we can prove thatτRτFA≤τFA.(2)
LetA∈PU.(i)
Note that(62)τRτTA≤τTA⇔∀α∈0,1,α<τRτTAimpliesα≤τTA.We assume that(63)α<τRτTA=∧x,y∈A×Ac∨x,y∈B×BcτTB.Then, for anyx,y∈A×Ac, there is Bxy∈PU such that x,y∈Bxy×Bxyc and α<τTBxy. Putting By=∪x∈ABxy, by (OSvnt3), we have(64)τTBy=τT∪x∈ABxy≥∧x∈AτTBxy≥α.Note thatA=∩y∈AcBy (indeed, if z∈A, then, for any y∈Ac,z∈Bzy⊆By, and so z∈∩y∈AcBy; hence, A⊆∩y∈AcBy; if z∉A, then, for any x∈A, we have x,z∈A×Ac, and then z∉Bxz so z∉Bz, which means that z∉∩y∈AcBy; hence, ∩y∈AcBy⊆A); then it follows by OSvnt2s that(65)τTA=τT∩y∈AcBy≥∧y∈AcτTBy≥α.Therefore,τRτTA≤τTA.(ii)
Note that(66)τRτIA≥τIA⇔∀α∈0,1,α>τRτIAimpliesα≥τIA.We assume that(67)τRτIA=∨x,y∈A×Ac∧x,y∈B×BcτIB<α.Then, for anyx,y∈A×Ac, there is Bxy∈PU such that x,y∈Bxy×Bxyc and α>τIBxy. Putting By=∪x∈ABxy, by OSvnt3, we have(68)τIBy=τI∪x∈ABxy≤∨x∈AτIBxy≤α.Note thatA=∩y∈AcBy; then it follows by OSvnt2s that(69)τIA=τI∩y∈AcBy≤∨y∈AcτIBy≤α.Therefore,τRτIA≥τIA.(iii)
Similar to (ii), we can prove thatτRτFA≥τFA.(i)–(iii) show thatτRτ⊑τ, and so τRτ=τ by (1).(3)
IfτRτ=τ, then it follows by Theorems 4 and 5 that τ is quasidiscrete.
From Theorems6 and 7, we obtain the following corollary.Corollary 1.
There is a one-to-one correspondence between reflexive and transitiveSvnas and quasidiscrete OSvnts with the same underlying set.Remark 5.
We can give a similar discussion onSvnas 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 ofU 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.



---

*Source: 5522021-2021-07-17.xml*