# New Rough Approximations Based onE-Neighborhoods

Authors: Tareq M. Al-shami; Wen Qing Fu; E. A. Abo-Tabl
Journal: Complexity
 (2021)

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## Abstract

This paper puts forward some rough approximations which are motivated from topology. Given a subsetR⊆U×U, we can use 8 types of E-neighborhoods to construct approximations of an arbitrary X⊆U on the one hand. On the other hand, we can also construct approximations relying on a topology which is induced by an E-neighborhood. Properties of these approximations and relationships between them are studied. For convenience of use, we also give some useful and easy-to-understand examples and make a comparison between our approximations and those in the published literature.


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## Body

## 1. Introduction and Preliminaries

The problem of imperfect knowledge became a crucial issue for computer scientists, especially in the area of information system and artificial intelligence [1, 2]. There are various approaches to manipulate and understand imperfect knowledge, among which is rough set theory. Rough set was proposed by Pawlak [3, 4] in 1982 which has been generalized in many ways [5–13]. What we are concerned about are those methods whose ideas are motivated from topology, for example, methods constructing the lower and upper approximations by using different kinds of neighborhoods, such as right and left neighborhoods [10, 14], minimal right neighborhoods [15], and intersection and union neighborhoods (see [16, 17]). In fact, a combination of rough set theory and topological theory became the main goal of many studies (see [18–25]).The present paper is a continuation to these works where we initiate new types of neighborhoods (namely,Ej-neighborhoods) to give lower and upper approximations of an arbitrary set X directly or indirectly. In Section 2, we apply Nj-neighborhoods to establish the concepts of Ej-neighborhoods and discuss the main properties. With the help of examples, we show the relationships between them as well as with Nj-neighborhoods and Pj-adhesion neighborhoods. In Section 3, we formulate and study the concepts of Ej-lower and Ej-upper approximations, Ej-boundary region, Ej-positive and Ej-negative regions, and Ej-accuracy measure of a subset and make comparisons between them with respect to different types of j. In Section 4, we study the previous concepts from a topological view and explore their main properties. In Section 5, we give some conclusions and make a plan for future works.Now, we recall some basic properties and results of rough set theory, particularly those related to some types of neighborhood systems.Definition 1 (see [3, 5, 16]).
(1)
A subsetR⊆U×U (also called a binary relation on U) is said to be an equivalence relation if it is reflexive (i.e., v,v∈R for each v∈U), symmetric (i.e., u,v∈R if v,u∈R), and transitive (i.e., u,w∈R whenever u,v∈R and v,w∈R). It is said to be a preorder (or quasi-order) if it is reflexive and transitive. It is said to be a partial order if it is an antisymmetric (i.e., u=v whenever u,v∈R and v,u∈R) preorder. It is said to be a diagonal if R=v,v:v∈U. It said to be serial if for every v∈U, w∈U such that v,w∈R (also written as vRw).(2)
(See [4, 26] for a special case). For an equivalence relation R on U and a subset X⊆U, the two related sets R¯X=∪A∈U/R:A⊆X and R¯X=∪A∈U/R:A⊆X≠∅ are called the Pawlak lower approximation and upper approximation of X, respectively (where U/R=x:x∈U is the set of all equivalence classes).Proposition 1 (see [3, 4] for a special case).
The lower approximations and the upper approximations have the following properties (E,F∪Eii∈I⊆2U, the power set of U):(1)
R¯∅=R¯∅=∅ and R¯U=R¯U=U.(2)
R¯E⊆E⊆R¯E and R¯E=R¯Ecc.(3)
R¯∪i∈IEi=∪i∈IR¯Ei and R¯∪i∈IEi=∪i∈IR¯Ei. Particularly, R¯E⊆R¯F and R¯E⊆R¯F if E⊆F.(4)
R¯R¯E=R¯E and R¯R¯E=R¯E.Definition 2.
A subsetJ⊆2U is called a topology on U (and U,J is called a topological space) if it is closed under arbitrary union and finite intersection. A topology satisfying that every open set is also closed is called a clopen topology. We will use Ao to denote the interior of A (i.e., the union of all open sets that are contained in A) and A− to denote the closure of A (i.e., the intersection of all closed sets containing A) in this paper.
LetJR=A∈2U:x⊆A∀x∈A. Then, U,JR is a topological space, R¯X=Xo, R¯X=X−, and x is the smallest open neighborhood of x∀x∈U. This inspires many people to define the lower approximation and the upper approximation by neighborhoods; actually, our approach in this paper is also motivated from topology. In the following, we write J=r,l,r,l,i,u,i,u.Definition 3 (see [10, 14, 16]).
LetR⊆U2 and j∈J.(1)
The 8 kinds ofj-neighborhoods are defined as follows: Nrv=w∈U:vRw, Nlv=w∈U:wRv, Nrv=∩Nrw:v∈Nrw (if v∈Nrw for some w∈U or ∅, otherwise), Nlv=∩Nlw:v∈Nlw (if v∈Nlw for some w∈U or ∅, otherwise), Niv=Nrv∩Nlv, Nuv=Nrv∪Nlv, Niv=Nrv∩Nlv, and Nuv=Nrv∪Nlv(2)
The tripleU,R,λj is called a j-neighborhood space (in brief, j-NS), where λj is a mapping from U to 2U which associated each v∈U with a j-neighborhoodDefinition 4 (see [6]).
LetR⊆U2. The j-adhesion neighborhoods are defined as follows:(1)
Prv=w∈U:Nrw=Nrv(2)
Plv=w∈U:Nlw=Nlv(3)
Prv=w∈U:Nrw=Nrv(4)
Plv=w∈U:Nlw=Nlv(5)
Pix=Prv∩Plv(6)
Pux=Prv∪Plv(7)
Piv=Prv∩Plv(8)
Puv=Prv∪Plv


## 2.E-Neighborhoods

In this section, we introduce the notions ofE-neighborhoods using j-neighborhoods and study their properties.Definition 5.
LetR⊆U2. The E-neighborhoods are defined as follows:(1)
Erx=y∈U:Nry∩Nrx≠∅(2)
Elx=y∈U:Nly∩Nlx≠∅(3)
Eix=Erx∩Elx(4)
Eux=Erx∪Elx(5)
Erx=y∈U:Nry∩Nrx≠∅(6)
Elx=y∈U:Nly∩Nlx≠∅(7)
Eix=Erx∩Elx(8)
Eux=Erx∪Elx
We give the following example to illustrate how we calculate different types of neighborhoods. Also, we will benefit from this example to clarify some obtained results.Example 1.
LetU=v,w,x,y and R=v,v,y,y,v,x,v,y,y,w,w,y. Then, the j-neighborhoods and E-neighborhoods of a point are as in Table 1.Table 1
j-neighborhoods and E-neighborhoods.
vwxyNrv,x,yy∅w,yNlvyvv,w,yNivy∅w,yNuv,x,yyvv,w,yNrv,x,yw,yv,x,yyNlvv,w,y∅yNivw,y∅yNuv,x,yv,w,yv,x,yyPrvwxyPlv,xwv,xyPivwxyPuv,xwv,xyPrv,xwv,xyPlvwxyPivwxyPuv,xwv,xyErv,w,yv,w,y∅v,w,yElv,x,yw,yv,x,yUEiv,yw,y∅v,w,yEuUv,w,yv,x,yUErUUUUElv,wv,w,y∅w,yEiv,wv,w,y∅w,yEuUUUUTheorem 1.
E-neighborhoods have the following properties (R⊆U2,v∈U):(1)
Eiv⊆Erv∩Elv⊆Erv∪Elv⊆Euv(2)
Eiv⊆Erv∩Elv⊆Erv∪Elv⊆Euv(3)
v∈Ejx iff x∈Ejvj∈J(4)
IfR is reflexive, then Ejv⊆Ejv and Pjv∪Njv⊆Ejvj∈J(5)
IfR is symmetric, then Erv=Elv=Eiv=Euv and Erx=Elv=Eiv=Euv(6)
IfR is transitive, then Ejv⊆Ejvj∈r,lu,i(7)
IfR is serial, then Pjv⊆Ejvj∈J(8)
IfR is symmetric and transitive, then Ejv⊆Njv and Ejv⊆Ejw (if v∈Ejw) for each j∈J(9)
IfR is preorder, then Ejv⊆Ejvj∈r,l,u,i(10)
IfR is a equivalence relation, then for each j∈J, all Ejv are identical, Ejv=Njv=Pjv, and v∈Ejw iff Ejv=EjwProof.
(3) Obviously,v∈Ejx⇔Njv∩Njx≠∅⇔x∈Ejv for each j∈r,l,r,l. Then, v∈Ejx⇔x∈Ejv for each j∈i,u,i,u.(4) Step 1: sinceR is reflexive, ∩v∈NrxNrx⊆Nrv and ∩v∈NlxNlx⊆Nlv. This implies that Erv⊆Erv and Elv⊆Elv. Consequently, Eiv⊆Eiv and Euv⊆Euv.Step 2: letx∈Njv. By reflexivity of R, we have x⊆Njx∩Njv. Therefore, x∈Ejv; thus, Njv⊆Ejv. Also, let x∈Pjv. Then, Njx=Njv. Since R is reflexive, Njx∩Njv≠∅. Therefore, x∈Ejv; thus, Pjv⊆Ejv. Hence, we obtain the desired result.(5) SinceR is symmetric, Nrv=Nlv. Therefore, Nrv∩Nrz≠∅⇔Nlv∩Nlz≠∅; thus, Erv=Elv. Consequently, Erv=Elx=Eix=Eux. Similarly, Erx=Elv=Eiv=Euv.(6) Leta∈Nrv. Then, vRa. For each Nrx containing v, we have xRv. Since R is transitive, xRa. Therefore, a∈Nrx; thus, a∈∩v∈NrxNrx=Nrv. Hence, Nrv⊆Nrv; consequently, Erv⊆Erv. Similarly, Elv⊆Elv. This implies that Eiv⊆Eiv and Euv⊆Euv.(7) It follows from (4) and the factv∈Ejx⇔Njv∩Njx≠∅⇔x∈Ejvj∈r,l,r,l.(8) Step 1: we prove the casej=r. Let x∈Erv. Then, Nrx∩Nrv≠∅, i.e., there exists z∈Nrx∩Nrv, xRz, and vRz. Since R is symmetric and transitive, vRx. Therefore, x∈Nrv; thus, Erv⊆Nrv.Step 2: sinceR is symmetric, Erv=Elv=Eiv=Euv by (5). We only prove the case j=l. Let v∈Elw. Then, Nlv∩Nlw≠∅; therefore, there is a∈U such that aRv and aRw. Now, let x∈Elv. Then, Nlx∩Nlv≠∅, i.e., there exists a b∈U such that bRx and bRv. Note that aRb; this means that aRx; consequently, a∈Nlx; also, a∈Nlw; thus, Nlx∩Nlx≠∅. Hence, x∈Elw, as required.(10) Step 1: by (4) and (8),Ejv=Njv and Pjv⊆Ejv. It remains to prove Ejv⊆Pjv. Since R is symmetric, Erv=Elv=Eiv=Euv by (5). Let x∈Elv. Then, Nlx∩Nlv≠∅. Since R is equivalence, Nlx=Nlv. Therefore, x∈Plv; consequently, Elv⊆Plv. Thus, Pjv=Ejv is proved.Step 2: we only prove the casej=l. Let v∈Elw. Then, Nlv∩Nlw≠∅. Since R is equivalence, Nlv=Nlw≠∅. Therefore, Ejv=Ejw. Conversely, Ejv=Ejw. Since R is reflexive, v∈Nlv. Hence, v∈Ejw.


## 3. Rough Approximations UsingE-Neighborhoods Directly

LetU,R,λj be j-NS and ℛj=Ejv:v∈Uj∈J. We devote this section to formulate the following concepts using E-neighborhoods directly: ℛj-lower approximation, ℛj-upper approximation, ℛj-boundary region, ℛj-positive region, ℛj-negative region, and ℛj-accuracy measure of a subset X. We will also illustrate the relationships between them and reveal the main properties with the help of examples.Definition 6.
LetX⊆U and j∈J. Then, ℛj−X=x∈U:Ejx⊆X and ℛj+X=x∈U:Ejx⊆X≠∅ are called ℛj-lower approximation and ℛj-upper approximation, respectively; ℬjX=ℛj+X−ℛj−X, POSjX=ℛj−X, and NEGjX=U−ℛj+X are called ℛj-boundary, ℛj-positive, and ℛj-negative regions of X, respectively; ℳjX=ℛj−X∩X/ℛj+X∩X is called the ℛj-accuracy measure of X (if 0<ℛj+X∪X<ℵ0), where X denotes the cardinality of a set X and ℵ0 denotes the cardinality of the set of natural numbers.Theorem 2.
ℛj-approximations have the following properties (R⊆U2, E,F∪Eii∈I⊆2U, j∈J):(1)
ℛj+∅=∅⊆ℛj−∅.(2)
ℛj+U⊆U=ℛj−U.(3)
ℛj−∩i∈IEi=∩i∈Iℛj−Ei and ℛj+∩i∈IEi=∩i∈Iℛj+Ei. Particularly, ℛj−E⊆ℛj−F and ℛj+E⊆ℛj+F if E⊆F.(4)
ℛj−E=ℛj+Ecc and ℛj+E=ℛj−Ecc.(5)
In general,ℛj−ℛj−E≠ℛj−E and ℛj+ℛj+E≠ℛj+E.Proof.
We only prove (3) and (4).(3) Step 1: obviously,ℛj−∩i∈IEi⊆∩i∈Iℛj−Ei. Conversely, for each x∈∩i∈Iℛj−Ei (i.e., x∈ℛj−Ei for each i∈I), we have Ejx⊆Ei∀i∈I, and thus, Ejx⊆∩i∈IEi. Hence, x∈ℛj−∩i∈IEi, as required.Step 2: obviously,∪i∈Iℛj+Ei⊆ℛj+∪i∈IEi. Conversely, for each x∈ℛj+∪i∈IEi, we have Ejx∩∪i∈IEi≠∅, i.e., Ejx∩Ei0≠∅ for some i0∈I. Thus, x∈ℛj+Ei0⊆∪i∈Iℛj+Ei, as required.(4)x∈ℛj−Ec⇔Ejx⊆Ec⇔Ejx∩E=∅⇔x∉ℛj+E⇔x∈ℛj+Ec. The second equality immediately comes by putting E=Fc in the first.Example 2.
Forj-NS U,R,λj in Example 1, the ℛj-lower approximations and ℛj-upper approximations (j∈r,l,i,u), the ℛi-accuracy measure, the lower approximations X− and upper approximations X+ in the sense of [6] for P<r>, and the lower approximations X− and upper approximations X+ in the sense of [10] for Nr are given in Table 2 (where Y=v,w,y and Z=v,x,y). From Table 2, we can see the approximations in this paper and those in [6] (resp., [10]) are incomparable.Table 2
ℛj-lower approximations and ℛj-upper approximations (j∈r,l,i,u).
Xℛu−Xℛu+Xℛr−Xℛr+Xℛl−Xℛl+Xℛi−Xℛi+XℳiXX−X+X−X+v∅UxY∅Zxv,y0∅v,xxvw∅YxY∅w,yxw,y0wwxyx∅Zx∅∅Zx∅1∅v,xxvy∅UxY∅UxY0yyw,xYv,w∅UxY∅Uv,xY1/3wv,w,xxv,yv,x∅UxY∅Yxv,y1/3v,xv,xxvv,y∅UxY∅Uv,xY1/3yZw,xYw,x∅UxY∅Uxw,y1wv,w,xxv,yw,y∅UxYwUw,xY1/3w,yw,yw,x,yYx,y∅UxY∅UxY1/4yZw,xYv,w,x∅UxY∅UxY1/4v,w,xv,w,xxv,yYYUUYwUUY1w,yUw,x,yYZZUxYv,xUv,xY1ZZv,w,xYw,x,y∅UxYwUw,xY2/3w,yUw,x,yYUUUUYUUUY1UUUY∅∅∅x∅∅∅x∅


## 4. Rough Approximations Induced byE-Topologies

We will construct rough approximations usingE-neighborhoods indirectly in this section. Let U,R,λj be j-NS j∈J. We first employ Ej-neighborhoods to generate a topology JR,j (called an Ej-topology) and then call the interior JR,j−X and the closure JR,j+X of a subset X⊆U the JR,j-lower approximation and JR,j-upper approximation of X, respectively. These kinds of approximations are compared with those in Section 3.Theorem 3.
(1)
JR,j=A∈2U:Ejx⊆A∀x∈A is a topology on U satisfying U−A∈JR,j whenever A∈JR,jj∈J(2)
BothJR,N,j=A∈2U:Njx⊆A∀x∈A and JR,P,j=A∈2U:Pjx⊆A∀x∈A are topologies on Uj∈J(3)
JR,u⊆JR,r∩JR,l⊆JR,r∪JR,l⊆JR,i(4)
JR,u⊆JR,r∩JR,l⊆JR,l∪JR,l⊆JR,i(5)
IfR is reflexive, then JR,P,j⊆JR,jj∈J(6)
IfR is serial, then JR,P,j⊆JR,jj∈r,l,u,i(7)
IfR is a equivalence relation, then JR,j is constant for all j∈JProof.
We only prove (1). Obviously,JR,j is a topology on U. For each A∈JR,j and each x∈U−A, we need to prove Ejx⊆U−A. Without loss of generality, we assume Ejx≠∅. Suppose v∈Ejx∩A. Then, x∈Ejv (by Theorem 1 (3)) and Ejv⊆A (as A∈JR,j), and thus, x∈A. This is a contradiction. Therefore, Ejx∩A=∅, i.e., Ejx⊆U−A.Definition 7.
LetX⊆U and j∈J. Then, JR,j−X=Xo (the interior of X in U,JR,j) and JR,j+X=X− (the closure of X in U,JR,j) are called the JR,j-lower approximation and JR,j-upper approximation, respectively; JR,j+X−JR,j−X, JR,j+X, and U−JR,j+X are called JR,j-boundary, JR,j-positive, and JR,j-negative regions of X, respectively; ℳJ,jX=JR,j−X/JR,j+X is called the JR,j-accuracy measure of X (if 0<JR,j+X<ℵ0).
The relation betweenℳJ,jX and ℳjXj∈J is given by the following.Theorem 4.
ℳJ,jX≤ℳjX for each j∈J (if 0<JR,j+X<ℵ0).Proof.
(i)
Step 1: for eachz∈JR,j−X, we have z∈ℛj−X (because JR,j−X=Xo and Ejz⊆Xo⊆X), and thus, JR,j−X⊆ℛj−X∩X, which implies(1)JR,j−X≤ℛj−X∩X.(ii)
Step 2: letz∈ℛj+X∪X. If z∈X, then z∈JR,j−X. If z∉X, then z∈ℛj+X, and thus, Ejz∩X≠∅. This means there exists v∈U−z such that v∈Ejz and v∈X. Consequently, for any V∈JR,j containing z, we have v∈V. Therefore, V∩X≠∅, and thus, z∈JR,j+X (because JR,j+X=X−). It follows that ℛj+X∪X⊆JR,j+X, and thus,(2)1JR,j+X≤1ℛj+X∪X.
From (1) and (2), we can see that ℳJ,jX=JR,j−X/JR,j+X≤ℛj−X∩X/ℛj+X∪X=ℳjXj∈J.Example 3.
(1)
Now, we exemplify an application of rough approximations introduced in this paper. LetX=x1,x2,…,x50 be a group of people who have just reached Xian Yang Airport (but are not allowed to outbound station) from two countries (by two planes involving 200 people denoted by a set U=x1,x2,…,x200) and will attend a meeting holding in Xi’an. Assume that x1 and x2 are actually infected of asymptomatic infection of new coronavirus, Ejx∈2U consists of x and all y∈U who contacted x after x had contacted one of x1 and x2x∈U−x1,x2, and Ejx1=x1 and Ejx2=x2. To insure the safety of this meeting, ℛj−X=x∈U|Ejx⊆X=x1,x2∪x∈X| each person contacting x directly after x contacts one of x1 and x2 is in X can be looked to be the set of all people who must run a nucleic acid test, and ℛj+X=X∪x∈U|Ejx∩X≠∅=X∪x∈U−X| some person contacting x directly after x contacts one of x1 and x2 is in X can be looked to be the set of all people who should run a nucleic acid test.(2)
Forj-NS U,R,λj in Example 1, the JR,j-lower approximations and JR,j-upper approximations (j∈r,l,i,u) and the ℳJ,j-accuracy measure are given in Table 3.Table 3
JR,j-lower approximations and JR,j-upper approximations (j∈r,l,i,u).
XJR,u−XJR,u+XJR,r−XJR,r+XJR,l−XJR,l+XJR,i−XJR,i+XℳJ,iXℳiXv∅U∅v,w,y∅U∅v,w,y00w∅U∅v,w,y∅U∅v,w,y00x∅UxU∅UxU1/41y∅U∅v,w,y∅U∅v,w,y00v,w∅U∅v,w,y∅U∅v,w,y01/3v,x∅UxU∅UxU1/41/3v,y∅U∅v,w,y∅U∅v,w,y01/3w,x∅UxU∅UxU1/41w,y∅U∅v,w,y∅U∅v,w,y01/3x,y∅UxU∅UxU1/41/4v,w,x∅UxU∅UxU1/41/4v,w,y∅Uv,w,yv,w,y∅Uv,w,yv,w,y11v,x,y∅UxU∅UxU1/41w,x,y∅UxU∅UxU1/42/3UUUUUUUU11∅∅∅∅∅∅∅∅∅


## 5. Concluding Remarks

Motivated by topology, this article has initiated two new rough approximations by introducing a new class of neighborhood systems (calledEj-neighborhoods) using j-neighborhoods. We have probed the main features and formulated the concepts of Ej-lower and Ej-upper approximations and Ej-accuracy measure which are had contacted one of induced from different types of j and compared them. We complete this work by studying these concepts from a topological view and comparing them. In all comparisons, we obtain higher accurate approximations in the case of j=i.In the upcoming works, we will study new types of neighborhoods in rough set theory and use them to define a topological structure. Also, we will investigate theEj-neighborhoods and approximations on the fuzzy rough set motivated from fuzzy control problems and fractional-order nonlinear systems.



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