# Rough Approximation Spaces via Maximal Union Neighborhoods and Ideals with a Medical Application

Authors: Mona Hosny; Tareq M. Al-shami; Abdelwaheb Mhemdi
Journal: Journal of Mathematics
 (2022)

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## Abstract

One of the most popular and important tools to deal with imperfect knowledge is the rough set theory. It starts from dividing the universe to obtain blocks utilizing an equivalence relation. To make it more flexibility and expand its scope of applications, many generalized rough set models have been proposed and studied. To contribute to this area, we introduce new generalized rough set models inspired by “maximal union neighborhoods and ideals.” These models are created with the aim to help decision-makers to analysis and evaluate the given data more accurately by decreasing the ambiguity regions. We confirm this aim by illustrating that the current models improve the approximations operators (lower and upper) and accuracy measures more than some existing method approaches. We point out that almost all major properties with respect to rough set model can be kept using the current models. One of the interesting obtained characterizations of the current models is preserving the monotonic property, which enables us to evaluate the vagueness in the data and enhance the confidence in the outcomes. Moreover, we compare the current approximation spaces with the help of concrete examples. Finally, we show the performance of the current models to discuss the information system of dengue fever disease and eliminate the ambiguity of the medical diagnosis, which produces an accurate decision.


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## Body

## 1. Introduction

Rough set theory was introduced via Pawlak [1, 2], as one of the representative granular computing models. It offers a useful mathematical instrument to cope with ambiguity/uncertainty. Presently, rough set theory has been successfully applied to several areas such as approximate reasoning [3], machine learning [4], incomplete information systems [5], and decision analysis [6]. Pawlak [1, 2] expressed the classical rough set theory by two new sets, obtained from an equivalence relation, called lower and upper approximations. To know the vagueness size and measure the completeness of data, the concepts of boundary regions and accuracy measures were presented. While the boundary region defined as the difference between the approximation operators (upper and lower), the accuracy measure represents a number obtained from the quotient of the cardinalities of lower approximation and upper approximation. To remove the strict condition imposed in the classical rough set model, many scholars replaced the equivalence relation by similarity relation [7], tolerance relation [8], or binary relation [9, 10].Additionally, to extend the applications of this theory, Yao [11, 12] proposed new classes of blocks (or granular computing) induced from right, left, and the intersection (union) of left and right neighborhoods with respect to any binary relation. This matter forms a revolution in rough set theory and opens a door for studying many types of neighborhood systems which are identical under an equivalence relation. Following this line, different types of neighborhood systems have been established with the aim to introduce several generalized rough set models (or approximation spaces), minimal left and minimal right neighborhoods [13], neighborhoods produced by the intersection of minimal left and right neighborhoods [14], and core neighborhoods [15]. Abu-Donia [16, 17] investigated some sorts of approximation operators using a family of binary relations instead of one binary relation. Ciucci [18] investigated different types of orthopair systems. In 2018, Dai et al. [19] familiarized the maximal right neighborhoods under a similarity relation and then applied it to present three different sorts of approximation spaces. Recently, Al-shami [20] has discussed new seven kinds of rough maximal neighborhoods and benefited to introduce new approximation spaces and rank suspected individuals of COVID-19. He proved that the accuracy measures kept the monotonic property with respect to any relation. With the desire to minimize the boundary region and maximize the accuracy values, the systems of containment neighborhoods [21] and subset neighborhoods [22] were studied and explored under arbitrary relation.In 2013, Kandi et al. [23] made use of an abstract structure called “ideal” to define new rough paradigms called “ideal approximation spaces.” This paradigm proved its efficiency to enlarge the knowledge obtained from the information system, and hence, high the degree of its completeness. Moreover, it was applied to model some uncertainty phenomena as illustrated in [24–26]. Recently, Hosny [27] has inserted the ideal structures via topological approximation spaces and showed how this technique deletes some objects from the upper approximations and/or adds new objects to the lower approximations, which means improving the approximation operators more than their counterparts induced from topological approximation spaces.More recently, Al-shami and Hosny [28, 29] have displayed new ideal approximation spaces with respect to maximal left neighborhood under any binary relation and showed their advantages compared to the previous ones. Following this line of research, we propose new generalized rough set models inspired by “maximal union neighborhoods and ideals.” By these models, we successfully decrease the ambiguity regions which helps us to evaluate the given data more accurately. The good performance of the current approach is illustrated by making comparisons with the approach introduced in [20].This manuscript is structured as follows. Section2 is dedicated to the basic concepts and properties on rough neighborhood systems and ideals. The goal of Section 3 is to introduce new four ideals approximation spaces and their essential characterizations are explored. In Section 4, with the help of illustrated examples, the comparisons between these models are studied as well as the comparison of the first kind of these models with its counterpart in [20] is investigated. The significance of the proposed paradigms with a medical diagnosis of dengue fever disease is demonstrated in Section 5. Finally, a summary of the obtained findings is given and upcoming works are proposed in Section 6.


## 2. Preliminaries

In this section, we briefly recall the essential of approximation spaces defined with respect to equivalence relations or arbitrary binary relations. Also, we mention the main properties of ideal structures.Definition 1 (see [1, 2]).
TakeR as an equivalence relation on a universe E and let eR be the equivalence class containing e. For any subset F of E, the lower and upper approximations and accuracy measure are defined, respectively, by
(1)apr¯F=e∈E:eR⊆F;(2)apr¯F=e∈E:eR∩F≠ϕ;and(3)AccRF=apr¯Fapr¯F.The main properties of these approximations (known as Pawlak’s properties) are the following.(L1) apr¯Fc=apr¯Fc, where Fc is the complement of F.(L2) apr¯E=E.(L3) apr¯ϕ=ϕ.(L4) apr¯F⊆F.(L5) apr¯F∩G=apr¯F∩apr¯G(L6) apr¯F∪G⊇apr¯F∪apr¯G(L7) F⊆G⇒apr¯F⊆apr¯G.(L8) apr¯apr¯F=apr¯F.(L9) apr¯F⊆apr¯apr¯F.(U1) apr¯Fc=apr¯Fc.(U2) apr¯E=E.(U3) apr¯ϕ=ϕ.(U4) F⊆apr¯F.(U5) apr¯F∪G=apr¯F∪apr¯G.(U6) apr¯F∩G⊆apr¯F∩apr¯G.(U7) F⊆G⇒apr¯F⊆apr¯G.(U8) apr¯apr¯F=apr¯F.(U9) apr¯apr¯F⊆apr¯F.Definition 2 (see [20]).
If the relationsR1 and R2 are equivalence on a universe E such that R1⊆R2. Then, the approximations inspired by these relations have the property of monotonicity (monotonic property) if AccR2F≤AccR1F for each F⊆E.Definition 3 (see [11, 19, 20]).
The next neighborhoods of an elemente of a finite set E≠∅ are defined with respect to any arbitrary binary relation R as follows.
(1)
The right neighborhood ofe, symbolized by Nre, is given by Nre=a∈E:e,a∈R(2)
The left neighborhood ofe, symbolized by Nle, is given by Nre=a∈E:a,e∈R(3)
μre is the union of all right neighborhoods containing e(4)
μle is the union of all left neighborhoods containing e(5)
μue=μre∪μleTheorem 4 [20].
LetE be a universal set and R1,R2 be two binary relations on E. If R1⊆R2, then, μ1ue⊆μ2ue,∀e∈E.Definition 5 (see [20]).
LetR be a binary relation on a nonempty set E. The lower and upper approximations, boundary region, accuracy, and roughness of a nonempty subset F of E are defined, respectively, by
(4)R⋆F=e∈E:μue⊆F,(5)R⋆F=e∈E:μue∩F≠ϕ,(6)BR⋆F=R⋆F−R⋆F,(7)AccR⋆F=R⋆F∩FR⋆F∪F,(8)RoughR⋆F=1−AccR⋆F.Definition 6 (see [30]).
An idealJ on a set E≠ϕ is nonempty collection of subsets of E that is closed under finite unions and subsets; i.e., it satisfies the following conditions:
(1)
F∈J and H∈J⇒F∪H∈J(2)
F∈J and H⊆F⇒H∈JDefinition 7 (see [26]).
For two ideals on a nonempty setE, the smallest collection generating by J1,J2, denoted by J1∨J2, is given by
(9)J1∨J2=G∪F:G∈J1,F∈J2.Proposition 8 (see [26]).
IfJ1,J2 are ideals on a nonempty set E, and F,H are subsets of E. Then, the collection J1∨J2 satisfies the next conditions:
(1)
J1∨J2≠ϕ(2)
F∈J1∨J2,H⊆F⇒H∈J1∨J2(3)
F,H∈J1∨J2⇒F∪H∈J1∨J2
It means that the collectionJ1∨J2 is an ideal on E.


## 3. Some Novel Rough Set Paradigms Produced byμue-Neighborhoods and Ideals

In this section, we display four paradigms of approximation spaces induced fromR˘<x>R˘-neighborhoods and ideals. With the aid of elucidative examples and counterexamples, we point out some directions that are invalid in the obtained findings and relationships.


### 3.1. The First Method of the Improvement of the Approximations and Accuracy Measure of Subsets

Definition 9.
LetR and J be, respectively, binary relation and ideal on a nonempty set E. The first kind of the improvement of lower and upper approximations, boundary region, accuracy, and roughness of a nonempty subset F of E induced from R and J is defined, respectively, by
(10)R⋆JF=e∈E:μue∩Fc∈J,(11)R⋆JF=e∈E:μue∩F∈J,(12)BNDR⋆JF=R⋆JF−R⋆JF,(13)ACCR⋆JF=R⋆JF∩FR⋆JF∪F,(14)RoughR⋆JF=1−ACCR⋆JF.Proposition 10.
ConsiderF,H⊆E and let J,K be ideals and R be a binary relation on E. Then,
(1)
R⋆Jϕ=ϕ(2)
F⊆H⇒R⋆JF⊆R⋆JH(3)
R⋆JF∩H⊆R⋆JF∩R⋆JH(4)
R⋆JF∪H=R⋆JF∪R⋆JH(5)
R⋆JF=R⋆JFcc(6)
IfF∈J, then R⋆JF=ϕ(7)
IfJ⊆K, then R⋆KF⊆R⋆JF(8)
IfJ=PE, then R⋆JF=ϕ(9)
R⋆J∩KF=R⋆JF∪R⋆KF(10)
R⋆J∨KF=R⋆JF∩R⋆KFProof.
(1)
R⋆Jϕ=e∈E:μue∩ϕ∈J.=ϕ(2)
Lete∈R⋆JF. Then, μue∩F∈J. Since, F⊆H and J is an ideal. Thus, μue∩H∈J. Therefore, e∈R⋆JH. Hence, R⋆JF⊆R⋆JH(3)
Immediately by part (2)(4)
R⋆JF∪R⋆JH⊆R⋆JF∪H by part (2). Let e∈R⋆JF∪H. Then, μue∩F∪H∈J. It follows that μue∩F∪μue∩H∈J. Therefore, μue∩F∈I or μue∩H∈J, that means e∈R⋆JF or e∈R⋆JH. Then, e∈R⋆JF∪R⋆JH. Thus, R⋆JF∪H⊆R⋆JF∪R⋆JH. Hence, R⋆JF∪H=R⋆JF∪R⋆JH(5)
R⋆JFcc=e∈E:μue∩F∈Jc.=e∈E:μue∩F∈J.=R⋆JF(6)
Immediately obtains by Definition9(7)
Lete∈R⋆KF. Then, μue∩F∈K. Since, J⊆K. Thus, μue∩F∈J. Therefore, e∈R⋆JF. Hence, R⋆KF⊆R⋆JF(8)
Immediately obtains by Definition9(9)
R⋆J∩KF=e∈E:μue∩F∈J∩K.=e∈E:μue∩F∈Jore∈E:μue∩F∈K.=e∈E:μue∩F∈J∪e∈E:μue∩F∈K.=R⋆JF∪R⋆KF(10)
R⋆J∨KF=e∈E:μue∩F∈J∨K.=e∈E:μue∩F∈J∪K.=e∈E:μue∩F∈Jande∈E:μue∩F∈K.=e∈E:μue∩F∈J∩e∈E:μue∩F∈K.=R⋆JF∩R⋆KFProposition 11.
ConsiderF,H⊆E and let J,K be ideals and R be a binary relation on E. Then,
(1)
R⋆JE=E(2)
F⊆H⇒R⋆JF⊆R⋆JH(3)
R⋆JF∪R⋆JH⊆R⋆JF∪H(4)
R⋆JF∩H=R⋆JF∩R⋆JH(5)
R⋆JF=R⋆JFcc(6)
IfFc∈J, then R⋆JF=E(7)
IfJ⊆K, then R⋆JF⊆R⋆KF(8)
IfJ=PE, then R⋆JF=E(9)
R⋆J∩KF=R⋆JF∩R⋆KF(10)
R⋆J∨KF=R⋆JF∪R⋆KFProof.
(1)
R⋆JE=e∈E:μue∩ϕ∈J.=E(2)
Lete∈R⋆JF. Then, μue∩Fc∈J. Since, Hc⊆Fc and J is an ideal. Thus, μue∩Hc∈J. Therefore, e∈R⋆JH. Hence, R⋆JF⊆R⋆JH(3)
Immediately by part (2)(4)
R⋆JF∩R⋆JH⊇R⋆JF∩H by part (2). Let e∈R⋆JF∩R⋆JH. Then, μue∩Fc∈J and μue∩Hc∈J. It follows that μue∩Fc∪Hc∈J. So, μue∩F∩Hc∈J. Therefore, e∈R⋆JF∩H. Thus, R⋆JF∩R⋆JH⊆R⋆JF∩H. Hence, R⋆JF∩R⋆JH=R⋆JF∩H(5)
R⋆JFcc=e∈E:μue∩Fc∈Jc.=e∈E:μue∩Fc∈J.=R⋆JF(6)
Immediately obtains by Definition9(7)
Lete∈R⋆JF. Then, μue∩Fc∈J. Since, J⊆K. Thus, μue∩Fc∈K. Therefore, e∈R⋆JF. Hence, R⋆JF⊆R⋆KF(8)
Immediately obtains by Definition9(9)
R⋆J∩KF=e∈E:μue∩Fc∈J∩K.=e∈E:μue∩Fc∈Jande∈E:μue∩Fc∈K.=e∈E:μue∩Fc∈J∩e∈E:μue∩Fc∈K.=R⋆JF∩R⋆KF(10)
R⋆J∨KF=e∈E:μue∩Fc∈J∨K.=e∈E:μue∩Fc∈J∪K.=e∈E:μue∩Fc∈Jore∈E:μue∩Fc∈K.=e∈E:μue∩Fc∈J∪e∈E:μue∩Fc∈K.=R⋆JF∪R⋆KFRemark 12.
By the next example, we elucidate that(1)
The converse of parts 2, 6, 7, and 8 in Proposition10 and Proposition 11 is generally incorrect(2)
The inclusion relations of part 3 in Proposition10 and Proposition 11 are generally properExample 1.
(i)
LetE=e1,e2,e3,e4,J=ϕ,e1,e2,e3,e1,e2,e1,e3,e2,e3,e1,e2,e3 and R=e1,e2,e1,e3,e2,e3,e2,e4,e3,e1,e3,e4 be a binary relation defined on E. By calculations, we obtain μue1=e1,e2,e4,μue2=e1,e2,e3,μue3=e2,e3,e4,μue4=e1,e3,e4. For part 2, take F=e1,e2 and H=e1,e4, then(a)
R⋆JF=ϕ,R⋆JH=e1,e3,e4. Therefore, R⋆JF⊆R⋆JH, but F⊈H(b)
R⋆JF=e2,R⋆JH=E. Therefore, R⋆JF⊆R⋆JH, but F⊈H(ii)
LetE=e1,e2,e3,e4,K=ϕ,e2,J=ϕ,e1 and R=e2,e2,e3,e3,e4,e4 be a binary relation defined on E. By calculations, we obtain μue1=ϕ,μue2=e2,μue3=e3,μue4=e4(1)
For part 6, take(a)
F=e1,e2, then, R⋆KF=ϕ. Therefore, R⋆KF=ϕ, but F∈K.(b)
F=e3,e4, then, R⋆KF=E. Therefore, R⋆KF=E, but Fc∈K.(2)
For part 7, take(a)
F=e1,e2, then, R⋆JF=e2,R⋆KF=ϕ. Therefore, R⋆KF⊆R⋆JF, but J⊈K(b)
F=e3,e4, then, R⋆JF=e1,e3,e4,R⋆KF=E. Therefore, R⋆JF⊆R⋆KF, but J⊈K(3)
For part 8, take(a)
F=e1,e2, then, R⋆KF=ϕ, but K≠PE(b)
F=e3,e4, then, R⋆KF=E, but K≠PE(iii)
LetE=e1,e2,e3,e4,J=ϕ,e4 and R=Δ∪e2,e1,e3,e1,e4,e1 be a binary relation defined on E, (where Δ is the identity relation and equal to e1,e1,e2,e2,e3,e3,e4,e4. By calculations, we obtain μue1=μue2=μue3=μue4=E. For part 3, take F=e1,e4,H=e2,e3. Hence(a)
F∩H=ϕ, then R⋆JF=R⋆JH=E,R⋆JF∩H=ϕ. Therefore, R⋆JF∩R⋆JH=E≠ϕ=R⋆JF∩H(b)
F∪H=E, then R⋆JF=R⋆JH=ϕ,R⋆JF∪H=E. Therefore, R⋆JF∪R⋆JH=ϕ≠E=R⋆JF∪HRemark 13.
Some Pawlak’s properties are not kept by this method as we demonstrate in the following.(i)
In Example1 (i) take(1)
F=e1,e2, then R⋆JF=ϕ. Hence, F⊈R⋆JF(2)
F=e1,e4, then R⋆JF=E. Hence, R⋆JF⊈F(3)
F=E, then R⋆JE=e1,e3,e4. Hence, R⋆JE≠E(4)
F=ϕ, then R⋆Jϕ=e2. Hence, R⋆Jϕ≠ϕ(ii)
In Example1 (i) take J=ϕ,e1(1)
F=e1,e4, then R⋆JF=e1,e3,e4,R⋆JR⋆JF=E. Hence, R⋆JF≠R⋆JR⋆JF(2)
F=e2,e3, then R⋆JF=e2,R⋆JR⋆JF=ϕ. Hence, R⋆JF≠R⋆JR⋆JF(3)
F=e1,e2, then R⋆JF=e1,e2,e3 and R⋆JR⋆JF=e2. Hence, R⋆JF⊈R⋆JR⋆JF(4)
F=e3,e4, then R⋆JF=e4 and R⋆JR⋆JF=e1,e3,e4. Hence, R⋆JR⋆JF⊈R⋆JFProposition 14.
ConsiderR and J are, respectively, relation and ideal on E≠ϕ and let ϕ≠F⊆E. Then,
(1)
0≤ACCR⋆JF≤1(2)
ACCR⋆JE=1.Proof.
To prove 1, it is obvious thatR⋆JF∪F≠ϕ for every nonempty subset F of E. Hence, ϕ⊆R⋆JF∩F⊆R⋆JF∪F. Therefore, 0≤R⋆JF∩F≤R⋆JF∪F. So, 0≤R⋆JF∩F/R⋆JF∪F≤1. It means that, 0≤ACCR⋆JF≤1.
Case 2 can be proved easily.Theorem 15.
ConsiderR as a relation on E and let J,K be ideals on E such that J⊆K. For each F⊆E, the next properties hold.
(1)
BNDR⋆KF⊆BNDR⋆JF(2)
ACCR⋆JF≤ACCR⋆KF(3)
RoughR⋆KF≤RoughR⋆JFProof.
(1)
Lete∈BNDR⋆KF. Then, e∈R⋆KF−R⋆KF. So, e∈R⋆KF and e∈R⋆KFc. Hence, e∈R⋆JF and e∈R⋆JFc by Propositions 10 and 11 part 7. It follows that e∈BNDR⋆JF. Therefore, BNDR⋆KF⊆BNDR⋆JF(2)
ACCR⋆JF=R⋆JF∩F/R⋆JF∪F.≤R⋆KF∩F/R⋆KF∪F.=ACCR⋆KF(3)
Directly follows from 2Remark 16.
In Theorem15, the converses of parts 1, 2, and 3 are generally false. To elucidate that, take F=e3,e4 as a subset of Example 1 (ii). Then
(1)
BNDR⋆KF=ϕ⊆ϕ=BNDR⋆JF, but J⊈K(2)
ACCR⋆JF=1≤1=ACCR⋆KF, but J⊈K(3)
RoughR⋆KF=0≤0=RoughR⋆JF, but J⊈KTheorem 17.
Letϕ≠F⊆E,J be an ideal on E and R1,R2 be two binary relations on E. If R1⊆R2, then
(1)
R1⋆JF⊆R2⋆JF(2)
R2⋆JF⊆R1⋆JF(3)
BNDR1⋆JF⊆BNDR2⋆JF(4)
ACCR2⋆JF≤ACCR1⋆JF(5)
RoughR1⋆JF≤RoughR2⋆JFProof.
(1)
Lete∈R1⋆JF. Then, μ1ue∩F∈J. Since, μ1ue⊆μ2ue (by Theorem 4 [20]). It follows that μ2ue∩F∈J. Thus, e∈R2⋆JF. Hence, R1⋆JF⊆R2⋆JF(2)
Lete∈R2⋆JF. Then, μ2ue∩Fc∈J. Since, μ1ue⊆μ2ue (by Theorem 4 [20]). It follows that μ1ue∩Fc∈J. Thus, e∈R1⋆JF. Hence, R2⋆JF⊆R1⋆JF(3)
Lete∈BNDR1⋆JF. Then, e∈R1⋆JF−R1⋆JF. So, e∈R1⋆JF and e∈R1⋆JFc. Thus, e∈R2⋆JF and e∈R1⋆JFc by parts 1 and 2. Hence, e∈BNDR2⋆JF. Therefore, BNDR1⋆JF⊆BNDR2⋆JF(4)
ACCR2⋆JF=R2⋆JF∩F/R2⋆JF∪F.≤R1⋆JF∩F/R1⋆JF∪F.=ACCR1⋆JF(5)
Straightforward by 4
In Theorem17, the relations of inclusion and less than are generally proper as we illustrate by the next example.Example 2.
LetE=e1,e2,e3,e4,J=ϕ,e2,e3,e4,e2,e3,e2,e4,e3,e4,e2,e3,e4 and R1=Δ∪e1,e4,e4,e1,R2=R1∪e1,e3,e3,e1 be two binary relation defined on E thus μ1ue1=μ1ue4=e1,e4,μ1ue2=e2,μ1ue3=e3,μ2ue1=μ2ue3=μ2ue4=e1,e3,e4,μ2ue2=e2. Take
(a)
F=e1,e4, then(1)
R1⋆JF=e4≠e1,e3,e4=R2⋆JF(2)
ACCR1⋆JF=2/3≠1/2=ACCR2⋆JF(3)
RoughR2⋆JF=1/2≠1/3=RoughR1⋆JF(4)
F=e4,e3, then R1⋆JF=e2,e3,e4≠e2=R2⋆JF


### 3.2. The Second Method of the Improvement of the Approximations and Accuracy Measure of Subsets

Definition 18.
LetR and J be, respectively, binary relation and ideal on a nonempty set E. The second kind of the improvement of lower and upper approximations, boundary region, accuracy, and roughness of a nonempty subset F of E induced from R and J is defined, respectively, by
(15)R⋆⋆JF=e∈F:μue∩Fc∈J,(16)R⋆⋆JF=F∪R⋆JF,(17)BNDR⋆⋆JF=R⋆⋆JF−R⋆⋆JF,(18)ACCR⋆⋆JF=R⋆⋆JFR⋆⋆JF,(19)RoughR⋆⋆JF=1−ACCR⋆⋆JF.Proposition 19.
ConsiderF,H⊆E and let J,K be ideals and R be a binary relation on E. Then,
(1)
F⊆R⋆⋆JF equality hold if F=ϕ or E(2)
F⊆H⇒R⋆⋆JF⊆R⋆⋆JH(3)
R⋆⋆JF⊆R⋆⋆JR⋆⋆JF(4)
R⋆⋆JF∩H⊆R⋆⋆JF∩R⋆⋆JH(5)
R⋆⋆JF∪H=R⋆⋆JF∪R⋆⋆JH(6)
R⋆⋆JF=R⋆⋆JFcc(7)
IfF∈J, then R⋆⋆JF=F(8)
IfJ⊆K, then R⋆⋆KF⊆R⋆⋆JF(9)
IfJ=PE, then R⋆⋆JF=F(10)
R⋆⋆J∩KF=R⋆⋆JF∪R⋆⋆KF(11)
R⋆⋆J∨KF=R⋆⋆JF∩R⋆⋆KFProof.
Similar to the technique provided in Proposition10.Proposition 20.
ConsiderF,H⊆E and let J,K be ideals and R be a binary relation on E. Then,
(1)
R⋆⋆JF⊆F equality hold if F=ϕ or E(2)
F⊆H⇒R⋆⋆JF⊆R⋆⋆JH(3)
R⋆⋆JR⋆⋆JF⊆R⋆⋆JF(4)
R⋆⋆JF∪R⋆⋆JH⊆R⋆⋆JF∪H(5)
R⋆⋆JF∩H=R⋆⋆JF∩R⋆⋆JH(6)
R⋆⋆JF=R⋆⋆JFcc(7)
IfFc∈J, then, R⋆⋆JF=F(8)
IfJ⊆K, then, R⋆⋆JF⊆R⋆⋆KF(9)
IfJ=PE, then, R⋆⋆JF=F(10)
R⋆⋆J∩KF=R⋆⋆JF∩R⋆⋆KF(11)
R⋆⋆J∨KF=R⋆⋆JF∪R⋆⋆KFProof.
Similar to Proposition11.Remark 21.
(i)
In Proposition19 and Proposition 20, the converse of parts 7 and 9 is generally incorrect. To show that, consider Example 1 (i).
(a)
For part 7, take(1)
F=e1,e3,e4, then R⋆⋆JF=F, but F∈J(2)
F=e2, then R⋆⋆JF=F, but Fc∈J(b)
For part 9, take(1)
F=e1,e3,e4, then R⋆⋆JF=F, but J≠PE(2)
F=e2, then R⋆⋆JF=F, but J≠PE(ii)
In Proposition19 and Proposition 20, the converse of part 8 is generally incorrect. To elucidate that consider Example 1 (ii)(1)
IfF=e1,e2, then, R⋆⋆KF=e1,e2⊆e1,e2=R⋆⋆JF, but J⊈K(2)
IfF=e3,e4, then, R⋆⋆JF=e3,e4⊆e3,e4=R⋆⋆KF, but J⊈K(iii)
In Example1 (i), take J=ϕ,e1. So, the converse of part 2 in Proposition 19 and Proposition 20 is generally incorrect. Additionally, the inclusion relations of parts 3 and 4 in Proposition 19 and Proposition 20 generally proper(a)
For part 2, takeF=e1,e4,H=e2,e3, then(1)
R⋆⋆JF=e1,e3,e4⊆E=R⋆⋆JH=E, but F⊈H(2)
R⋆⋆JF=ϕ⊆e2=R⋆⋆JH, but F⊈H(b)
For part 3, take(1)
F=e1,e4, then, R⋆⋆JF=e1,e3,e4,R⋆⋆JR⋆⋆JF=E. Therefore, R⋆⋆JF=e1,e3,e4≠E=R⋆⋆JR⋆⋆JF(2)
F=e2,e3, then, R⋆⋆JF=e2,R⋆⋆JR⋆⋆JF=ϕ. Therefore, R⋆⋆JF=e2≠ϕ=R⋆⋆JR⋆⋆JF(c)
For part 4, takeF=e1,e4,H=e2,e3 thus(1)
F∩H=ϕ, then, R⋆⋆JF=e1,e3,e4,R⋆⋆JH=E,R⋆⋆JF∩H=ϕ. Therefore, R⋆⋆JF∩R⋆⋆JH=e1,e3,e4≠ϕ=R⋆⋆JF∩H(2)
F∪H=E, then, R⋆⋆JF=ϕ,R⋆⋆JH=e2,R⋆⋆JF∪H=E. Therefore, R⋆⋆JF∪R⋆⋆JH=e2≠E=R⋆⋆JF∪HRemark 22.
Some properties of the lower and upper approximations given in the first method are not kept by this method as we demonstrate in the following.(i)
In Example1 (i) take(1)
F=u1∈J, then, R⋆⋆JF=F. Hence, if F∈J⇏R⋆⋆JF=ϕ(2)
Fc=u1∈J, then, R⋆⋆JF=F. Hence, if Fc∈J⇏R⋆⋆JF=U(ii)
In Example1 (ii) take(1)
K=PE,F=e1,e3, then, R⋆⋆KF=F. Hence, if K=PE⇏R⋆⋆KF=ϕ(2)
K=PE,F=e2,e4, then, R⋆⋆KF=F. Hence, if K=PE⇏R⋆⋆KF=ERemark 23.
Some Pawlak’s properties are not kept by this method as we demonstrate in the following. In Example1 (i) take J=ϕ,e1,(1)
F=e1,e2, then, R⋆⋆JF=e1,e2,e3 and R⋆⋆JR⋆⋆JF=e2. Hence, R⋆⋆JF⊈R⋆⋆JR⋆⋆JF(2)
F=e3,e4, then, R⋆⋆JF=e4 and R⋆⋆JR⋆⋆JF=e1,e3,e4. Hence, R⋆⋆JR⋆⋆JF⊈R⋆⋆JFProposition 24.
ConsiderR and J are, respectively, binary relation and ideal on E≠ϕ and let ϕ≠F⊆E. Then,
(1)
0≤ACCR⋆⋆JF≤1(2)
ACCR⋆⋆JE=1Proof.
As given in Proposition14.Theorem 25.
ConsiderR as a binary relation on E and let J,K be ideals on E such that J⊆K. For each F⊆E, the next properties hold.
(1)
BNDR⋆⋆KF⊆BNDR⋆⋆JF(2)
ACCR⋆⋆JF≤ACCR⋆⋆KF(3)
RoughR⋆⋆KF≤RoughR⋆⋆JFProof.
Following similar arguments given in Theorem15.Remark 26.
In Theorem25, the converses of parts 1, 2, and 3 are generally incorrect. To validate this notice, consider Example 1 (ii) and take F=e2,e4. Then,
(1)
BNDR⋆⋆KF=ϕ⊆ϕ=BNDR⋆⋆JF, but J⊈K(2)
ACCR⋆⋆KF=1≤1=ACCR⋆⋆JF, but J⊈K(3)
RoughR⋆⋆KF=0≤0=RoughR⋆⋆JF, but J⊈KTheorem 27.
Letϕ≠F⊆E,J be an ideal on E and R1,R2 be two binary relations on E. If R1⊆R2, then
(1)
R1⋆⋆JF⊆R2⋆⋆JF(2)
R2⋆⋆JF⊆R1⋆⋆JF(3)
BNDR1⋆⋆JF⊆BNDR2⋆⋆JF(4)
ACCR2⋆⋆JF≤ACCR1⋆⋆JF(5)
RoughR1⋆⋆JF≤RoughR2⋆⋆JFProof.
Similar to Theorem17.Remark 28.
In Theorem27, the inclusion and less than relations are generally proper. To illustrate that, consider Example 2 and take F=e1,e2. Then,
(1)
R1⋆⋆JF=e1,e4≠E=R2⋆⋆JF(2)
BNDR1⋆⋆JF=e4≠e1,e3,e4=BNDR2⋆⋆JF(3)
ACCR1⋆⋆JF=2/3≠1/4=ACCR2⋆⋆JF(4)
RoughR1⋆⋆JF=1/3≠3/4=RoughR2⋆⋆JF


### 3.3. The Third Method of the Improvement of the Approximations and Accuracy Measure of Subsets

Definition 29.
LetR and J be, respectively, binary relation and ideal on a nonempty set E. The third kind of the improvement of lower and upper approximations, boundary region, accuracy, and roughness of a nonempty subset F of E induced from R and J is defined, respectively, by
(20)R⋆⋆⋆JF=∪e∈Eμue:μue∩Fc∈J,(21)R⋆⋆⋆JF=R⋆⋆⋆JFcc,(22)BNDR⋆⋆⋆JF=R⋆⋆⋆JF−R⋆⋆⋆JF,(23)ACCR⋆⋆⋆JF=R⋆⋆⋆JF∩FR⋆⋆⋆JF∪F,(24)RoughR⋆⋆⋆JF=1−ACCR⋆⋆⋆JF.Proposition 30.
ConsiderF,H⊆E and let J,K be ideals and R be a binary relation on E. Then,
(1)
F⊆H⇒R⋆⋆⋆JF⊆R⋆⋆⋆JH(2)
R⋆⋆⋆JF∪R⋆⋆⋆JH⊆R⋆⋆⋆JF∪H(3)
R⋆⋆⋆JF∩H⊆R⋆⋆⋆JF∩R⋆⋆⋆JH(4)
R⋆⋆⋆JF=R⋆⋆⋆JFcc(5)
IfJ⊆K, then R⋆⋆⋆JF⊆R⋆⋆⋆KF(6)
R⋆⋆⋆J∩KF=R⋆⋆⋆JF∩R⋆⋆⋆KFProof.
(1)
LetF⊆H and e∈R⋆⋆⋆JF. Then, ∃y∈E such that e∈μuy∩Fc∈J. Hence, e∈μuy∩Hc∈J (by Hc⊆Fc, and the properties of ideal). Thus, e∈R⋆⋆⋆JH. Therefore, R⋆⋆⋆JF⊆R⋆⋆⋆JH(2)
Immediately by part (4)(3)
Immediately by part (4)(4)
Immediately obtains by Definition29(5)
LetJ⊆K and e∈R⋆⋆⋆JF. Then, ∃y∈E such that e∈μuy∩Fc∈J⊆K. So, e∈R⋆⋆⋆KF, and hence, R⋆⋆⋆JF⊆R⋆⋆⋆KF(6)
R⋆⋆⋆J∩KF=∪e∈Eμue:μue∩Fc∈J∩K.=∪e∈Eμue:μue∩Fc∈Jand∪e∈Eμue:μue∩Fc∈K.=∪e∈Eμue:μue∩Fc∈J∩∪e∈Eμue:μue∩Fc∈K.=R⋆⋆⋆JF∩R⋆⋆⋆KFProposition 31.
ConsiderF,H⊆E and let J,K be ideals and R be a binary relation on E. Then,
(1)
F⊆H⇒R⋆⋆⋆JF⊆R⋆⋆⋆JH(2)
R⋆⋆⋆JF∩H⊆R⋆⋆⋆JF∩R⋆⋆⋆JH(3)
R⋆⋆⋆JF∪R⋆⋆⋆JH⊆R⋆⋆⋆JF∪H(4)
R⋆⋆⋆JF=R⋆⋆⋆JFcc(5)
IfJ⊆K, then R⋆⋆⋆KF⊆R⋆⋆⋆JF(6)
R⋆⋆⋆J∩KF=R⋆⋆⋆JF∪R⋆⋆⋆KFProof.
(1)
LetF⊆H. Thus, Hc⊆Fc, and hence, R⋆⋆⋆JHc⊆R⋆⋆⋆JFc (by no. (4) in Proposition 30). So, R⋆⋆⋆JFcc⊆R⋆⋆⋆JHcc. Consequently, R⋆⋆⋆JF⊆R⋆⋆⋆JH(2)
Immediately by part (4)(3)
Immediately by part (4)(4)
Immediately obtains by Definition29(5)
LetJ⊆K and e∈R⋆⋆⋆KF. Then, e∈R⋆⋆⋆KFcc⊆R⋆⋆⋆JFcc, (by no. (5) in Proposition 30). Thus, e∈R⋆⋆⋆JFcc=R⋆⋆⋆JF. Therefore, R⋆⋆⋆KF⊆R⋆⋆⋆JF(6)
R⋆⋆⋆J∩KF=R⋆⋆⋆J∩KFcc.=R⋆⋆⋆JFc∩R⋆⋆⋆KFcc (by no. (6) in Proposition 30). =R⋆⋆⋆JFcc∪R⋆⋆⋆KFcc.=R⋆⋆⋆JF∪R⋆⋆⋆KFRemark 32.
(1)
The converse of part 1 in Proposition30 and Proposition 31 is generally incorrect. Example 1 (i) points out this fact(a)
IfF=e1,H=e4, then, R⋆⋆⋆JF=ϕ,R⋆⋆⋆JH=e4. Therefore, R⋆⋆⋆JF⊆R⋆⋆⋆JH, but F⊈H(b)
IfF=e2,H=e1,e3,e4, then, R⋆⋆⋆JF=e1,e2,e3,R⋆⋆⋆JH=E. Therefore, R⋆⋆⋆JF⊆R⋆⋆⋆JH, but F⊈H(2)
The inclusion relations of part 2 in Proposition30 and Proposition 31 are generally proper. To show that consider Example 1 (iii) and take F=e1,e4,H=e2,e3. Then(a)
R⋆⋆⋆JF=R⋆⋆⋆JH=E,R⋆⋆⋆JF∩H=ϕ. Therefore, R⋆⋆⋆JF∩R⋆⋆⋆JH=E≠ϕ=R⋆⋆⋆JF∩H(b)
R⋆⋆⋆JF=R⋆⋆⋆JH=ϕ,R⋆⋆⋆JF∪H=E. Therefore, R⋆⋆⋆JF∪R⋆⋆⋆JH=ϕ≠E=R⋆⋆⋆JF∪H(3)
In Example1 (i), if J=ϕ,e1. So, the inclusion relations of part 3 in Proposition 30 and Proposition 31 are generally proper as we illustrate in the following(a)
F=e1,e2,e4,H=e1,e3,e4,F∩H=e1,e4, then, R⋆⋆⋆JF=F,R⋆⋆⋆JH=H,R⋆⋆⋆JF∩H=ϕ. Therefore, R⋆⋆⋆JF∩R⋆⋆⋆JH=e1,e4≠ϕ=R⋆⋆⋆JF∩H(b)
F=e3,H=e2,F∪H=e2,e3, then, R⋆⋆⋆JF=F,R⋆⋆⋆JH=H,R⋆⋆⋆JF∪H=E. Therefore, R⋆⋆⋆JF∪R⋆⋆⋆JH=e2,e3≠E=R⋆⋆⋆JF∪H(4)
In Proposition30 and Proposition 31, the converse of part 5 is generally incorrect. To show that consider Example 1 (ii) and take(a)
F=e1,e2, then, R⋆⋆⋆JF=F,R⋆⋆⋆KF=e1. Therefore, R⋆⋆⋆KF⊆R⋆⋆⋆JF, but J⊈K(b)
F=e1,e2,e3, then, R⋆⋆⋆JF=R⋆⋆⋆KF=e2,e3. Therefore, R⋆⋆⋆JF⊆R⋆⋆⋆KF, but J⊈KRemark 33.
Some properties of the lower and upper approximations given in the second method are not kept by this method as we demonstrate in the following.(i)
In Example1 (i) take(1)
F=e1, then R⋆⋆⋆JF=ϕ. Hence, F⊈R⋆⋆⋆JF(2)
F=e2, then R⋆⋆⋆JF=e1,e2,e3. Hence, R⋆⋆⋆JF⊈F(3)
F=E, then R⋆⋆⋆JE=e4. Hence, R⋆⋆⋆JE≠E(4)
F=ϕ, then R⋆⋆⋆Jϕ=e1,e2,e3. Hence, R⋆⋆⋆Jϕ≠ϕExample 3.
LetE=e1,e2,e3,e4,J=ϕ,e2 and R=e2,e2 be a binary relation defined on E thus μue2=e2,μue1=μue3=μue4=ϕ. Take
(1)
F=E, then R⋆⋆⋆JE=e2. Hence, R⋆⋆⋆JE≠E(2)
F=ϕ, then R⋆⋆⋆Jϕ=e1,e3,e4. Hence, R⋆⋆⋆Jϕ≠ϕRemark 34.
Some properties of the lower and upper approximations given in the first/second method are not kept by this method as we demonstrate in the following. In Example1, take
(1)
F=e1,e3,e4, then, Fc∈J, then R⋆⋆⋆JF=e2. Hence, if Fc∈J⇏R⋆⋆⋆JF=E or F(2)
F=e2∈J, then, R⋆⋆⋆JF=e1,e3,e4. Hence, if F∈J⇏R⋆⋆⋆JF=ϕ or F(3)
F=e1,e3,e4,J=PE, then, R⋆⋆⋆JF=e2. Hence, if J=PE⇏R⋆⋆⋆JF=E, or F(4)
F=e2,J=PE, then, R⋆⋆⋆JF=e1,e3,e4. Hence, if J=PE⇏R⋆⋆⋆JF=ϕ, or FRemark 35.
Some Pawlak’s properties are not kept by this method as we demonstrate in the following. In Example1 (i), if J=ϕ,e1,(1)
F=e1,e2, then R⋆⋆⋆JF=e2,R⋆⋆⋆JR⋆⋆⋆JF=ϕ. Hence, R⋆⋆⋆JF⊈R⋆⋆⋆JR⋆⋆⋆JF(2)
F=e3,e4, then R⋆⋆⋆JF=e1,e3,e4,R⋆⋆⋆JR⋆⋆⋆JF=E. Hence, R⋆⋆⋆JR⋆⋆⋆JF⊈R⋆⋆⋆JFProposition 36.
ConsiderR and J are, respectively, binary relation and ideal on E≠ϕ and let ϕ≠F⊆E. Then,
(1)
0≤ACCR⋆⋆⋆JF≤1(2)
ACCR⋆⋆JE=1Proof.
It is similar to Proposition14.Theorem 37.
ConsiderR as a binary relation on E and let J,K be ideals on E such that J⊆K. For each F⊆E, the next properties hold.
(1)
BNDR⋆⋆⋆KF⊆BNDR⋆⋆⋆JF(2)
ACCR⋆⋆⋆JF≤ACCR⋆⋆⋆KF(3)
RoughR⋆⋆⋆KF≤RoughR⋆⋆⋆JFProof.
Similar to Theorem15.Remark 38.
In Theorem37, the converse of parts 1, 2, and 3 is generally false. To validate that take Example 1 (ii) and let F=e3,e4. Then,
(1)
BNDR⋆⋆⋆KF=e1⊆e1,e2=BNDR⋆⋆⋆JF, but J⊈K(2)
ACCR⋆⋆⋆JF=1/2≤2/3=ACCR⋆⋆⋆KF, but J⊈K(3)
RoughR⋆⋆⋆KF=1/3≤1/2=RoughR⋆⋆⋆JF, but J⊈KThe lower and upper approximations, boundary region, accuracy, and roughness generated by the third type does not have the monotonicity. The next example confirm this fact.Example 4.
LetE=e1,e2,e3,e4,e5,e6,e7,J=ϕ,e2,R1,R2 be two binary relations on E where R1=Δ∪e1,e3,e3,e1,e3,e7,e4,e6,e5,e7,e6,e4,e7,e3,e7,e5,R2=R1∪e1,e4,e1,e5,e2,e6,e4,e1,e5,e1,e6,e2. Thus, μ1ue1=e1,e3,e7,μ1ue2=e2,μ1ue3=μ1ue7=e1,e3,e5,e7,μ1ue4=μ1ue6=e4,e6,μ1ue5=e3,e5,e7,μ2ue1=e1,e3,e4,e5,e6,e7,μ2ue2=e2,e4,e6,μ2ue3=μ2ue5=e1,e3,e4,e5,e7,μ2ue4=e1,e2,e3,e4,e5,e6,μ2ue6=e1,e2,e4,e6,μ2ue7=e1,e3,e5,e7. Take
(1)
F=e1,e2,e3,e4,e5,e6, then, R1⋆⋆⋆JF=e2,e4,e6,R2⋆⋆⋆JF=e1,e2,e3,e4,e5,e6. Therefore, R1⋆⋆⋆JFU^R2⋆⋆⋆JF(2)
F=e7, then, R1⋆⋆⋆JF=e1,e3,e5,e7,R2⋆⋆⋆JF=e7. Therefore, R1⋆⋆⋆JF⊈R2⋆⋆⋆JF
F=e1,e2,e3,e4,e5,e6, then R1⋆⋆JF=e2,e4,e6,R1⋆⋆⋆JF=E,R2⋆⋆JF=F,R2⋆⋆⋆JF=E. Therefore,
(a)
BNDR1⋆⋆⋆JF=e1,e3,e5,e7⊈e7=BNDR2⋆⋆⋆JF(b)
ACCR1⋆⋆⋆JF=3/7<6/7=ACCR2⋆⋆⋆JF(c)
RoughR1⋆⋆⋆JF=4/7>1/7=RoughR2⋆⋆⋆JF
Although,R1⊆R2.


### 3.4. The Fourth Method of the Improvement of the Approximations and Accuracy Measure of Subsets

Definition 39.
LetR and J be, respectively, binary relation and ideal on a nonempty set E. The fourth kind of the improvement of lower and upper approximations, boundary region, accuracy, and roughness of a nonempty subset F of E induced from R and J is defined, respectively, by
(25)R⋆⋆⋆⋆JF=∪e∈Eμue:μue∩F∈J,(26)R⋆⋆⋆⋆JF=R⋆⋆⋆⋆JFcc,(27)BNDR⋆⋆⋆⋆JF=R⋆⋆⋆⋆JF−R⋆⋆⋆⋆JF,(28)ACCR⋆⋆⋆⋆JF=R⋆⋆⋆⋆JF∩FR⋆⋆⋆⋆JF∪F,(29)RoughR⋆⋆⋆⋆JF=1−ACCR⋆⋆⋆⋆JF.Proposition 40.
ConsiderF,H⊆E and let J,K be ideals and R be a binary relation on E. Then,
(1)
R⋆⋆⋆⋆Jϕ=ϕ(2)
F⊆H⇒R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆JH(3)
R⋆⋆⋆⋆JF∩H⊆R⋆⋆⋆⋆JF∩R⋆⋆⋆⋆JH(4)
R⋆⋆⋆⋆JF∪H=R⋆⋆⋆⋆JF∪R⋆⋆⋆⋆JH(5)
R⋆⋆⋆⋆JF=R⋆⋆⋆⋆JFcc(6)
IfF∈J, then R⋆⋆⋆⋆JF=ϕ(7)
IfJ⊆K, then R⋆⋆⋆⋆KF⊆R⋆⋆⋆⋆JF(8)
IfJ=PE, then R⋆⋆⋆⋆JF=ϕ(9)
R⋆⋆⋆⋆J∩KF=R⋆⋆⋆⋆JF∪R⋆⋆⋆⋆KF(10)
R⋆⋆⋆⋆J∨KF=R⋆⋆⋆⋆JF∩R⋆⋆⋆⋆KFProof.
(1)
R⋆⋆⋆⋆Jϕ=∪e∈Eμue:μue∩ϕ∈J=ϕ(2)
LetF⊆H and e∈R⋆⋆⋆⋆JF. Then, ∃y∈E such that e∈μuy and μuy∩F∈J. Thus, μuy∩H∈J. So, e∈R⋆⋆⋆⋆JH. Consequently, R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆JH(3)
Immediately by part (2)(4)
R⋆⋆⋆⋆JF∪H=∪e∈Eμue:μue∩F∪H∈J.=∪e∈Eμue:μue∩F∈J∪∪e∈Eμue:μue∩H∈J.=∪e∈Eμue:μue∩F∈Jor∪e∈Eμue:μue∩H∈J.=R⋆⋆⋆⋆JF∪R⋆⋆⋆⋆JH(5)
R⋆⋆⋆⋆JFcc=R⋆⋆⋆⋆JFcc.=R⋆⋆⋆⋆JF(6)
Immediately obtains by Definition39(7)
LetJ⊆K,e∈R⋆⋆⋆⋆KF. Then, ∃y∈E such that e∈μuy and μuy∩F∈K. Thus, μuy∩F∈J as J⊆K. So, e∈R⋆⋆⋆⋆JF. Hence, R⋆⋆⋆⋆KF⊆R⋆⋆⋆⋆JF(8)
Immediately obtains by Definition39(9)
R⋆⋆⋆⋆J∩KF=∪e∈Eμue:μue∩F∈J∩K.=∪e∈Eμue:μue∩F∈Jor∪e∈Eμue:μue∩F∈K.=∪e∈Eμue:μue∩F∈J∪∪e∈Eμue:μue∩F∈K.=R⋆⋆⋆⋆JF∪R⋆⋆⋆⋆KF(10)
R⋆⋆⋆⋆J∨KF=∪e∈Eμue:μue∩F∈J∨K.=∪e∈Eμue:μue∩F∈J∪K.=∪e∈Eμue:μue∩F∈Jand∪e∈Eμue:μue∩F∈K.=∪e∈Eμue:μue∩F∈J∩∪e∈Eμue:μue∩F∈K.=R⋆⋆⋆⋆JF∩R⋆⋆⋆⋆KFProposition 41.
ConsiderF,H⊆E and let J,K be ideals and R be a binary relation on E. Then,
(1)
R⋆⋆⋆⋆JE=E(2)
F⊆H⇒R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆JH(3)
R⋆⋆⋆⋆JF∪R⋆⋆⋆⋆JH⊆R⋆⋆⋆⋆JF∪H(4)
R⋆⋆⋆⋆JF∩H=R⋆⋆⋆⋆JF∩R⋆⋆⋆⋆JH(5)
R⋆⋆⋆⋆JF=R⋆⋆⋆⋆JFcc(6)
IfFc∈J, then, R⋆⋆⋆⋆JF=E(7)
IfJ⊆K, then, R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆KF(8)
IfJ=PE, then, R⋆⋆⋆⋆JF=E(9)
R⋆⋆⋆⋆J∩KF=R⋆⋆⋆⋆JF∩R⋆⋆⋆⋆KF(10)
R⋆⋆⋆⋆J∨KF=R⋆⋆⋆⋆JF∪R⋆⋆⋆⋆KFProof.
(1)
R⋆⋆⋆⋆JE=R⋆⋆⋆⋆Jϕc=ϕc=E by Proposition 40 part (1).(2)
LetF⊆H. Thus, Hc⊆Fc, and hence, R⋆⋆⋆⋆JHc⊆R⋆⋆⋆⋆JFc (by Proposition 40 part (2)). Then, R⋆⋆⋆⋆JFcc⊆R⋆⋆⋆⋆JHcc. So, R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆JH(3)
Immediately by part (2)(4)
R⋆⋆⋆⋆JF∩H=R⋆⋆⋆⋆JF∩Hcc.=R⋆⋆⋆⋆JFc∪Hcc.=R⋆⋆⋆⋆JFc∪R⋆⋆⋆⋆JHcc(by no. (4) in Proposition 40). =R⋆⋆⋆⋆JFcc∩R⋆⋆⋆⋆JHcc.=R⋆⋆⋆⋆JF∩R⋆⋆⋆⋆JH(5)
Immediately obtains by Definition39(6)
LetFc∈J, then R⋆⋆⋆⋆JF=R⋆⋆⋆⋆JFcc=ϕc=E by Proposition 40 part (6)(7)
LetJ⊆K. Then, R⋆⋆⋆⋆KFc⊆R⋆⋆⋆⋆JFc by Proposition 40 part (7). Thus, R⋆⋆⋆⋆JFcc⊆R⋆⋆⋆⋆KFcc. Hence, R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆KF(8)
LetJ=PE, then, R⋆⋆⋆⋆JF=R⋆⋆⋆⋆JFcc=ϕc=E by Proposition 40 part (8)(9)
R⋆⋆⋆⋆J∩KF=R⋆⋆⋆⋆J∩KFcc.=R⋆⋆⋆⋆JFc∪R⋆⋆⋆⋆KFcc (by Proposition 40 part (9)). =R⋆⋆⋆⋆JFcc∩R⋆⋆⋆⋆KFcc.=R⋆⋆⋆⋆JF∩R⋆⋆⋆⋆KF(10)
R⋆⋆⋆⋆J∨KF=R⋆⋆⋆⋆J∨KFcc.=R⋆⋆⋆⋆JFc∩R⋆⋆⋆⋆KFcc (by Proposition 40 part (10)). =R⋆⋆⋆⋆JFcc∪R⋆⋆⋆⋆KFcc.=R⋆⋆⋆⋆JF∪R⋆⋆⋆⋆KFRemark 42.
(1)
In Proposition40 and Proposition 41, the converse of part 2 is generally false. To validate that consider Example 1 (i)(a)
IfF=e1,H=e4, then, R⋆⋆⋆⋆JF=ϕ,R⋆⋆⋆⋆JH=E. Therefore, R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆JH, but F⊈H(b)
IfF=e1,e2,e3 and H=e2,e3,e4, then R⋆⋆⋆⋆JF=ϕ,R⋆⋆⋆⋆JH=E. Therefore, R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆JH, but F⊈H(2)
In Proposition40 and Proposition 41, the converse of parts 6, 7, and 8 is generally false. To show that consider Example 1 (ii).
(a)
For part 6 take(i)
F=e1,e2, then, R⋆⋆⋆⋆KF=ϕ. Therefore, R⋆⋆⋆⋆KF=ϕ, but F∈K(ii)
F=e3,e4, then, R⋆⋆⋆⋆KF=E. Therefore, R⋆⋆⋆⋆KF=E, but Fc∈K.(b)
For part 7 take(i)
F=e1,e2, then, R⋆⋆⋆⋆JF=e2,R⋆⋆⋆⋆KF=ϕ. Therefore, R⋆⋆⋆⋆KF⊆R⋆⋆⋆⋆JF, but J⊈K(ii)
F=e3,e4, then, R⋆⋆⋆⋆JF=e1,e3,e4,R⋆⋆⋆⋆KF=E. Therefore, R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆KF, but J⊈K(c)
For part 8 take(i)
F=e1,e2, then, R⋆⋆⋆⋆KF=ϕ, but K≠PE(ii)
F=e3,e4, then, R⋆⋆⋆⋆KF=E, but K≠PE(3)
In Proposition40 and Proposition 41, the inclusion relations of part 3 are generally proper. To demonstrate that consider Example 1 (iii), and let F=e1,e4 and H=e2,e3. Then(a)
R⋆⋆⋆⋆JF=R⋆⋆⋆⋆JH=E,R⋆⋆⋆⋆JF∩H=ϕ. Therefore, R⋆⋆⋆⋆JF∩R⋆⋆⋆⋆JH=E≠ϕ=R⋆⋆⋆JF∩H(b)
R⋆⋆⋆⋆JF=R⋆⋆⋆⋆JH=ϕ,R⋆⋆⋆⋆JF∪H=E. Therefore, R⋆⋆⋆⋆JF∪R⋆⋆⋆⋆JH=ϕ≠E=R⋆⋆⋆⋆JF∪HRemark 43.
Some properties of the lower and upper approximations given in the second method are not kept by this method as we demonstrate in the following.(i)
In Example1 (i) take(a)
F=e1, then, R⋆⋆⋆⋆JF=ϕ. Hence, F⊈R⋆⋆⋆⋆JF(b)
F=e2,e3,e4, then, R⋆⋆⋆⋆JF=E. Hence, R⋆⋆⋆⋆JF⊈F(ii)
In Example1 (ii) take(a)
F=E, then, R⋆⋆⋆⋆JE=e2,e3,e4. Hence, R⋆⋆⋆⋆JE≠E(b)
F=ϕ, then, R⋆⋆⋆⋆Jϕ=e1. Hence, R⋆⋆⋆⋆Jϕ≠ϕProposition 44.
ConsiderR and J are, respectively, binary relation and ideal on E≠ϕ and let ϕ≠F⊆E. Then,
(1)
0≤ACCR⋆⋆⋆⋆JF≤1(2)
ACCR⋆⋆⋆⋆JE=1Proof.
Similar to Proposition14.Theorem 45.
ConsiderR as a binary relation on E and let J,K be ideals on E such that J⊆K. For each F⊆E, the next properties hold.
(1)
BNDR⋆⋆⋆⋆KF⊆BNDR⋆⋆⋆⋆JF(2)
ACCR⋆⋆⋆⋆JF≤ACCR⋆⋆⋆⋆KF(3)
RoughR⋆⋆⋆⋆KF≤RoughR⋆⋆⋆⋆JFProof.
Similar to Theorem15.Remark 46.
Example1 (ii) shows that the converse of parts 1, 2, and 3 in Theorem 45 is not necessary to be true in general. Take, F=e3,e4, then
(1)
BNDR⋆⋆⋆⋆KF=ϕ⊆ϕ=BNDR⋆⋆⋆⋆JF, but J⊈K(2)
ACCR⋆⋆⋆⋆JF=1≤1=ACCR⋆⋆⋆⋆KF, but J⊈K(3)
RoughR⋆⋆⋆⋆KF=0≤0=RoughR⋆⋆⋆⋆JF, but J⊈KTheorem 47.
Letϕ≠F⊆E,J be an ideal on E and R1,R2 be two binary relations on E. If R1⊆R2, then
(1)
R1⋆⋆⋆⋆JF⊆R2⋆⋆⋆⋆JF(2)
R2⋆⋆⋆⋆JF⊆R1⋆⋆⋆⋆JF(3)
BNDR1⋆⋆⋆⋆JF⊆BNDR2⋆⋆⋆⋆JF(4)
ACCR2⋆⋆⋆⋆JF≤ACCR1⋆⋆⋆⋆JF(5)
RoughR1⋆⋆⋆⋆JF≤RoughR2⋆⋆⋆⋆JFProof.
(1)
Lete∈R1⋆⋆⋆⋆JF. Then, ∃y∈E such that e∈μ1uy∩F∈J. Since, μ1uy⊆μ2uy (by Theorem 4 [20]). It follows that e∈μ2uy∩F∈J. Thus, e∈R2⋆⋆⋆⋆JF. Hence, R1⋆⋆⋆⋆JF⊆R2⋆⋆⋆⋆JF(2)
e∈R2⋆⋆⋆⋆JF=R2⋆⋆⋆⋆JFcc.⊆R1⋆⋆⋆⋆JFcc (by part (1)).=R1⋆⋆⋆⋆JF(3)
Lete∈BNDR1⋆⋆⋆⋆JF. Then, e∈R1⋆⋆⋆⋆JF−R1⋆⋆⋆⋆JF. So, e∈R1⋆⋆⋆⋆JF and e∈R1⋆⋆⋆⋆JFc. Thus, e∈R2⋆⋆⋆⋆JF and e∈R2⋆⋆⋆⋆JFc by parts 1 and 2. Hence, e∈BNDR2⋆⋆⋆⋆JF. Therefore, BNDR1⋆⋆⋆⋆JF⊆BNDR2⋆⋆⋆⋆JF(4)
ACCR2⋆⋆⋆⋆JF=R2⋆⋆⋆⋆JF∩F/R2⋆⋆⋆⋆JF∪F≤R1⋆⋆⋆⋆JF∩F/R1⋆⋆⋆⋆JF∪F=ACCR1⋆⋆⋆⋆JF(5)
Straightforward by (4)Remark 48.
In Theorem47, the inclusion and less than relations are generally proper. To show this matter consider Example 2 and take
(i)
F=e1,e2, then(1)
R1⋆⋆⋆⋆JF=e1,e4≠e1,e3,e4=R2⋆⋆⋆⋆JF(2)
ACCR1⋆⋆⋆⋆JF=2/3≠1/2=ACCR2⋆⋆⋆⋆JF(3)
RoughR1⋆⋆⋆⋆JF=1/3≠1/2=RoughR2⋆⋆⋆⋆JF(ii)
F=e2,e3, then, R1⋆⋆⋆⋆JF=F≠e2=R2⋆⋆⋆⋆JF


## 3.1. The First Method of the Improvement of the Approximations and Accuracy Measure of Subsets

Definition 9.
LetR and J be, respectively, binary relation and ideal on a nonempty set E. The first kind of the improvement of lower and upper approximations, boundary region, accuracy, and roughness of a nonempty subset F of E induced from R and J is defined, respectively, by
(10)R⋆JF=e∈E:μue∩Fc∈J,(11)R⋆JF=e∈E:μue∩F∈J,(12)BNDR⋆JF=R⋆JF−R⋆JF,(13)ACCR⋆JF=R⋆JF∩FR⋆JF∪F,(14)RoughR⋆JF=1−ACCR⋆JF.Proposition 10.
ConsiderF,H⊆E and let J,K be ideals and R be a binary relation on E. Then,
(1)
R⋆Jϕ=ϕ(2)
F⊆H⇒R⋆JF⊆R⋆JH(3)
R⋆JF∩H⊆R⋆JF∩R⋆JH(4)
R⋆JF∪H=R⋆JF∪R⋆JH(5)
R⋆JF=R⋆JFcc(6)
IfF∈J, then R⋆JF=ϕ(7)
IfJ⊆K, then R⋆KF⊆R⋆JF(8)
IfJ=PE, then R⋆JF=ϕ(9)
R⋆J∩KF=R⋆JF∪R⋆KF(10)
R⋆J∨KF=R⋆JF∩R⋆KFProof.
(1)
R⋆Jϕ=e∈E:μue∩ϕ∈J.=ϕ(2)
Lete∈R⋆JF. Then, μue∩F∈J. Since, F⊆H and J is an ideal. Thus, μue∩H∈J. Therefore, e∈R⋆JH. Hence, R⋆JF⊆R⋆JH(3)
Immediately by part (2)(4)
R⋆JF∪R⋆JH⊆R⋆JF∪H by part (2). Let e∈R⋆JF∪H. Then, μue∩F∪H∈J. It follows that μue∩F∪μue∩H∈J. Therefore, μue∩F∈I or μue∩H∈J, that means e∈R⋆JF or e∈R⋆JH. Then, e∈R⋆JF∪R⋆JH. Thus, R⋆JF∪H⊆R⋆JF∪R⋆JH. Hence, R⋆JF∪H=R⋆JF∪R⋆JH(5)
R⋆JFcc=e∈E:μue∩F∈Jc.=e∈E:μue∩F∈J.=R⋆JF(6)
Immediately obtains by Definition9(7)
Lete∈R⋆KF. Then, μue∩F∈K. Since, J⊆K. Thus, μue∩F∈J. Therefore, e∈R⋆JF. Hence, R⋆KF⊆R⋆JF(8)
Immediately obtains by Definition9(9)
R⋆J∩KF=e∈E:μue∩F∈J∩K.=e∈E:μue∩F∈Jore∈E:μue∩F∈K.=e∈E:μue∩F∈J∪e∈E:μue∩F∈K.=R⋆JF∪R⋆KF(10)
R⋆J∨KF=e∈E:μue∩F∈J∨K.=e∈E:μue∩F∈J∪K.=e∈E:μue∩F∈Jande∈E:μue∩F∈K.=e∈E:μue∩F∈J∩e∈E:μue∩F∈K.=R⋆JF∩R⋆KFProposition 11.
ConsiderF,H⊆E and let J,K be ideals and R be a binary relation on E. Then,
(1)
R⋆JE=E(2)
F⊆H⇒R⋆JF⊆R⋆JH(3)
R⋆JF∪R⋆JH⊆R⋆JF∪H(4)
R⋆JF∩H=R⋆JF∩R⋆JH(5)
R⋆JF=R⋆JFcc(6)
IfFc∈J, then R⋆JF=E(7)
IfJ⊆K, then R⋆JF⊆R⋆KF(8)
IfJ=PE, then R⋆JF=E(9)
R⋆J∩KF=R⋆JF∩R⋆KF(10)
R⋆J∨KF=R⋆JF∪R⋆KFProof.
(1)
R⋆JE=e∈E:μue∩ϕ∈J.=E(2)
Lete∈R⋆JF. Then, μue∩Fc∈J. Since, Hc⊆Fc and J is an ideal. Thus, μue∩Hc∈J. Therefore, e∈R⋆JH. Hence, R⋆JF⊆R⋆JH(3)
Immediately by part (2)(4)
R⋆JF∩R⋆JH⊇R⋆JF∩H by part (2). Let e∈R⋆JF∩R⋆JH. Then, μue∩Fc∈J and μue∩Hc∈J. It follows that μue∩Fc∪Hc∈J. So, μue∩F∩Hc∈J. Therefore, e∈R⋆JF∩H. Thus, R⋆JF∩R⋆JH⊆R⋆JF∩H. Hence, R⋆JF∩R⋆JH=R⋆JF∩H(5)
R⋆JFcc=e∈E:μue∩Fc∈Jc.=e∈E:μue∩Fc∈J.=R⋆JF(6)
Immediately obtains by Definition9(7)
Lete∈R⋆JF. Then, μue∩Fc∈J. Since, J⊆K. Thus, μue∩Fc∈K. Therefore, e∈R⋆JF. Hence, R⋆JF⊆R⋆KF(8)
Immediately obtains by Definition9(9)
R⋆J∩KF=e∈E:μue∩Fc∈J∩K.=e∈E:μue∩Fc∈Jande∈E:μue∩Fc∈K.=e∈E:μue∩Fc∈J∩e∈E:μue∩Fc∈K.=R⋆JF∩R⋆KF(10)
R⋆J∨KF=e∈E:μue∩Fc∈J∨K.=e∈E:μue∩Fc∈J∪K.=e∈E:μue∩Fc∈Jore∈E:μue∩Fc∈K.=e∈E:μue∩Fc∈J∪e∈E:μue∩Fc∈K.=R⋆JF∪R⋆KFRemark 12.
By the next example, we elucidate that(1)
The converse of parts 2, 6, 7, and 8 in Proposition10 and Proposition 11 is generally incorrect(2)
The inclusion relations of part 3 in Proposition10 and Proposition 11 are generally properExample 1.
(i)
LetE=e1,e2,e3,e4,J=ϕ,e1,e2,e3,e1,e2,e1,e3,e2,e3,e1,e2,e3 and R=e1,e2,e1,e3,e2,e3,e2,e4,e3,e1,e3,e4 be a binary relation defined on E. By calculations, we obtain μue1=e1,e2,e4,μue2=e1,e2,e3,μue3=e2,e3,e4,μue4=e1,e3,e4. For part 2, take F=e1,e2 and H=e1,e4, then(a)
R⋆JF=ϕ,R⋆JH=e1,e3,e4. Therefore, R⋆JF⊆R⋆JH, but F⊈H(b)
R⋆JF=e2,R⋆JH=E. Therefore, R⋆JF⊆R⋆JH, but F⊈H(ii)
LetE=e1,e2,e3,e4,K=ϕ,e2,J=ϕ,e1 and R=e2,e2,e3,e3,e4,e4 be a binary relation defined on E. By calculations, we obtain μue1=ϕ,μue2=e2,μue3=e3,μue4=e4(1)
For part 6, take(a)
F=e1,e2, then, R⋆KF=ϕ. Therefore, R⋆KF=ϕ, but F∈K.(b)
F=e3,e4, then, R⋆KF=E. Therefore, R⋆KF=E, but Fc∈K.(2)
For part 7, take(a)
F=e1,e2, then, R⋆JF=e2,R⋆KF=ϕ. Therefore, R⋆KF⊆R⋆JF, but J⊈K(b)
F=e3,e4, then, R⋆JF=e1,e3,e4,R⋆KF=E. Therefore, R⋆JF⊆R⋆KF, but J⊈K(3)
For part 8, take(a)
F=e1,e2, then, R⋆KF=ϕ, but K≠PE(b)
F=e3,e4, then, R⋆KF=E, but K≠PE(iii)
LetE=e1,e2,e3,e4,J=ϕ,e4 and R=Δ∪e2,e1,e3,e1,e4,e1 be a binary relation defined on E, (where Δ is the identity relation and equal to e1,e1,e2,e2,e3,e3,e4,e4. By calculations, we obtain μue1=μue2=μue3=μue4=E. For part 3, take F=e1,e4,H=e2,e3. Hence(a)
F∩H=ϕ, then R⋆JF=R⋆JH=E,R⋆JF∩H=ϕ. Therefore, R⋆JF∩R⋆JH=E≠ϕ=R⋆JF∩H(b)
F∪H=E, then R⋆JF=R⋆JH=ϕ,R⋆JF∪H=E. Therefore, R⋆JF∪R⋆JH=ϕ≠E=R⋆JF∪HRemark 13.
Some Pawlak’s properties are not kept by this method as we demonstrate in the following.(i)
In Example1 (i) take(1)
F=e1,e2, then R⋆JF=ϕ. Hence, F⊈R⋆JF(2)
F=e1,e4, then R⋆JF=E. Hence, R⋆JF⊈F(3)
F=E, then R⋆JE=e1,e3,e4. Hence, R⋆JE≠E(4)
F=ϕ, then R⋆Jϕ=e2. Hence, R⋆Jϕ≠ϕ(ii)
In Example1 (i) take J=ϕ,e1(1)
F=e1,e4, then R⋆JF=e1,e3,e4,R⋆JR⋆JF=E. Hence, R⋆JF≠R⋆JR⋆JF(2)
F=e2,e3, then R⋆JF=e2,R⋆JR⋆JF=ϕ. Hence, R⋆JF≠R⋆JR⋆JF(3)
F=e1,e2, then R⋆JF=e1,e2,e3 and R⋆JR⋆JF=e2. Hence, R⋆JF⊈R⋆JR⋆JF(4)
F=e3,e4, then R⋆JF=e4 and R⋆JR⋆JF=e1,e3,e4. Hence, R⋆JR⋆JF⊈R⋆JFProposition 14.
ConsiderR and J are, respectively, relation and ideal on E≠ϕ and let ϕ≠F⊆E. Then,
(1)
0≤ACCR⋆JF≤1(2)
ACCR⋆JE=1.Proof.
To prove 1, it is obvious thatR⋆JF∪F≠ϕ for every nonempty subset F of E. Hence, ϕ⊆R⋆JF∩F⊆R⋆JF∪F. Therefore, 0≤R⋆JF∩F≤R⋆JF∪F. So, 0≤R⋆JF∩F/R⋆JF∪F≤1. It means that, 0≤ACCR⋆JF≤1.
Case 2 can be proved easily.Theorem 15.
ConsiderR as a relation on E and let J,K be ideals on E such that J⊆K. For each F⊆E, the next properties hold.
(1)
BNDR⋆KF⊆BNDR⋆JF(2)
ACCR⋆JF≤ACCR⋆KF(3)
RoughR⋆KF≤RoughR⋆JFProof.
(1)
Lete∈BNDR⋆KF. Then, e∈R⋆KF−R⋆KF. So, e∈R⋆KF and e∈R⋆KFc. Hence, e∈R⋆JF and e∈R⋆JFc by Propositions 10 and 11 part 7. It follows that e∈BNDR⋆JF. Therefore, BNDR⋆KF⊆BNDR⋆JF(2)
ACCR⋆JF=R⋆JF∩F/R⋆JF∪F.≤R⋆KF∩F/R⋆KF∪F.=ACCR⋆KF(3)
Directly follows from 2Remark 16.
In Theorem15, the converses of parts 1, 2, and 3 are generally false. To elucidate that, take F=e3,e4 as a subset of Example 1 (ii). Then
(1)
BNDR⋆KF=ϕ⊆ϕ=BNDR⋆JF, but J⊈K(2)
ACCR⋆JF=1≤1=ACCR⋆KF, but J⊈K(3)
RoughR⋆KF=0≤0=RoughR⋆JF, but J⊈KTheorem 17.
Letϕ≠F⊆E,J be an ideal on E and R1,R2 be two binary relations on E. If R1⊆R2, then
(1)
R1⋆JF⊆R2⋆JF(2)
R2⋆JF⊆R1⋆JF(3)
BNDR1⋆JF⊆BNDR2⋆JF(4)
ACCR2⋆JF≤ACCR1⋆JF(5)
RoughR1⋆JF≤RoughR2⋆JFProof.
(1)
Lete∈R1⋆JF. Then, μ1ue∩F∈J. Since, μ1ue⊆μ2ue (by Theorem 4 [20]). It follows that μ2ue∩F∈J. Thus, e∈R2⋆JF. Hence, R1⋆JF⊆R2⋆JF(2)
Lete∈R2⋆JF. Then, μ2ue∩Fc∈J. Since, μ1ue⊆μ2ue (by Theorem 4 [20]). It follows that μ1ue∩Fc∈J. Thus, e∈R1⋆JF. Hence, R2⋆JF⊆R1⋆JF(3)
Lete∈BNDR1⋆JF. Then, e∈R1⋆JF−R1⋆JF. So, e∈R1⋆JF and e∈R1⋆JFc. Thus, e∈R2⋆JF and e∈R1⋆JFc by parts 1 and 2. Hence, e∈BNDR2⋆JF. Therefore, BNDR1⋆JF⊆BNDR2⋆JF(4)
ACCR2⋆JF=R2⋆JF∩F/R2⋆JF∪F.≤R1⋆JF∩F/R1⋆JF∪F.=ACCR1⋆JF(5)
Straightforward by 4
In Theorem17, the relations of inclusion and less than are generally proper as we illustrate by the next example.Example 2.
LetE=e1,e2,e3,e4,J=ϕ,e2,e3,e4,e2,e3,e2,e4,e3,e4,e2,e3,e4 and R1=Δ∪e1,e4,e4,e1,R2=R1∪e1,e3,e3,e1 be two binary relation defined on E thus μ1ue1=μ1ue4=e1,e4,μ1ue2=e2,μ1ue3=e3,μ2ue1=μ2ue3=μ2ue4=e1,e3,e4,μ2ue2=e2. Take
(a)
F=e1,e4, then(1)
R1⋆JF=e4≠e1,e3,e4=R2⋆JF(2)
ACCR1⋆JF=2/3≠1/2=ACCR2⋆JF(3)
RoughR2⋆JF=1/2≠1/3=RoughR1⋆JF(4)
F=e4,e3, then R1⋆JF=e2,e3,e4≠e2=R2⋆JF


## 3.2. The Second Method of the Improvement of the Approximations and Accuracy Measure of Subsets

Definition 18.
LetR and J be, respectively, binary relation and ideal on a nonempty set E. The second kind of the improvement of lower and upper approximations, boundary region, accuracy, and roughness of a nonempty subset F of E induced from R and J is defined, respectively, by
(15)R⋆⋆JF=e∈F:μue∩Fc∈J,(16)R⋆⋆JF=F∪R⋆JF,(17)BNDR⋆⋆JF=R⋆⋆JF−R⋆⋆JF,(18)ACCR⋆⋆JF=R⋆⋆JFR⋆⋆JF,(19)RoughR⋆⋆JF=1−ACCR⋆⋆JF.Proposition 19.
ConsiderF,H⊆E and let J,K be ideals and R be a binary relation on E. Then,
(1)
F⊆R⋆⋆JF equality hold if F=ϕ or E(2)
F⊆H⇒R⋆⋆JF⊆R⋆⋆JH(3)
R⋆⋆JF⊆R⋆⋆JR⋆⋆JF(4)
R⋆⋆JF∩H⊆R⋆⋆JF∩R⋆⋆JH(5)
R⋆⋆JF∪H=R⋆⋆JF∪R⋆⋆JH(6)
R⋆⋆JF=R⋆⋆JFcc(7)
IfF∈J, then R⋆⋆JF=F(8)
IfJ⊆K, then R⋆⋆KF⊆R⋆⋆JF(9)
IfJ=PE, then R⋆⋆JF=F(10)
R⋆⋆J∩KF=R⋆⋆JF∪R⋆⋆KF(11)
R⋆⋆J∨KF=R⋆⋆JF∩R⋆⋆KFProof.
Similar to the technique provided in Proposition10.Proposition 20.
ConsiderF,H⊆E and let J,K be ideals and R be a binary relation on E. Then,
(1)
R⋆⋆JF⊆F equality hold if F=ϕ or E(2)
F⊆H⇒R⋆⋆JF⊆R⋆⋆JH(3)
R⋆⋆JR⋆⋆JF⊆R⋆⋆JF(4)
R⋆⋆JF∪R⋆⋆JH⊆R⋆⋆JF∪H(5)
R⋆⋆JF∩H=R⋆⋆JF∩R⋆⋆JH(6)
R⋆⋆JF=R⋆⋆JFcc(7)
IfFc∈J, then, R⋆⋆JF=F(8)
IfJ⊆K, then, R⋆⋆JF⊆R⋆⋆KF(9)
IfJ=PE, then, R⋆⋆JF=F(10)
R⋆⋆J∩KF=R⋆⋆JF∩R⋆⋆KF(11)
R⋆⋆J∨KF=R⋆⋆JF∪R⋆⋆KFProof.
Similar to Proposition11.Remark 21.
(i)
In Proposition19 and Proposition 20, the converse of parts 7 and 9 is generally incorrect. To show that, consider Example 1 (i).
(a)
For part 7, take(1)
F=e1,e3,e4, then R⋆⋆JF=F, but F∈J(2)
F=e2, then R⋆⋆JF=F, but Fc∈J(b)
For part 9, take(1)
F=e1,e3,e4, then R⋆⋆JF=F, but J≠PE(2)
F=e2, then R⋆⋆JF=F, but J≠PE(ii)
In Proposition19 and Proposition 20, the converse of part 8 is generally incorrect. To elucidate that consider Example 1 (ii)(1)
IfF=e1,e2, then, R⋆⋆KF=e1,e2⊆e1,e2=R⋆⋆JF, but J⊈K(2)
IfF=e3,e4, then, R⋆⋆JF=e3,e4⊆e3,e4=R⋆⋆KF, but J⊈K(iii)
In Example1 (i), take J=ϕ,e1. So, the converse of part 2 in Proposition 19 and Proposition 20 is generally incorrect. Additionally, the inclusion relations of parts 3 and 4 in Proposition 19 and Proposition 20 generally proper(a)
For part 2, takeF=e1,e4,H=e2,e3, then(1)
R⋆⋆JF=e1,e3,e4⊆E=R⋆⋆JH=E, but F⊈H(2)
R⋆⋆JF=ϕ⊆e2=R⋆⋆JH, but F⊈H(b)
For part 3, take(1)
F=e1,e4, then, R⋆⋆JF=e1,e3,e4,R⋆⋆JR⋆⋆JF=E. Therefore, R⋆⋆JF=e1,e3,e4≠E=R⋆⋆JR⋆⋆JF(2)
F=e2,e3, then, R⋆⋆JF=e2,R⋆⋆JR⋆⋆JF=ϕ. Therefore, R⋆⋆JF=e2≠ϕ=R⋆⋆JR⋆⋆JF(c)
For part 4, takeF=e1,e4,H=e2,e3 thus(1)
F∩H=ϕ, then, R⋆⋆JF=e1,e3,e4,R⋆⋆JH=E,R⋆⋆JF∩H=ϕ. Therefore, R⋆⋆JF∩R⋆⋆JH=e1,e3,e4≠ϕ=R⋆⋆JF∩H(2)
F∪H=E, then, R⋆⋆JF=ϕ,R⋆⋆JH=e2,R⋆⋆JF∪H=E. Therefore, R⋆⋆JF∪R⋆⋆JH=e2≠E=R⋆⋆JF∪HRemark 22.
Some properties of the lower and upper approximations given in the first method are not kept by this method as we demonstrate in the following.(i)
In Example1 (i) take(1)
F=u1∈J, then, R⋆⋆JF=F. Hence, if F∈J⇏R⋆⋆JF=ϕ(2)
Fc=u1∈J, then, R⋆⋆JF=F. Hence, if Fc∈J⇏R⋆⋆JF=U(ii)
In Example1 (ii) take(1)
K=PE,F=e1,e3, then, R⋆⋆KF=F. Hence, if K=PE⇏R⋆⋆KF=ϕ(2)
K=PE,F=e2,e4, then, R⋆⋆KF=F. Hence, if K=PE⇏R⋆⋆KF=ERemark 23.
Some Pawlak’s properties are not kept by this method as we demonstrate in the following. In Example1 (i) take J=ϕ,e1,(1)
F=e1,e2, then, R⋆⋆JF=e1,e2,e3 and R⋆⋆JR⋆⋆JF=e2. Hence, R⋆⋆JF⊈R⋆⋆JR⋆⋆JF(2)
F=e3,e4, then, R⋆⋆JF=e4 and R⋆⋆JR⋆⋆JF=e1,e3,e4. Hence, R⋆⋆JR⋆⋆JF⊈R⋆⋆JFProposition 24.
ConsiderR and J are, respectively, binary relation and ideal on E≠ϕ and let ϕ≠F⊆E. Then,
(1)
0≤ACCR⋆⋆JF≤1(2)
ACCR⋆⋆JE=1Proof.
As given in Proposition14.Theorem 25.
ConsiderR as a binary relation on E and let J,K be ideals on E such that J⊆K. For each F⊆E, the next properties hold.
(1)
BNDR⋆⋆KF⊆BNDR⋆⋆JF(2)
ACCR⋆⋆JF≤ACCR⋆⋆KF(3)
RoughR⋆⋆KF≤RoughR⋆⋆JFProof.
Following similar arguments given in Theorem15.Remark 26.
In Theorem25, the converses of parts 1, 2, and 3 are generally incorrect. To validate this notice, consider Example 1 (ii) and take F=e2,e4. Then,
(1)
BNDR⋆⋆KF=ϕ⊆ϕ=BNDR⋆⋆JF, but J⊈K(2)
ACCR⋆⋆KF=1≤1=ACCR⋆⋆JF, but J⊈K(3)
RoughR⋆⋆KF=0≤0=RoughR⋆⋆JF, but J⊈KTheorem 27.
Letϕ≠F⊆E,J be an ideal on E and R1,R2 be two binary relations on E. If R1⊆R2, then
(1)
R1⋆⋆JF⊆R2⋆⋆JF(2)
R2⋆⋆JF⊆R1⋆⋆JF(3)
BNDR1⋆⋆JF⊆BNDR2⋆⋆JF(4)
ACCR2⋆⋆JF≤ACCR1⋆⋆JF(5)
RoughR1⋆⋆JF≤RoughR2⋆⋆JFProof.
Similar to Theorem17.Remark 28.
In Theorem27, the inclusion and less than relations are generally proper. To illustrate that, consider Example 2 and take F=e1,e2. Then,
(1)
R1⋆⋆JF=e1,e4≠E=R2⋆⋆JF(2)
BNDR1⋆⋆JF=e4≠e1,e3,e4=BNDR2⋆⋆JF(3)
ACCR1⋆⋆JF=2/3≠1/4=ACCR2⋆⋆JF(4)
RoughR1⋆⋆JF=1/3≠3/4=RoughR2⋆⋆JF


## 3.3. The Third Method of the Improvement of the Approximations and Accuracy Measure of Subsets

Definition 29.
LetR and J be, respectively, binary relation and ideal on a nonempty set E. The third kind of the improvement of lower and upper approximations, boundary region, accuracy, and roughness of a nonempty subset F of E induced from R and J is defined, respectively, by
(20)R⋆⋆⋆JF=∪e∈Eμue:μue∩Fc∈J,(21)R⋆⋆⋆JF=R⋆⋆⋆JFcc,(22)BNDR⋆⋆⋆JF=R⋆⋆⋆JF−R⋆⋆⋆JF,(23)ACCR⋆⋆⋆JF=R⋆⋆⋆JF∩FR⋆⋆⋆JF∪F,(24)RoughR⋆⋆⋆JF=1−ACCR⋆⋆⋆JF.Proposition 30.
ConsiderF,H⊆E and let J,K be ideals and R be a binary relation on E. Then,
(1)
F⊆H⇒R⋆⋆⋆JF⊆R⋆⋆⋆JH(2)
R⋆⋆⋆JF∪R⋆⋆⋆JH⊆R⋆⋆⋆JF∪H(3)
R⋆⋆⋆JF∩H⊆R⋆⋆⋆JF∩R⋆⋆⋆JH(4)
R⋆⋆⋆JF=R⋆⋆⋆JFcc(5)
IfJ⊆K, then R⋆⋆⋆JF⊆R⋆⋆⋆KF(6)
R⋆⋆⋆J∩KF=R⋆⋆⋆JF∩R⋆⋆⋆KFProof.
(1)
LetF⊆H and e∈R⋆⋆⋆JF. Then, ∃y∈E such that e∈μuy∩Fc∈J. Hence, e∈μuy∩Hc∈J (by Hc⊆Fc, and the properties of ideal). Thus, e∈R⋆⋆⋆JH. Therefore, R⋆⋆⋆JF⊆R⋆⋆⋆JH(2)
Immediately by part (4)(3)
Immediately by part (4)(4)
Immediately obtains by Definition29(5)
LetJ⊆K and e∈R⋆⋆⋆JF. Then, ∃y∈E such that e∈μuy∩Fc∈J⊆K. So, e∈R⋆⋆⋆KF, and hence, R⋆⋆⋆JF⊆R⋆⋆⋆KF(6)
R⋆⋆⋆J∩KF=∪e∈Eμue:μue∩Fc∈J∩K.=∪e∈Eμue:μue∩Fc∈Jand∪e∈Eμue:μue∩Fc∈K.=∪e∈Eμue:μue∩Fc∈J∩∪e∈Eμue:μue∩Fc∈K.=R⋆⋆⋆JF∩R⋆⋆⋆KFProposition 31.
ConsiderF,H⊆E and let J,K be ideals and R be a binary relation on E. Then,
(1)
F⊆H⇒R⋆⋆⋆JF⊆R⋆⋆⋆JH(2)
R⋆⋆⋆JF∩H⊆R⋆⋆⋆JF∩R⋆⋆⋆JH(3)
R⋆⋆⋆JF∪R⋆⋆⋆JH⊆R⋆⋆⋆JF∪H(4)
R⋆⋆⋆JF=R⋆⋆⋆JFcc(5)
IfJ⊆K, then R⋆⋆⋆KF⊆R⋆⋆⋆JF(6)
R⋆⋆⋆J∩KF=R⋆⋆⋆JF∪R⋆⋆⋆KFProof.
(1)
LetF⊆H. Thus, Hc⊆Fc, and hence, R⋆⋆⋆JHc⊆R⋆⋆⋆JFc (by no. (4) in Proposition 30). So, R⋆⋆⋆JFcc⊆R⋆⋆⋆JHcc. Consequently, R⋆⋆⋆JF⊆R⋆⋆⋆JH(2)
Immediately by part (4)(3)
Immediately by part (4)(4)
Immediately obtains by Definition29(5)
LetJ⊆K and e∈R⋆⋆⋆KF. Then, e∈R⋆⋆⋆KFcc⊆R⋆⋆⋆JFcc, (by no. (5) in Proposition 30). Thus, e∈R⋆⋆⋆JFcc=R⋆⋆⋆JF. Therefore, R⋆⋆⋆KF⊆R⋆⋆⋆JF(6)
R⋆⋆⋆J∩KF=R⋆⋆⋆J∩KFcc.=R⋆⋆⋆JFc∩R⋆⋆⋆KFcc (by no. (6) in Proposition 30). =R⋆⋆⋆JFcc∪R⋆⋆⋆KFcc.=R⋆⋆⋆JF∪R⋆⋆⋆KFRemark 32.
(1)
The converse of part 1 in Proposition30 and Proposition 31 is generally incorrect. Example 1 (i) points out this fact(a)
IfF=e1,H=e4, then, R⋆⋆⋆JF=ϕ,R⋆⋆⋆JH=e4. Therefore, R⋆⋆⋆JF⊆R⋆⋆⋆JH, but F⊈H(b)
IfF=e2,H=e1,e3,e4, then, R⋆⋆⋆JF=e1,e2,e3,R⋆⋆⋆JH=E. Therefore, R⋆⋆⋆JF⊆R⋆⋆⋆JH, but F⊈H(2)
The inclusion relations of part 2 in Proposition30 and Proposition 31 are generally proper. To show that consider Example 1 (iii) and take F=e1,e4,H=e2,e3. Then(a)
R⋆⋆⋆JF=R⋆⋆⋆JH=E,R⋆⋆⋆JF∩H=ϕ. Therefore, R⋆⋆⋆JF∩R⋆⋆⋆JH=E≠ϕ=R⋆⋆⋆JF∩H(b)
R⋆⋆⋆JF=R⋆⋆⋆JH=ϕ,R⋆⋆⋆JF∪H=E. Therefore, R⋆⋆⋆JF∪R⋆⋆⋆JH=ϕ≠E=R⋆⋆⋆JF∪H(3)
In Example1 (i), if J=ϕ,e1. So, the inclusion relations of part 3 in Proposition 30 and Proposition 31 are generally proper as we illustrate in the following(a)
F=e1,e2,e4,H=e1,e3,e4,F∩H=e1,e4, then, R⋆⋆⋆JF=F,R⋆⋆⋆JH=H,R⋆⋆⋆JF∩H=ϕ. Therefore, R⋆⋆⋆JF∩R⋆⋆⋆JH=e1,e4≠ϕ=R⋆⋆⋆JF∩H(b)
F=e3,H=e2,F∪H=e2,e3, then, R⋆⋆⋆JF=F,R⋆⋆⋆JH=H,R⋆⋆⋆JF∪H=E. Therefore, R⋆⋆⋆JF∪R⋆⋆⋆JH=e2,e3≠E=R⋆⋆⋆JF∪H(4)
In Proposition30 and Proposition 31, the converse of part 5 is generally incorrect. To show that consider Example 1 (ii) and take(a)
F=e1,e2, then, R⋆⋆⋆JF=F,R⋆⋆⋆KF=e1. Therefore, R⋆⋆⋆KF⊆R⋆⋆⋆JF, but J⊈K(b)
F=e1,e2,e3, then, R⋆⋆⋆JF=R⋆⋆⋆KF=e2,e3. Therefore, R⋆⋆⋆JF⊆R⋆⋆⋆KF, but J⊈KRemark 33.
Some properties of the lower and upper approximations given in the second method are not kept by this method as we demonstrate in the following.(i)
In Example1 (i) take(1)
F=e1, then R⋆⋆⋆JF=ϕ. Hence, F⊈R⋆⋆⋆JF(2)
F=e2, then R⋆⋆⋆JF=e1,e2,e3. Hence, R⋆⋆⋆JF⊈F(3)
F=E, then R⋆⋆⋆JE=e4. Hence, R⋆⋆⋆JE≠E(4)
F=ϕ, then R⋆⋆⋆Jϕ=e1,e2,e3. Hence, R⋆⋆⋆Jϕ≠ϕExample 3.
LetE=e1,e2,e3,e4,J=ϕ,e2 and R=e2,e2 be a binary relation defined on E thus μue2=e2,μue1=μue3=μue4=ϕ. Take
(1)
F=E, then R⋆⋆⋆JE=e2. Hence, R⋆⋆⋆JE≠E(2)
F=ϕ, then R⋆⋆⋆Jϕ=e1,e3,e4. Hence, R⋆⋆⋆Jϕ≠ϕRemark 34.
Some properties of the lower and upper approximations given in the first/second method are not kept by this method as we demonstrate in the following. In Example1, take
(1)
F=e1,e3,e4, then, Fc∈J, then R⋆⋆⋆JF=e2. Hence, if Fc∈J⇏R⋆⋆⋆JF=E or F(2)
F=e2∈J, then, R⋆⋆⋆JF=e1,e3,e4. Hence, if F∈J⇏R⋆⋆⋆JF=ϕ or F(3)
F=e1,e3,e4,J=PE, then, R⋆⋆⋆JF=e2. Hence, if J=PE⇏R⋆⋆⋆JF=E, or F(4)
F=e2,J=PE, then, R⋆⋆⋆JF=e1,e3,e4. Hence, if J=PE⇏R⋆⋆⋆JF=ϕ, or FRemark 35.
Some Pawlak’s properties are not kept by this method as we demonstrate in the following. In Example1 (i), if J=ϕ,e1,(1)
F=e1,e2, then R⋆⋆⋆JF=e2,R⋆⋆⋆JR⋆⋆⋆JF=ϕ. Hence, R⋆⋆⋆JF⊈R⋆⋆⋆JR⋆⋆⋆JF(2)
F=e3,e4, then R⋆⋆⋆JF=e1,e3,e4,R⋆⋆⋆JR⋆⋆⋆JF=E. Hence, R⋆⋆⋆JR⋆⋆⋆JF⊈R⋆⋆⋆JFProposition 36.
ConsiderR and J are, respectively, binary relation and ideal on E≠ϕ and let ϕ≠F⊆E. Then,
(1)
0≤ACCR⋆⋆⋆JF≤1(2)
ACCR⋆⋆JE=1Proof.
It is similar to Proposition14.Theorem 37.
ConsiderR as a binary relation on E and let J,K be ideals on E such that J⊆K. For each F⊆E, the next properties hold.
(1)
BNDR⋆⋆⋆KF⊆BNDR⋆⋆⋆JF(2)
ACCR⋆⋆⋆JF≤ACCR⋆⋆⋆KF(3)
RoughR⋆⋆⋆KF≤RoughR⋆⋆⋆JFProof.
Similar to Theorem15.Remark 38.
In Theorem37, the converse of parts 1, 2, and 3 is generally false. To validate that take Example 1 (ii) and let F=e3,e4. Then,
(1)
BNDR⋆⋆⋆KF=e1⊆e1,e2=BNDR⋆⋆⋆JF, but J⊈K(2)
ACCR⋆⋆⋆JF=1/2≤2/3=ACCR⋆⋆⋆KF, but J⊈K(3)
RoughR⋆⋆⋆KF=1/3≤1/2=RoughR⋆⋆⋆JF, but J⊈KThe lower and upper approximations, boundary region, accuracy, and roughness generated by the third type does not have the monotonicity. The next example confirm this fact.Example 4.
LetE=e1,e2,e3,e4,e5,e6,e7,J=ϕ,e2,R1,R2 be two binary relations on E where R1=Δ∪e1,e3,e3,e1,e3,e7,e4,e6,e5,e7,e6,e4,e7,e3,e7,e5,R2=R1∪e1,e4,e1,e5,e2,e6,e4,e1,e5,e1,e6,e2. Thus, μ1ue1=e1,e3,e7,μ1ue2=e2,μ1ue3=μ1ue7=e1,e3,e5,e7,μ1ue4=μ1ue6=e4,e6,μ1ue5=e3,e5,e7,μ2ue1=e1,e3,e4,e5,e6,e7,μ2ue2=e2,e4,e6,μ2ue3=μ2ue5=e1,e3,e4,e5,e7,μ2ue4=e1,e2,e3,e4,e5,e6,μ2ue6=e1,e2,e4,e6,μ2ue7=e1,e3,e5,e7. Take
(1)
F=e1,e2,e3,e4,e5,e6, then, R1⋆⋆⋆JF=e2,e4,e6,R2⋆⋆⋆JF=e1,e2,e3,e4,e5,e6. Therefore, R1⋆⋆⋆JFU^R2⋆⋆⋆JF(2)
F=e7, then, R1⋆⋆⋆JF=e1,e3,e5,e7,R2⋆⋆⋆JF=e7. Therefore, R1⋆⋆⋆JF⊈R2⋆⋆⋆JF
F=e1,e2,e3,e4,e5,e6, then R1⋆⋆JF=e2,e4,e6,R1⋆⋆⋆JF=E,R2⋆⋆JF=F,R2⋆⋆⋆JF=E. Therefore,
(a)
BNDR1⋆⋆⋆JF=e1,e3,e5,e7⊈e7=BNDR2⋆⋆⋆JF(b)
ACCR1⋆⋆⋆JF=3/7<6/7=ACCR2⋆⋆⋆JF(c)
RoughR1⋆⋆⋆JF=4/7>1/7=RoughR2⋆⋆⋆JF
Although,R1⊆R2.


## 3.4. The Fourth Method of the Improvement of the Approximations and Accuracy Measure of Subsets

Definition 39.
LetR and J be, respectively, binary relation and ideal on a nonempty set E. The fourth kind of the improvement of lower and upper approximations, boundary region, accuracy, and roughness of a nonempty subset F of E induced from R and J is defined, respectively, by
(25)R⋆⋆⋆⋆JF=∪e∈Eμue:μue∩F∈J,(26)R⋆⋆⋆⋆JF=R⋆⋆⋆⋆JFcc,(27)BNDR⋆⋆⋆⋆JF=R⋆⋆⋆⋆JF−R⋆⋆⋆⋆JF,(28)ACCR⋆⋆⋆⋆JF=R⋆⋆⋆⋆JF∩FR⋆⋆⋆⋆JF∪F,(29)RoughR⋆⋆⋆⋆JF=1−ACCR⋆⋆⋆⋆JF.Proposition 40.
ConsiderF,H⊆E and let J,K be ideals and R be a binary relation on E. Then,
(1)
R⋆⋆⋆⋆Jϕ=ϕ(2)
F⊆H⇒R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆JH(3)
R⋆⋆⋆⋆JF∩H⊆R⋆⋆⋆⋆JF∩R⋆⋆⋆⋆JH(4)
R⋆⋆⋆⋆JF∪H=R⋆⋆⋆⋆JF∪R⋆⋆⋆⋆JH(5)
R⋆⋆⋆⋆JF=R⋆⋆⋆⋆JFcc(6)
IfF∈J, then R⋆⋆⋆⋆JF=ϕ(7)
IfJ⊆K, then R⋆⋆⋆⋆KF⊆R⋆⋆⋆⋆JF(8)
IfJ=PE, then R⋆⋆⋆⋆JF=ϕ(9)
R⋆⋆⋆⋆J∩KF=R⋆⋆⋆⋆JF∪R⋆⋆⋆⋆KF(10)
R⋆⋆⋆⋆J∨KF=R⋆⋆⋆⋆JF∩R⋆⋆⋆⋆KFProof.
(1)
R⋆⋆⋆⋆Jϕ=∪e∈Eμue:μue∩ϕ∈J=ϕ(2)
LetF⊆H and e∈R⋆⋆⋆⋆JF. Then, ∃y∈E such that e∈μuy and μuy∩F∈J. Thus, μuy∩H∈J. So, e∈R⋆⋆⋆⋆JH. Consequently, R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆JH(3)
Immediately by part (2)(4)
R⋆⋆⋆⋆JF∪H=∪e∈Eμue:μue∩F∪H∈J.=∪e∈Eμue:μue∩F∈J∪∪e∈Eμue:μue∩H∈J.=∪e∈Eμue:μue∩F∈Jor∪e∈Eμue:μue∩H∈J.=R⋆⋆⋆⋆JF∪R⋆⋆⋆⋆JH(5)
R⋆⋆⋆⋆JFcc=R⋆⋆⋆⋆JFcc.=R⋆⋆⋆⋆JF(6)
Immediately obtains by Definition39(7)
LetJ⊆K,e∈R⋆⋆⋆⋆KF. Then, ∃y∈E such that e∈μuy and μuy∩F∈K. Thus, μuy∩F∈J as J⊆K. So, e∈R⋆⋆⋆⋆JF. Hence, R⋆⋆⋆⋆KF⊆R⋆⋆⋆⋆JF(8)
Immediately obtains by Definition39(9)
R⋆⋆⋆⋆J∩KF=∪e∈Eμue:μue∩F∈J∩K.=∪e∈Eμue:μue∩F∈Jor∪e∈Eμue:μue∩F∈K.=∪e∈Eμue:μue∩F∈J∪∪e∈Eμue:μue∩F∈K.=R⋆⋆⋆⋆JF∪R⋆⋆⋆⋆KF(10)
R⋆⋆⋆⋆J∨KF=∪e∈Eμue:μue∩F∈J∨K.=∪e∈Eμue:μue∩F∈J∪K.=∪e∈Eμue:μue∩F∈Jand∪e∈Eμue:μue∩F∈K.=∪e∈Eμue:μue∩F∈J∩∪e∈Eμue:μue∩F∈K.=R⋆⋆⋆⋆JF∩R⋆⋆⋆⋆KFProposition 41.
ConsiderF,H⊆E and let J,K be ideals and R be a binary relation on E. Then,
(1)
R⋆⋆⋆⋆JE=E(2)
F⊆H⇒R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆JH(3)
R⋆⋆⋆⋆JF∪R⋆⋆⋆⋆JH⊆R⋆⋆⋆⋆JF∪H(4)
R⋆⋆⋆⋆JF∩H=R⋆⋆⋆⋆JF∩R⋆⋆⋆⋆JH(5)
R⋆⋆⋆⋆JF=R⋆⋆⋆⋆JFcc(6)
IfFc∈J, then, R⋆⋆⋆⋆JF=E(7)
IfJ⊆K, then, R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆KF(8)
IfJ=PE, then, R⋆⋆⋆⋆JF=E(9)
R⋆⋆⋆⋆J∩KF=R⋆⋆⋆⋆JF∩R⋆⋆⋆⋆KF(10)
R⋆⋆⋆⋆J∨KF=R⋆⋆⋆⋆JF∪R⋆⋆⋆⋆KFProof.
(1)
R⋆⋆⋆⋆JE=R⋆⋆⋆⋆Jϕc=ϕc=E by Proposition 40 part (1).(2)
LetF⊆H. Thus, Hc⊆Fc, and hence, R⋆⋆⋆⋆JHc⊆R⋆⋆⋆⋆JFc (by Proposition 40 part (2)). Then, R⋆⋆⋆⋆JFcc⊆R⋆⋆⋆⋆JHcc. So, R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆JH(3)
Immediately by part (2)(4)
R⋆⋆⋆⋆JF∩H=R⋆⋆⋆⋆JF∩Hcc.=R⋆⋆⋆⋆JFc∪Hcc.=R⋆⋆⋆⋆JFc∪R⋆⋆⋆⋆JHcc(by no. (4) in Proposition 40). =R⋆⋆⋆⋆JFcc∩R⋆⋆⋆⋆JHcc.=R⋆⋆⋆⋆JF∩R⋆⋆⋆⋆JH(5)
Immediately obtains by Definition39(6)
LetFc∈J, then R⋆⋆⋆⋆JF=R⋆⋆⋆⋆JFcc=ϕc=E by Proposition 40 part (6)(7)
LetJ⊆K. Then, R⋆⋆⋆⋆KFc⊆R⋆⋆⋆⋆JFc by Proposition 40 part (7). Thus, R⋆⋆⋆⋆JFcc⊆R⋆⋆⋆⋆KFcc. Hence, R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆KF(8)
LetJ=PE, then, R⋆⋆⋆⋆JF=R⋆⋆⋆⋆JFcc=ϕc=E by Proposition 40 part (8)(9)
R⋆⋆⋆⋆J∩KF=R⋆⋆⋆⋆J∩KFcc.=R⋆⋆⋆⋆JFc∪R⋆⋆⋆⋆KFcc (by Proposition 40 part (9)). =R⋆⋆⋆⋆JFcc∩R⋆⋆⋆⋆KFcc.=R⋆⋆⋆⋆JF∩R⋆⋆⋆⋆KF(10)
R⋆⋆⋆⋆J∨KF=R⋆⋆⋆⋆J∨KFcc.=R⋆⋆⋆⋆JFc∩R⋆⋆⋆⋆KFcc (by Proposition 40 part (10)). =R⋆⋆⋆⋆JFcc∪R⋆⋆⋆⋆KFcc.=R⋆⋆⋆⋆JF∪R⋆⋆⋆⋆KFRemark 42.
(1)
In Proposition40 and Proposition 41, the converse of part 2 is generally false. To validate that consider Example 1 (i)(a)
IfF=e1,H=e4, then, R⋆⋆⋆⋆JF=ϕ,R⋆⋆⋆⋆JH=E. Therefore, R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆JH, but F⊈H(b)
IfF=e1,e2,e3 and H=e2,e3,e4, then R⋆⋆⋆⋆JF=ϕ,R⋆⋆⋆⋆JH=E. Therefore, R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆JH, but F⊈H(2)
In Proposition40 and Proposition 41, the converse of parts 6, 7, and 8 is generally false. To show that consider Example 1 (ii).
(a)
For part 6 take(i)
F=e1,e2, then, R⋆⋆⋆⋆KF=ϕ. Therefore, R⋆⋆⋆⋆KF=ϕ, but F∈K(ii)
F=e3,e4, then, R⋆⋆⋆⋆KF=E. Therefore, R⋆⋆⋆⋆KF=E, but Fc∈K.(b)
For part 7 take(i)
F=e1,e2, then, R⋆⋆⋆⋆JF=e2,R⋆⋆⋆⋆KF=ϕ. Therefore, R⋆⋆⋆⋆KF⊆R⋆⋆⋆⋆JF, but J⊈K(ii)
F=e3,e4, then, R⋆⋆⋆⋆JF=e1,e3,e4,R⋆⋆⋆⋆KF=E. Therefore, R⋆⋆⋆⋆JF⊆R⋆⋆⋆⋆KF, but J⊈K(c)
For part 8 take(i)
F=e1,e2, then, R⋆⋆⋆⋆KF=ϕ, but K≠PE(ii)
F=e3,e4, then, R⋆⋆⋆⋆KF=E, but K≠PE(3)
In Proposition40 and Proposition 41, the inclusion relations of part 3 are generally proper. To demonstrate that consider Example 1 (iii), and let F=e1,e4 and H=e2,e3. Then(a)
R⋆⋆⋆⋆JF=R⋆⋆⋆⋆JH=E,R⋆⋆⋆⋆JF∩H=ϕ. Therefore, R⋆⋆⋆⋆JF∩R⋆⋆⋆⋆JH=E≠ϕ=R⋆⋆⋆JF∩H(b)
R⋆⋆⋆⋆JF=R⋆⋆⋆⋆JH=ϕ,R⋆⋆⋆⋆JF∪H=E. Therefore, R⋆⋆⋆⋆JF∪R⋆⋆⋆⋆JH=ϕ≠E=R⋆⋆⋆⋆JF∪HRemark 43.
Some properties of the lower and upper approximations given in the second method are not kept by this method as we demonstrate in the following.(i)
In Example1 (i) take(a)
F=e1, then, R⋆⋆⋆⋆JF=ϕ. Hence, F⊈R⋆⋆⋆⋆JF(b)
F=e2,e3,e4, then, R⋆⋆⋆⋆JF=E. Hence, R⋆⋆⋆⋆JF⊈F(ii)
In Example1 (ii) take(a)
F=E, then, R⋆⋆⋆⋆JE=e2,e3,e4. Hence, R⋆⋆⋆⋆JE≠E(b)
F=ϕ, then, R⋆⋆⋆⋆Jϕ=e1. Hence, R⋆⋆⋆⋆Jϕ≠ϕProposition 44.
ConsiderR and J are, respectively, binary relation and ideal on E≠ϕ and let ϕ≠F⊆E. Then,
(1)
0≤ACCR⋆⋆⋆⋆JF≤1(2)
ACCR⋆⋆⋆⋆JE=1Proof.
Similar to Proposition14.Theorem 45.
ConsiderR as a binary relation on E and let J,K be ideals on E such that J⊆K. For each F⊆E, the next properties hold.
(1)
BNDR⋆⋆⋆⋆KF⊆BNDR⋆⋆⋆⋆JF(2)
ACCR⋆⋆⋆⋆JF≤ACCR⋆⋆⋆⋆KF(3)
RoughR⋆⋆⋆⋆KF≤RoughR⋆⋆⋆⋆JFProof.
Similar to Theorem15.Remark 46.
Example1 (ii) shows that the converse of parts 1, 2, and 3 in Theorem 45 is not necessary to be true in general. Take, F=e3,e4, then
(1)
BNDR⋆⋆⋆⋆KF=ϕ⊆ϕ=BNDR⋆⋆⋆⋆JF, but J⊈K(2)
ACCR⋆⋆⋆⋆JF=1≤1=ACCR⋆⋆⋆⋆KF, but J⊈K(3)
RoughR⋆⋆⋆⋆KF=0≤0=RoughR⋆⋆⋆⋆JF, but J⊈KTheorem 47.
Letϕ≠F⊆E,J be an ideal on E and R1,R2 be two binary relations on E. If R1⊆R2, then
(1)
R1⋆⋆⋆⋆JF⊆R2⋆⋆⋆⋆JF(2)
R2⋆⋆⋆⋆JF⊆R1⋆⋆⋆⋆JF(3)
BNDR1⋆⋆⋆⋆JF⊆BNDR2⋆⋆⋆⋆JF(4)
ACCR2⋆⋆⋆⋆JF≤ACCR1⋆⋆⋆⋆JF(5)
RoughR1⋆⋆⋆⋆JF≤RoughR2⋆⋆⋆⋆JFProof.
(1)
Lete∈R1⋆⋆⋆⋆JF. Then, ∃y∈E such that e∈μ1uy∩F∈J. Since, μ1uy⊆μ2uy (by Theorem 4 [20]). It follows that e∈μ2uy∩F∈J. Thus, e∈R2⋆⋆⋆⋆JF. Hence, R1⋆⋆⋆⋆JF⊆R2⋆⋆⋆⋆JF(2)
e∈R2⋆⋆⋆⋆JF=R2⋆⋆⋆⋆JFcc.⊆R1⋆⋆⋆⋆JFcc (by part (1)).=R1⋆⋆⋆⋆JF(3)
Lete∈BNDR1⋆⋆⋆⋆JF. Then, e∈R1⋆⋆⋆⋆JF−R1⋆⋆⋆⋆JF. So, e∈R1⋆⋆⋆⋆JF and e∈R1⋆⋆⋆⋆JFc. Thus, e∈R2⋆⋆⋆⋆JF and e∈R2⋆⋆⋆⋆JFc by parts 1 and 2. Hence, e∈BNDR2⋆⋆⋆⋆JF. Therefore, BNDR1⋆⋆⋆⋆JF⊆BNDR2⋆⋆⋆⋆JF(4)
ACCR2⋆⋆⋆⋆JF=R2⋆⋆⋆⋆JF∩F/R2⋆⋆⋆⋆JF∪F≤R1⋆⋆⋆⋆JF∩F/R1⋆⋆⋆⋆JF∪F=ACCR1⋆⋆⋆⋆JF(5)
Straightforward by (4)Remark 48.
In Theorem47, the inclusion and less than relations are generally proper. To show this matter consider Example 2 and take
(i)
F=e1,e2, then(1)
R1⋆⋆⋆⋆JF=e1,e4≠e1,e3,e4=R2⋆⋆⋆⋆JF(2)
ACCR1⋆⋆⋆⋆JF=2/3≠1/2=ACCR2⋆⋆⋆⋆JF(3)
RoughR1⋆⋆⋆⋆JF=1/3≠1/2=RoughR2⋆⋆⋆⋆JF(ii)
F=e2,e3, then, R1⋆⋆⋆⋆JF=F≠e2=R2⋆⋆⋆⋆JF


## 4. Comparison the Proposed Methods and Their Advantages Compared to the Previous Ones

Herein, we first compare between the current purposed methods and demonstrate that the method given in Subsection3.3 is the best in terms of develop the approximation operators and values of accuracy. Then, we clarify that the first approach produces accuracy measures of subsets higher than their counterparts displayed in [20].Theorem 49.
LetF⊆E,J be an ideal on E and R be a binary relation on E. Then,
(1)
R⋆JF⊆R⋆⋆JF(2)
R⋆⋆JF⊆R⋆JF(3)
BNDR⋆⋆JF⊆BNDR⋆JF(4)
ACCR⋆⋆JF=ACCR⋆JF(5)
RoughR⋆⋆JF=RoughR⋆JFProof.
Immediately by using the Definitions9 and 18.Remark 50.
Example2 shows that the inclusion and less than in Theorem 49 cannot be replaced by equality relation in general. Take F=e1,e2,e3, then
(1)
R1⋆JF=e1,e4≠E=R1⋆⋆JF(2)
R⋆⋆1JF=e1,e2,e3≠E=R⋆1JF(3)
BNDR1⋆JF=ϕ≠e4=BNDR1⋆⋆JFTheorem 51.
LetF⊆E,J be an ideal on E and R be a reflexive relation on E. Then,
(1)
R⋆⋆JF⊆R⋆JF⊆R⋆⋆⋆JF(2)
R⋆⋆⋆JF⊆R⋆JF⊆R⋆⋆JF(3)
BNDR⋆⋆⋆JF⊆BNDR⋆JF⊆BNDR⋆⋆JF(4)
ACCR⋆⋆JF≤ACCR⋆JF≤ACCR⋆⋆⋆JF(5)
RoughR⋆⋆⋆JF≤RoughR⋆JF≤RoughR⋆⋆JFProof.
(1)
By Theorem49, we have R⋆⋆JF⊆R⋆JF. To prove, R⋆JF⊆R⋆⋆⋆JF. Let e∈R⋆JF, then, μue∩Fc∈J. Hence, μue⊆R⋆⋆⋆JF. Since, R is a reflexive relation, thus, e∈μue⊆R⋆⋆⋆JF. Therefore, e∈R⋆⋆⋆JF(2)
To prove,R⋆⋆⋆JF⊆R⋆J. Let e∈R⋆⋆⋆JF=R⋆⋆⋆Fcc, then, x∈R⋆⋆⋆Fc. Hence, by Definition 29, we get μue∩F∈J. It follows that e∈R⋆JF. By Theorem 49, we have R⋆JF⊆R⋆⋆JF(3)
Straightforward from (4) and (2)
In Theorem51, the inclusion and less than relation in Theorem 51 cannot be replaced by equality relation in general as the next example demonstrates.Example 52.
LetE=e1,e2,e3,e4,J=ϕ,e4, and R=Δ∪e1,e4,e3,e4,e4,e2 be a binary relation defined on E. By calculations, we obtain μue1=μue3=e1,e3,e4,μue2=e2,e4,μue4=E. For part 3, take F=e1,e3,e4,then
(1)
R⋆⋆⋆JF=e1,e3⊊e1,e3,e4=R⋆JF(2)
R⋆JF=e1,e3⊊e1,e3,e4=R⋆⋆⋆JF(3)
BNDR⋆⋆⋆JF=ϕ⊊e4=BNDR⋆JF(4)
ACCR⋆JF=2/3⪇1=ACCR⋆⋆⋆JF(5)
RoughR⋆⋆⋆JF=0⪇1/3=RoughR⋆JFTheorem 53.
LetF⊆E,J be an ideal on E and R be a reflexive relation on E. Then,
(1)
R⋆⋆⋆⋆JF⊆R⋆JF⊆R⋆⋆⋆JF(2)
R⋆⋆⋆JF⊆R⋆J⊆R⋆⋆⋆⋆JF(3)
BNDR⋆⋆⋆JF⊆BNDR⋆JF⊆BNDR⋆⋆⋆⋆JF(4)
ACCR⋆⋆⋆⋆JF≤ACCR⋆JF≤ACCR⋆⋆⋆JF(5)
RoughR⋆⋆⋆JF≤RoughR⋆JF≤RoughR⋆⋆⋆⋆JFProof.
(1)
By Theorem51, we have R⋆JF⊆R⋆⋆⋆JF. To prove, R⋆⋆⋆⋆JF⊆R⋆JF, let e∈R⋆⋆⋆⋆JF=R⋆⋆⋆⋆JFcc. Then, x∈R⋆⋆⋆⋆JFc. Thus, by Definition 39, μue∩Fc∈J. It follows that μue⊆R⋆JF. Since, R is a reflexive relation, then e∈μue⊆R⋆JF. Therefore, e∈R⋆JF(2)
By Theorem51, we have R⋆⋆⋆JF⊆R⋆JF. To prove R⋆JF⊆R⋆⋆⋆⋆JF, let e∈R⋆JF, then μue∩F∈J. It follows that μue⊆R⋆⋆⋆⋆JF. Since, R is a reflexive relation, then e∈μue⊆R⋆⋆⋆⋆JF. Therefore, e∈R⋆⋆⋆⋆JF(3)
Straightforward from (4) and (2)Remark 54.
In Theorem53, the inclusion and less than relations are generally proper. To show this matter, consider Example 52 and take F=e1,e3,e4. Then,
(1)
R⋆J=e1,e3,e4⊊E=R⋆⋆⋆⋆JF(2)
R⋆⋆⋆⋆JF=ϕ⊊e1,e3=R⋆JF(3)
BNDR⋆JF=e4⊊E=BNDR⋆⋆⋆⋆JF(4)
ACCR⋆⋆⋆⋆JF=0⪇2/3=ACCR⋆JF(5)
RoughR⋆JF=1/3⪇1=RoughR⋆⋆⋆⋆JFRemark 55.
It follows from Theorem51 and Theorem 53 that the best method to improve the approximations and increase the accuracy values is that given in the third type in Subsection 3.3, since the boundary regions in this case are decreased (or canceled) by increasing the lower approximations and decreasing the upper approximations with the comparison of the other types in the other sections. Moreover, the accuracy is more accurate than the other types.
The following result elucidates that the first type of our approaches is better than the approximation spaces given in [20] (see Definition 5).Theorem 56.
LetF⊆E,J be an ideal on E and R be a binary relation on a nonempty set E. Then,
(1)
R⋆JF⊆R⋆F(2)
R⋆F⊆R⋆JF(3)
BNDR⋆JF⊆BR⋆F(4)
AccR⋆F≤ACCR⋆JF(5)
RoughR⋆JF≤RoughR⋆FProof.
(1)
Lete∈R⋆JF. Then, μue∩F∈J. Therefore, μue∩F≠ϕ. Hence, e∈R⋆⋆F, which means that R⋆JF⊆R⋆⋆F(2)
Lete∈R⋆⋆F. Then, μue⊆F. Therefore, μue∩Fc∈J. Hence, e∈R⋆JF, which means that R⋆⋆F⊆R⋆JF(3)
Immediately by parts 1 and 2Remark 57.
In Theorem56, the inclusion and less than relations are generally proper. To show this matter consider Example 2 and take F=e1,e2. Then,
(1)
R1⋆JF=e1,e4≠e1,e2,e4=R1⋆F(2)
R⋆1JF=E≠e2=R1⋆F(3)
BNDR1⋆JF=ϕ≠e1,e4=BR1⋆F(4)
AccR1⋆JF=2/3≠1/3=AccR1⋆F(5)
RoughR1⋆JF=1/3≠2/3=RoughR1⋆FAccording to Theorem56, it can be seen that the present methods reduce the boundary region by increasing the lower approximations and decreasing the upper approximations with the comparison of Al-shami’s methods [20]. This means that the current approximation spaces are proper generalizations of Al-shami’s approximations [20].One can easily prove the next result which show that Al-shami’s approximations [20] are special cases of the current approximations.Proposition 58.
If the idealJ is the empty set, then, the approximation spaces given herein and the approximation spaces given in [20] are identical.


## 5. Medical Application to Dengue Fever Disease

One of the global diseases that disturb humanity is dengue fever. According to the data from World Health Organization, it is common in many regions around the world, mainly in South America and Asia, and it causes about 60 million symptomatic infections and 13600 status deaths. Medically, this illness is transmitted to humans via virus-carrying dengue mosquitoes. Its symptoms begin from 3 to 4 days of infection. The average period of recovery lies from 2 days to a week.In this section, we analyze this illness via the structures of maximal union neighborhoods and ideals with respect to the first type of the proposed approach. We also compare the current approach with one of the recent approaches introduced in [20] and show that the present approach is more accurate.To start this discussion, we deal with the data of eight patients of dengue feverE=v1,v2,v3,v4,v5,v6,v7,v8 as displayed in Table 1. The set of symptoms of this illness (called “attributes”) is the set Σ=F,H,J,S, where F, H, J, and S, respectively, denote fever, headache, characteristic skin rash, and muscle and joint pains. The attributes have two values: 1 refers patient has symptoms, and 0 refers the patient has no symptoms. The made decision also has the same two values with the meaning of possessing dengue fever disease or not.Table 1
Information system of dengue fever.
EFHJSDengue feverv110111v200100v300101v401000v510010v611101v711000v811101We associate each patient with his/her symptoms by a mapf:E⟶2Σ such that fvi equals symptoms of patient vi. According to Table 1, we find the following.
(30)fv1=F,J,S,(31)fv2=fv3=J,(32)fv4=H,(33)fv5=F,S,(34)fv6=fv8=F,H,J,(35)fv7=F,H.The binary relation between these patients is specified by the system’s experts; assume this relation is given as follows.(36)viRvj⇔fvi∩fvj≥2.That is, the two patients are related if they have two similar symptoms at least. Of course, this relation is changed according to the standpoint of system’s experts. It can be noted the relation in 1 is a symmetry relation. This means that the right, left, and union neighborhoods are equal, i.e.,Nrv=Nlv=Nuv for each v∈E. On the other hand, R is not a reflexive because v2,v2∈R, also, it is not a transitive relation because v1,v7∈R in spite of v1,v6,v6,v7∈R. This means we cannot apply Pawlak’s approach to model this information system and because Pawlak’s approach applied only when the relation is equivalence.To build the ideal approximation space of dengue fever information system, we first list the relation as followsR=v1,v1,v1,v5,v1,v6,v1,v8,v5,v1,v5,v5,v6,v1,v6,v6,v6,v7,v6,v8,v7,v6,v7,v7,v7,v8,v8,v1,v8,v6,v8,v7,v8,v8.Then, we calculate the union neighborhoodNu of each element in E(37)Nuv1=v1,v5,v6,v8,(38)Nuv2=Nuv3=Nuv4=ϕ,(39)Nuv5=v1,v5,(40)Nuv6=Nuv8=v1,v6,v7,v8,(41)Nuv7=v6,v7,v8.After that, we calculate the maximal union neighborhoodμu of each element in E.
(42)μuv1=μuv6=μuv8=v1,v5,v6,v7,v8,(43)μuv2=μuv3=μuv4=ϕ,(44)μuv5=v1,v5,v6,v8,(45)μuv7=v1,v6,v7,v8,Finally, we consider the ideal isJ=ϕ,v5,v7,v5,v7.Now, we consider two sets: a set of patients infected with dengue feverA1=v1,v3,v6,v8, and a set of patients without infection with dengue fever A2=v2,v4,v5,v7. In what follows, we calculate their lower and upper approximations, boundary regions, and the accuracy measures utilizing a method of maximal union neighborhoods given in [20] and the first method given herein.
(i)
Al-shami’s approach [20] (see Definition 5)Case 1.
The set of patients infected with dengue feverA1=v1,v3,v6,v8.By computing, we find the lower and upper approximations areR⋆A1=v2,v3,v4 and R⋆A1=v1,v5,v6,v7,v8, respectively. This means that the boundary region is BR⋆A1=v1,v5,v6,v7,v8. Hence, the accuracy and roughness measures are AccR⋆A1=1/6 and RoughR⋆A1=5/6, respectively.
(ii)
Our approach given in Definition9By computing, we find the lower and upper approximations areR⋆JA1=v1,v2,v3,v4,v6,v7,v8 and R⋆JA1=ϕ, respectively. This means that the boundary region is BNDR⋆JA1=ϕ. Hence, the accuracy and roughness measures are ACCR⋆JA1=3/4 and RoughR⋆JA1=1/4, respectively.
(i)
Al-shami’s approach [20] (see Definition 5)Case 2.
The set without infection with dengue feverA2=v2,v4,v5,v7.By computing, we find the lower and upper approximations areR⋆A2=v2,v3,v4 and R⋆A2=v1,v5,v6,v7,v8, respectively. This means that the boundary region is BR⋆A2=v1,v5,v6,v7,v8. Hence, the accuracy and roughness measures are AccR⋆A2=2/7 and RoughR⋆A2=5/7, respectively.
(ii)
Our approach given in Definition9By computing, we find the lower and upper approximations areR⋆JA2=v2,v3,v4 and R⋆JA2=ϕ, respectively. This means that the boundary region is BNDR⋆JA2=ϕ. Hence, the accuracy and roughness measures are ACCR⋆JA2=RoughR⋆JA2=1/2.It follows from Case1 and Case 2 that the boundary region of the infected patients with dengue fever using approach given in [20] is v1,v5,v6,v7,v8, so we cannot decide whether these patients are infected with dengue fever or not. This matter expands the area of vagueness/uncertainty and affects the precision of made decision. On the other hand, by using the first method introduced in this work, we see the boundary region is the empty set, which means we decrease the vagueness in the data and then enhance the value of accuracy.According to the above discussion and computations, we see that there are different techniques applied to approximate the sets. Our approach “maximal union neighborhoods and ideal” is one of the preferable techniques since it minimizes (or cancels) the boundary region by enlarging the lower approximation and dwindling the upper approximation, which leads to increase the value of the accuracy compared to the other types such those given in [20]. Hence, our approach eliminates the ambiguity of the data in the practical problems; particularly, in the medical diagnosis which needs accurate decisions.


## 6. Conclusion

In recent decades, rough set theory forms an important instrument for many authors interested to decision-making issues. The studies on this theory are mostly subjected to approximation spaces induced from new types of neighborhood systems or their interaction with other structures like ideals.In this work, we have focused on the idea of approximation spaces with the aim to decrease the regions of boundary and increase the values of accuracy. Therefore, we have provided novel approximation spaces using the concepts of “maximal union neighborhoods” and “ideals.” We have scrutinized the basic properties of Pawlak rough set model for the given approximation spaces. Also, we have discussed these properties in the current approximation spaces with respect to different ideals. One of the interesting obtained characterizations of the current models is preserving the monotonic property, which enables us to evaluate the vagueness in the data and enhance the confidence for the outcomes. With the help of examples, we have compared between the given approximation spaces with respect to the accuracy values. To demonstrate the advantages of the followed approach, we have proved that the first kind of our approximation spaces produces higher accuracy than the approach studied in [20] as well as we applied it to the analysis of the information system of dengue fever illness.Our future roadmaps are to study these ideal approximation paradigms in the frame of soft rough set and fuzzy rough set. Also, we probe these ideal approximation paradigms from a topological view and elucidate the relationships between the two approaches. We can improve these ideal approximation paradigms via topological structures by using some generalizations of open sets such asα-open and b-open sets. Moreover, we will research these approximation paradigms using other kinds of maximal neighborhoods and apply to handle real-life problems.



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*Source: 5459796-2022-09-23.xml*