New Rough Approximations Based on E-Neighborhoods

/is paper puts forward some rough approximations which are motivated from topology. Given a subset R⊆ 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.


Introduction and Preliminaries
e problem of imperfect knowledge became a crucial issue for computer scientists, especially in the area of information system and artificial intelligence [1,2]. ere 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][6][7][8][9][10][11][12][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][19][20][21][22][23][24][25]). e present paper is a continuation to these works where we initiate new types of neighborhoods (namely, E j -neighborhoods) to give lower and upper approximations of an arbitrary set X directly or indirectly. In Section 2, we apply N j -neighborhoods to establish the concepts of E j -neighborhoods and discuss the main properties. With the help of examples, we show the relationships between them as well as with N j -neighborhoods and P j -adhesion neighborhoods. In Section 3, we formulate and study the concepts of E j -lower and E j -upper approximations, E j -boundary region, E j -positive and E j -negative regions, and E j -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 subset R ⊆ 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., , 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 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 } are called the Pawlak lower approximation and upper approximation of X, re- } is the set of all equivalence classes).
e lower approximations and the upper approximations have the following properties ( E, F { } ∪ E i i∈I ⊆ 2 U , the power set of U ): Definition 2. A subset J ⊆ 2 U 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 A o 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. Let is 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〉 { }.
(1) e 8 kinds of j-neighborhoods are defined as follows: Definition 4 (see [6]). Let R ⊆ U 2 . e j-adhesion neighborhoods are defined as follows:

E-Neighborhoods
In this section, we introduce the notions of E-neighborhoods using j-neighborhoods and study their properties.
e E-neighborhoods are defined as follows: 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.
en, the j-neighborhoods and E-neighborhoods of a point are as in Table 1.
Step 2: let x ∈ N j (v). By reflexivity of R, we have Similarly, It follows from (4) and the fact v Step 1: we prove the case j � r. Let x ∈ E r (v). en, N r (x) ∩ N r (v) ≠ ∅, i.e., there exists z ∈ N r (x) ∩ N r (v), xRz, and vRz. Since R is symmetric and transitive, vRx.
Step 2: since R is symmetric, (5). We only prove the case j � l. Let v ∈ E l (w). en, N l (v) ∩ N l (w) ≠ ∅; therefore, there is a ∈ U such that aRv and aRw. Now, let x ∈ E l (v). en, N l (x) ∩ N l (v) ≠ ∅, i.e., there exists a b ∈ U such that bRx and bRv. Note that aRb; this means that aRx; consequently, a ∈ N l (x); also, a ∈ N l (w); thus, N l (x) ∩ N l (x) ≠ ∅. Hence, x ∈ E l (w), as required. (10) Step 1: by (4) and (8), Step 2: we only prove the case j � l.

Rough Approximations Using E-Neighborhoods Directly
Let (U, R, λ j ) be j-NS and R j � E j (v): v ∈ U (j ∈ J). We devote this section to formulate the following concepts using E-neighborhoods directly: R j -lower approximation, R j -upper approximation, R j -boundary region, R j -positive region, R j -negative region, and R j -accuracy measure of a subset X. We will also illustrate the relationships between them and reveal the main properties with the help of examples.
where |X| denotes the cardinality of a set X and ℵ 0 denotes the cardinality of the set of natural numbers.

Theorem 2. R j -approximations have the following prop-
Proof. We only prove (3) and (4). (3) Step 1: obviously, e second equality immediately comes by putting E � F c in the first. □ Example 2. For j-NS (U, R, λ j ) in Example 1, the R j -lower approximations and R j -upper approximations (j ∈ r, l, i, u { }), the R i -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 N r 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.

Rough Approximations Induced by E-Topologies
We will construct rough approximations using E-neighborhoods indirectly in this section. Let (U, R, λ j ) be j-NS (j ∈ J). We first employ E j -neighborhoods to generate a topology J R,j (called an E j -topology) and then call the interior J − R,j (X) and the closure J + R,j (X) of a subset X ⊆ U the J R,j -lower approximation and J R,j -upper approximation of X, respectively. ese kinds of approximations are compared with those in Section 3.

Theorem 3
Proof. We only prove (1). Obviously, J R,j is a topology on U. For each A ∈ J R,j and each x ∈ U − A, we need to prove E j (x) ⊆ U − A. Without loss of generality, we assume en, x ∈ E j (v) (by eorem 1 (3)) and E j (v) ⊆ A (as A ∈ J R,j ), and thus, x ∈ A. is is a contradiction. erefore, U � x 1 , x 2 , . . . , x 200 ) and will attend a meeting holding in Xi'an. Assume that x 1 and x 2 are actually infected of asymptomatic infection of new coronavirus, E j (x) ∈ 2 U consists of x and all y ∈ U who contacted x after x had contacted one of x 1 and x 2 (x ∈ U − x 1 , x 2 ), and E j (x 1 ) � x 1 and E j (x 2 ) � x 2 . To insure the safety of this meeting, person contacting x directly after x contacts one of x 1 and x 2 is in X} can be looked to be the set of all people who must run a nucleic acid test, and { some person contacting x directly after x contacts one of x 1 and x 2 is in X} can be looked to be the set of all people who should run a nucleic acid test.
(2) For j-NS (U, R, λ j ) in Example 1, the J R,j -lower approximations and J R,j -upper approximations (j ∈ r, l, i, u { }) and the M J,j -accuracy measure are given in Table 3.

Concluding Remarks
Motivated by topology, this article has initiated two new rough approximations by introducing a new class of neighborhood systems (called E j -neighborhoods) using j-neighborhoods. We have probed the main features and formulated the concepts of E j -lower and E j -upper approximations and E j -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 Table 3: J R,j -lower approximations and J R,j -upper approximations (j ∈ r, l, i, u