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## Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(1): 43-49

Published online March 25, 2024

https://doi.org/10.5391/IJFIS.2024.24.1.43

© The Korean Institute of Intelligent Systems

## Rough Neighborhood Ideal and Its Applications

Abdel Fatah A. Azzam1,2

1Department of Mathematics, Faculty of Science and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj, Saudi Arabia
2Department of Mathematics, Faculty of Science, New Valley University, Elkharga, Egypt

Correspondence to :
Abdel Fatah A. Azzam (aa.azzam@psau.edu.sa)

Received: February 27, 2023; Revised: October 9, 2023; Accepted: November 28, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this study, we present a novel notion of an approximation space based on the idea of an ideal, called the rough neighborhood ideal, and its properties. Additionally, we combine two areas in which decision-making precision is improved using rough ideal expansions. An application scenario was presented to show how decision-making accuracy increased for coronavirus disease 2019 diagnosis. We also use variable precision to improve the accuracy of decision-making. Finally, we compared our strategy with another strategy.

Keywords: Ideal, Rough set, Variable precision, Topological rough neighbourhood ideal, Covid-19

### 1. Introduction and Preliminaries

It is generally recognized that the difficulty of choosing the best option in a specific medical case frequently involves the application of differential equations, mathematical modeling, functional analysis, and mathematical statistics. Until recently, researchers assumed that abstract topology was irrelevant. This assumption changed when it significantly contributed to the data analysis process in multiple fields [1]. Over the past 30 years, the notion of rough sets has been applied to research on medical information systems. This approach is based on a topological space with both open and closed sets. Pawlak used rough sets and the idea of approximation [2, 3] to define any set B ⊑ ℵ, where ℵ is a universal set that is not empty. The rough approximate set concept divides the field into three regions (lower, upper, and boundary) because it is motivated by the idea that the currently available knowledge makes it impossible to fully classify objects according to the conventional Pawlak rough set model [4]. The theory of rough sets begins with an equivalence relation, which appears to be a strict constraint restricting the application of rough sets. To address this unreasonableness, other expansions have been proposed under various relationships [57]. New forms of neighborhoods, such as minimal right (left) [8, 9] and intersection (union) neighborhoods [10], have been built to fulfill a range of goals, including increasing the accuracy of the set. To the best of our knowledge, ideas are defined using approximate sets. For example, we can state that two sets with different members are approximately equal if their upper and lower approximations are identical. Topological spaces are characterized by the comparison of sets in terms of their closures and interior points rather than their elements. Skowron [11] andWiweger [12] studied rough set theory by considering topological concepts. Lashin et al. [13] created a topology based on binary relations, which they used to expand the key concepts in rough set theory. To introduce multi-knowledge bases, Abu-Donia [14] used rough approximations and topologies. A topological vacuum was employed with the ideal concept introduced 60 years ago to build a larger topological vacuum to overcome and address some of the existing challenges [15, 16]. In this study, we attempt to utilize the ideal in a novel manner by constructing it based on a new neighborhood relationship. We then apply rough set theory to calculate any group’s accuracy and attempt to minimize variables and obtain decision-making accuracy in certain medical applications, such as coronavirus spread and factors affecting hospitals in Egypt. We also compared the two procedures after adding a variable to improve the decision-making accuracy.

If ® is an equivalent relation over a nonempty set ℵ, then the category in ® containing an element n ∈ ℵ is given by: [n] = {y : n®y} with classification ® and the upper and lower approximations are ®(E) = {n ∈ ℵ : [n] ⊑ E} and ®(E) = {n ∈ ℵ : [n] ⊓ E ≠ ∅} where E ⊑ ℵ.

### Definition 1.1

Yao [7] generalized this relationship by using after set n ® or before set ® n, where ® is a binary relation on ℵ such that n ® = {y ∈ ℵ : n®y}, ® n = {y ∈ ℵ : y ® n}, and defined new approximations for E ⊑ ℵ as ®(E) = {n ∈ ℵ : n ® ⊑ E} and ®(E) = {n ∈ ℵ : n ®⊓ E ≠ ∅} or ® (E) = {n ∈ ℵ : ®nE} and ®(E) = {n ∈ ℵ : ®nE ≠ ∅}.

### Definition 1.2

Abo-Table [17] generalized this relation by using the intersection of after set < n >®= ⊓{p ® : np ®}. ® is reflexive relation and defined a new approximation ®(E) = {n ∈ ℵ : < n >®E} and ®(E) = {n ∈ ℵ : < n >®E ≠ ∅}.

### Definition 1.3

Kandil et al. [18] used the ideal concept to generalize the rough approximation as ®(E) = {n ∈ ℵ : < n >®EcI} and ®(E) = {n ∈ ℵ : < n >®EI}, ® is reflexive relation and I is an ideal.

However, with this generalization, some problems arise in this relationship, where the Pawlak properties are not satisfied as: ®(E) ⊑ E®(E). And if I = (ℵ), then ®(E) = ∅, which is demonstrated in the following instance.

### Example 1.4

Let ℵ = {n, m, o, p} and ® = {(n,m), (n, p), (o,m), (o, p), (m, o), (m, n), (p, n), (p, o)} then n ® = {m, p} = o ®, m ® = {n, o} = p ®. Then from Yao’s definition [7], we have ®{n, o} = {m, p} and ®{m, p} = {n, o}.

Therefore, to generalize the rough approximation by an ideal induced neighborhood, we offer another method in Section 2 of this study.

### Definition 1.5 ([19])

A nonempty collection I(ℵ) ((ℵ) is the power set of ℵ) of space ℵ that is called ideal in this space if the following criteria are satisfied:

• (1) EI, gives (E) ⊑ I.

• (2) EI and CI; subsequently, ECI.

Given that ℵ carries topology τ with an ideal I on ℵ, the set operator()* : (ℵ) → (ℵ), which is a local function [20] of E. with respect to τ and I is defined as follows: E ⊑ ℵ, E*(I, τ ) = {y ∈ ℵ : OyEI for every Oyτ (y)} where τ (y) = {Oτ : yO}. Cl*() an operator for the Kuratowski closure in the τ*(I, τ ) topology known as the τ*topology, which is larger than the τ is defined as Cl*(E) = EE*(I, τ ) [17]. When there is no potential for confusion, we simply write as follows: E* for E*(I, τ ) and τ* for τ*(I, τ ). Space (ℵ, τ, I) is an ideal topological space if I is ideal for ℵ.

The ideal depends on two mappings, ()* and Cl*, which generate a unique ideal topological space finer than τ in the space ℵ denoted by τ* on ℵ, as discussed in [20, 21].

### Definition 1.6 ([19])

There exists a distinct topology on ℵ provided by τ* = {U ⊑ ℵ : Cl*(ℵ – U) = ℵ – U}, which corresponds to an ideal I in topological space (ℵ, τ ).

### Remark 1.7 ([20])

Let (ℵ, τ ) represent a topological space and I represent the ideal ℵ. Then and EI} is the open base for τ*.

### Definition 1.8 ([17])

All closed subsets of ℵ that contain E intersects at this point is defined as the closure of a subset E of a topological space (ℵ, τ ), represented by Cl(E).

### Definition 1.9 ([19])

Let (ℵ, τ ) be a space and E ⊑ ℵ. Int(E) represents the union of all the open sets included in E, which is the interior of E. It is also the largest open subset of ℵ in E.

### Definition 1.10 ([22, 23])

COVID-19 is a coronavirus disease that spread in 2019 and caused a global pandemic. This ailment is passed on to humans through animals. On December 31, 2019, pneumonia of unknown cause was discovered in Wuhan, China, and reported to the World Health Organization Country Office in China for the first time. Several fundamental public health instruments that were effectively employed in the fight against Ebola and COVID-19 are still in use today. Symptoms may appear 3 to 14 days after viral exposure. COVID-19 symptoms include high fever (T), cough (C), breathing difficulties (D), muscle and joint pain (P), and sore throat (S).

As of May 12, 2020, it had caused approximately 5 million symptomatic infections worldwide and 300,000 deaths. Consequently, we attempted to analyze this problem through neighborhoods, which reduced the condition attributes but increased the decision attribute accuracy.

### Definition 1.11

For a topological space (ℵ, τ ), it is natural to define the accuracy of subset E of ℵ as the accuracy Γ(E) = |Int(E)| / |Cl(E)|, where |Int(E)| is the cardinality of Int(E).

### Definition 1.12 (Variable precision)

• (1) It is clear for all N,E ⊑ ℵ, where ℵ is a finite universe that is not empty, that NE if E contains all of N’s elements.

• (2) In [1, 24], the authors show that NE with a ratio of error γ based on the measurement value M(N,E). In other words, Nγ E if and only if M(N,E) ≤ γ, where M(N,E)=1-card(NE)card(N) if card(N) > 0. M(N,E) = 0 if card(N) = 0, where 0 < γ ≤ 0.5 is the allowable classification error.

### 2. New Rough Set Approximations

In this section, the ideal concept is used to generalize the rough approximation. We demonstrate the results and compare the two approaches.

### Definition 2.1

Consider an approximation space (ℵ,®). ℵ is a universal set, ® is a binary relation on ℵ, and ℏ(n) = {y : n®y for all n, y ∈ ℵ}, where ℏ(n) is called a right neighbourhood of n for all n ∈ ℵ,

### Proposition 2.2

Let ℵ be a universe that is not empty and ® ⊑ ℵ × ℵ be a preordering relation on ℵ. Then, Ω = {ℏ(n) : n ∈ ℵ} is the base for topology on ℵ.

Proof

Because n ∈ ℏ(n) ∀ n ∈ ℵ, ® is reflexive. Then ℵ = ⊔n∈ℵ;ℏ(n), Let ℏ(n), ℏ(y) ∈ Ω such that ℏ(n) ⊓ ℏ(y) ≠ ∅ ∀ n, y ∈ ℵ that is, ∃ n ∈ ℵ such that n ∈ ℏ(n) and n ∈ ℏ(y). However, z ∈ ℏ(n) ⇒, ℏ(z) ⊑ ℏ(n) and z ∈ ℏ(y) ⇒ ℏ(z) ⊑ ℏ(y) because ® is transitive, ℏ(z) ⊑ ℏ(n) ⊓ ℏ(y). i.e., ∀ ℏ(n), ℏ(y) ∈ Ω such that ℏ(x) ⊓ ℏ(y) ∈ Ω. Then, Ω is the basis for the topology of ℵ.

### Definition 2.3

We consider an ideal on ℵ using ℏ(n) as follows: I® = {ℏ(n) ⊔ E : E is a subset of ℵ that satisfies the ideal alongside ℏ(n)}, and the lower and upper approximations can then be defined as follows using the ideal I®(E) = {II® : IE} and I®(E) = {II® : IE ≠ ∅}.

### Example 2.4

Let ℵ = {n, m, o, p} and any relation ® = {(n, n), (m, n), (m,m), (o, p), (p,m), (p, o)} and ℏ(n) = {n}, ℏ(m) = {n, m}, ℏ(o) = {p}, ℏ(p) = {m, o}, then we have I® = {{n}, {m}, {o}, {p}, {n, m}, {n, o}, {n, p}, {m, o}, {m, p}, {o, p}, {n, m, o}, {n, m, p}, {m, o, p}, {n, o, p}, ℵ, ∅}.

For E = {n, o}, we have I®(E) = {n, o} and I®(E) = ℵ. In addition, bI (E) = {m, p} and ΓI (E) = 0.5. For C = {n, p}, we have I®(C) = {n, p}, I®(C) = ℵ. Additionally, bI (C) = {m, o} and ΓI (C) = 0.5.

### Remark 2.5

By comparing Examples 2.4 and 1.1, we note that the problem in 1.4, has been solved, i.e., I®(E) ⊑ EI®(E).

### Example 2.6

Let ℵ = {n, m, o, p} and any relation ® = {(n, n), (m, n), (m,m), (o, p), (p,m), (p, o)} and ℏ(n) = {n}, ℏ(m) = {n, m}, ℏ(o) = {p}, ℏ(p) = {m, o}, then we have I® = {{n}, {m}, {o}, {p}, {n, m}, {n, o}, {n, p}, {m, o}, {m, p}, {o, p}, {n, m, o}, {n, m, p}, {m, o, p}, {n, o, p}, ℵ, ∅}.

For E = {m, p}, ®(E) = {m, p} and ®(E) = {m}, However, I®(E) = {o} and Γ(B)=23. But, I®(E) = {m, p} and I®(E) = ℵ for E = {m, p}; thus, we have bI (E) = {n, o} and ΓI(E)=12.

### Proposition 2.7

Let E be any subset of ℵ and let Ec be the set’s complement. The pair of ideal approximation operators satisfies certain useful properties with respect to the UI-upper approximation. Let E, CU.

• (1) I®[(Ec)]c = I®(E).

• (2) EI®(E).

• (3) I®(∅) = ∅.

• (4) ECI®(E) ⊑ I®(C).

• (5) I®(EC) = I®(E) ⊔ I®(C).

• (6) I®(EC) ⊑ I®(E) ⊓ I®(C).

• (7) I®(I®(E)) ⊑ I®(E).

Proof

(1) I®[(Ec)]c = {II® : IEc ≠ ∅}c = {II® : IEc = ∅} =, {II® : IE} = I®(E).

From Definition 2.3, (2), (3), and (4) are obvious.

(5) I®(EC) = {II® : I ⊓ (EC) ≠ ∅} = {II® : (IE)⊔(IC) ≠ ∅} = {II® : IE ≠ ∅}⊔{II® : IC ≠ ∅} = I®(E) ⊔ I®(C).

(6) Proof of (6) is clear from the fourth part.

(7) Let nI®(I®(E)). Then, nII®(E) ≠ ∅. Thus, nI®(E) and I®(I®(E)) ⊑ I®(E).

### Proposition 2.8

Let E be any subset of ℵ and Ec denote the complement of the set E. Some useful properties are satisfied with respect to I®-lower approximation using the pair of ideal the approximation operators. Let E,C ⊑ ℵ.

• (1) [I®(Ec)]c = I®(E).

• (2) I®(E) ⊑ E.

• (3) ECI®(E) ⊑ I®(C).

• (4) I®(EC) ⊒, I(E) ⊔ I®(C).

• (5) I®(EC) = I®(E) ⊓ .I®.(C).

• (6) I®(I®(E)) = I®(E).

• (7) I®(ℵ) = ℵ.

Proof

Straightforward.

### Definition 2.9

Let (ℵ,®, I®) be IAS and E ⊑ ℵ. Then E is called I®-definable if and only if I®(E) = I®(E).

### Theorem 2.10

Let (ℵ,®, I®) be IAS on ℵ and E ⊑ ℵ. Then nI®(E) ⇔ ∃II® : IE ≠ ∅, for every nE.

Proof

It is clear.

### Definition 2.11

Let (ℵ,®, I®) be IAS on ℵ and E ⊑ ℵ. Then:

• 1. The ideal boundary of E (I®-bnd(E)) is I-bnd(E) = I®(E)-I®(E);

• 2. The ideal internal edge of E(I®edg(E)) is I®edg(E) = EI®(E);

• 3. The ideal external edge of E(I®edg(E)) is I®edg(E) = I®(E) – E.

### Proposition 2.12

Let (ℵ,®, I®) be IAS, and E ⊑ ℵ. Then E is called I®-definable (exact) if and only if I®(E) = I®(E).

Proof

It is clear.

### Example 2.13

Let ℵ = {n, m, o, p} and ® be a binary relation of ℵ and ℏ(n) = {n,m}, ℏ(m) = {m, o}, ℏ(o) = {p}, ℏ(p) = {o}. Then, I® = {{n}, {m}, {o}, {p}, {n, m}, {n, o}, {n, p}, {m, o}, {m, p}, {o, p}, {n, m, p}, {n, m, o}, {m, o, p}, {n, o, p}, ℵ, ∅}.

Let E = {n, m}, C = {m, o}, D = {m, o, p}. Subsequently, I®(E) = {n, m}, I®(E) = ℵ, hence Ibnd(E) = {o, p}. For set C, we have I®(C) = C, I®(C) = ℵ. Hence, Ibnd(C) = {n, p}. Finally, for the set D, we have I®(D) = D and I®(D) = ℵ. Hence, Ibnd(D) = {n}.

The link between the ideal boundary edge, ideal internal edge, and ideal external edge is described as follows:

### Remark 2.14

Let (ℵ,®, I®) be the IAS. Then the universal set ℵ can be separated into three disjoint regions relative to any E ⊑ ℵ

• 1- positive region by I®(E);

• 2- boundary region by I®(E) – I®(E); and

• 3- negative region by EI®(E).

### Definition 2.15

Let (ℵ,®, I®) be an IAS and E ⊑ ℵ. Then E is called

• 1- I® internally definable if and only if I®(E) = E,

• 2- I is externally definable if and only if I®(E) = E.

### Proposition 2.16

Let (ℵ,®, I®) be an IAS and E ⊑ ℵ. Then E is called I®-definable if I® is internally definable, and I® is defined externally.

Proof

It is clear.

### Definition 2.17

Let (ℵ,®, I®) be an ideal topological space. Then for every E ⊑ ℵ we have:

• 1- E is called I®-internally (I®-externally, I®-totally) definable if and only if E is I®-open (I®-closed, I®-clopen) set in ideal topological space.

• 2- E is called I®-undefinable (rough) set if and only if E neither open ideal nor closed ideal in an ideal topological space.

Thus we concluded that the ideal structure of topology show that the members of the I® sets, (I®)c sets is I®-open sets, I®-closed sets are the basic tools to measure exactness and the roughness of sets.

### Proposition 2.18

Let (ℵ,®, I®) be an IAS and E ⊑ ℵ. Then

• 1- E is called I®-internally definable if and only if Ec is I® is externally definable; that is, I®(E) = E if and only if I®(Ec) = Ec.

• 2- E is called I®-definable if and only if: Ec is I®-definable; that is, I®(E) = I®(E) if and only if I®(Ec) = I®(Ec).

• 3- E is called I®-undefinable if and only if Ec is I®-undefinable; that is, I®(E) ≠ E and I®(E) ≠ E if and only if I®(Ec) ≠ Ec and I®(Ec) ≠ Ec.

Proof

It is evident.

### 2.1 Applications of Rough Set Models

Before presenting the experimental results, this subsection presents a preliminary investigation of the five symptoms of COVID-19. The investigation was carried out on August 5, 2020, at Kharga General Hospital in Kharga City, New Valley Governorate, Egypt. Due to comparable patients, the COVID-19 problem was explained using the information system data for only 14 occurrences, as shown in Table 1. The columns show the symptoms of COVID-19; yes (1) denotes the presence of symptoms, and no (0) denotes the absence of symptoms in the patient (condition attributes), where T is the temperature. C represents cough, D represents difficulty in breathing, P represents muscle and joint pain, and S represents sore throat. Attribute d denotes the decision of COVID-19. In Table 1, O = {o1, o2, o3, o4, o5}, represent the patients.

We converted Tables 1 and 2 based on the similarity of attributes concerning patients.

Now, we construct an ideal via the neighborhood relation, which is related to the problem studied. Notably, this relationship was determined by the opinions of the problem experts. In this case, ab ⇔ ℏ(a, b) > 0.8, where ℏ(a, b) denotes the degree of similarity between a and b. Table 2 shows that ℵ(o1) = {o1}, ℵ(o2) = {o2}, ℵ(o3) = {o3}, ℵ(o4) = {o4}, ℵ(o5) = {o5}.

The ideal on O is I® = {{o1}, {o2}, {o3}, {o4}, {o5}, {o1, o2}, {o1, o3}, {o1, o4}, {o1, o5}, {o2, o3}, {o2, o4}, {o2, o5}, {o3, o4}, {o3, o5}, {o4, o5}, {o1, o2, o3}, {o1, o2, o4}, {o1, o2, o5}, {o2, o3, o4}, {o2, o3, o5}, {o3, o4, o5}, {o1, o3, o4}, {o1, o3, o5}, {o1, o4, o5}, {o2, o4, o5}, {o1, o2, o3, o4}, {o1, o2, o3, o5}, {o2, o3, o4, o5}, {o1, o3, o4, o5}, {o1, o2, o4, o5}, ∅, O}.

In case 1: Let E1 = {o1, o3} be the set of patients with COVID-19. We can measure the accuracy of E1 using rough approximations (lower and upper)and Definition 2.3 as I®(E1) = {o1, o3}, I®(E1) = {o1, o2, o3, o4, o5} and Γ(E1)=25, bI (E1) = {o2, o4, o5}.

In case 2: Let E2 = {o2, o4, o5} be the set of patients without COVID-19. By rough approximations (lower and upper) and by Definition 2.3, we can measure the accuracy of E2 as I®(E2) = {o2, o4, o5}, I®(E2) = {o1, o2, o3, o4, o5} and Γ(E2)=35, bI (E2) = {o1, o3}.

### 2.2 Improving Accuracy Via New Approaches

All sectors, particularly in the medical field, must be considered when making decisions. As a result, we aim to increase decision-making precision. A comparison of these two approaches was presented. Consequently, the following approximation pairs were used:

Definition 2.19

Let ® be a binary relationship between ℵ and EI®γ(E)={GE:M(G,E)γ,GI®} and Iγ®(E)={FE:C(F,E)<1-γ}.

By applying Definition 2.19 to improve the accuracy of the decision-making, we resolve the problem in Table 2 and obtain.

In case 1: Let E1 = {o1, o3} be the set of patients with COVID-19. We can measure the accuracy of E1 using rough approximations (lower and upper) and Definition 2.3 as I®0.5(E1)={o1,o3},I0.5®(E1)={o1,o3} and Γ0.5(E1) = 1, bI0.5(E1)=.

In case 2: Let E2 = {o2, o4, o5} be the set of patients without COVID-19. By rough approximations (lower and upper) and by Definition 2.3, we can measure the accuracy of E2 as I®0.5(E2)={o2,o4,o5},I0.5®(E2)={o2,o4,o5} and Γ0.5(E2) = 1, bI0.5(E2)=.

Notably, there is a clear improvement after using variable precision, as shown in Table 3.

Theorem 2.20

Let (ℵ,®, I®) be an IAS and E ⊑ ℵ. Then

• 1- I®γ()= and Iγ®()=.

• 2- I®γ(E)E and EIγ®(E).

• 3- If EC, then Iγ®(E)Iγ®(C), and I®γ(E)I®γ(C), for all E, C ∈ ℵ.

• 4- I®γ(I®γ(E))=I®γ(E) and Iγ®(Iγ®(E))=Iγ®(E).

• 5- I®γ(\E)=\Iγ®(E) and Iγ®(\E)=\I®γ(E).

Proof

1- By definition of value measure M(C,D) and Definition 4.1, I®γ(E)={GB:M(G,E)γ,GI®}, GI®}, ℵ, ∅ ∈ I® and M(ℵ, ℵ) = 0, that is I®γ()=. In addition, M(G, ∅) = 1; thus, Iγ®()=.

Proving 2, 3, 4, and 5 directly from Definition 2.19.

In this study, new ideal notions for approximation spaces were introduced. We were able to address the issues in the Yao, Allam, and Kandil approximations because of these new notions. Furthermore, these new approximations based on variable precision assisted us in improving application decision-making. The current approximations generally assist in reducing the boundary region and obtaining the finest decision-making. These new approximations can also be used in the future in other fields such as engineering, artificial intelligence, and economics.

This study is supported via funding from Prince Sattam bin Abdulaziz University project number (Project No. PSAU/2024/R/1445).

### Conflict of Interest

Table. 1.

Table 1. System for original medical information.

Oo1o2o3o4o5
T10101
C10100
D10100
P11111
S01101
d10100

Table. 2.

Table 2. Similarities in patients symptoms.

o1o2o3o4o5
o1115452525
o2151254545
o3452511535
o4254515135
o5254535351

Table. 3.

Table 3. Comparison between the method without variables and our improved method.

Without variable methodDefinition 2.19’s present technique

®®+Γ®γ=0.5-®γ=0.5+Γγ=0.5
1 = {o1, o3}{o1, o3}{o1, o2, o3, o4, o5}25{o1, o3}{o1, o3}1
2 = {o2, o4, o5}{o2, o4, o5}{o1, o2, o3, o4, o5}35{o2, o4, o5}{o2, o4, o5}1

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Abdel Fatah A. Azzam is an associate professor at the Department of Mathematics, Faculty of Science and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj, Saudi Arabia, and at the Department of Mathematics, Faculty of Science, New Valley University, Elkharga, Egypt. He received his A.P. degree (associate professor) in topology on June 28, 2021. He received his B.Sc. degree in mathematics in 1992, M.Sc. degree in 2000, and Ph.D. degree in 2010 from the Faculty of Science, Tanta University, Egypt. His research interests are general topology, variable precision in rough set theory, theory of generalized closed sets, ideals topology, theory of rough sets, Fuzzy rough sets, digital topology, and Grill with topology. In these areas, he has published over 51 technical papers in refereed international journals or conference proceedings. He is also the referee of many researches in high-impact journals.

E-mail: aa.azzam@psau.edu.sa

### Article

#### Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(1): 43-49

Published online March 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.1.43

## Rough Neighborhood Ideal and Its Applications

Abdel Fatah A. Azzam1,2

1Department of Mathematics, Faculty of Science and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj, Saudi Arabia
2Department of Mathematics, Faculty of Science, New Valley University, Elkharga, Egypt

Correspondence to:Abdel Fatah A. Azzam (aa.azzam@psau.edu.sa)

Received: February 27, 2023; Revised: October 9, 2023; Accepted: November 28, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this study, we present a novel notion of an approximation space based on the idea of an ideal, called the rough neighborhood ideal, and its properties. Additionally, we combine two areas in which decision-making precision is improved using rough ideal expansions. An application scenario was presented to show how decision-making accuracy increased for coronavirus disease 2019 diagnosis. We also use variable precision to improve the accuracy of decision-making. Finally, we compared our strategy with another strategy.

Keywords: Ideal, Rough set, Variable precision, Topological rough neighbourhood ideal, Covid-19

### 1. Introduction and Preliminaries

It is generally recognized that the difficulty of choosing the best option in a specific medical case frequently involves the application of differential equations, mathematical modeling, functional analysis, and mathematical statistics. Until recently, researchers assumed that abstract topology was irrelevant. This assumption changed when it significantly contributed to the data analysis process in multiple fields [1]. Over the past 30 years, the notion of rough sets has been applied to research on medical information systems. This approach is based on a topological space with both open and closed sets. Pawlak used rough sets and the idea of approximation [2, 3] to define any set B ⊑ ℵ, where ℵ is a universal set that is not empty. The rough approximate set concept divides the field into three regions (lower, upper, and boundary) because it is motivated by the idea that the currently available knowledge makes it impossible to fully classify objects according to the conventional Pawlak rough set model [4]. The theory of rough sets begins with an equivalence relation, which appears to be a strict constraint restricting the application of rough sets. To address this unreasonableness, other expansions have been proposed under various relationships [57]. New forms of neighborhoods, such as minimal right (left) [8, 9] and intersection (union) neighborhoods [10], have been built to fulfill a range of goals, including increasing the accuracy of the set. To the best of our knowledge, ideas are defined using approximate sets. For example, we can state that two sets with different members are approximately equal if their upper and lower approximations are identical. Topological spaces are characterized by the comparison of sets in terms of their closures and interior points rather than their elements. Skowron [11] andWiweger [12] studied rough set theory by considering topological concepts. Lashin et al. [13] created a topology based on binary relations, which they used to expand the key concepts in rough set theory. To introduce multi-knowledge bases, Abu-Donia [14] used rough approximations and topologies. A topological vacuum was employed with the ideal concept introduced 60 years ago to build a larger topological vacuum to overcome and address some of the existing challenges [15, 16]. In this study, we attempt to utilize the ideal in a novel manner by constructing it based on a new neighborhood relationship. We then apply rough set theory to calculate any group’s accuracy and attempt to minimize variables and obtain decision-making accuracy in certain medical applications, such as coronavirus spread and factors affecting hospitals in Egypt. We also compared the two procedures after adding a variable to improve the decision-making accuracy.

If ® is an equivalent relation over a nonempty set ℵ, then the category in ® containing an element n ∈ ℵ is given by: [n] = {y : n®y} with classification $ℵ®$ and the upper and lower approximations are ®(E) = {n ∈ ℵ : [n] ⊑ E} and ®(E) = {n ∈ ℵ : [n] ⊓ E ≠ ∅} where E ⊑ ℵ.

### Definition 1.1

Yao [7] generalized this relationship by using after set n ® or before set ® n, where ® is a binary relation on ℵ such that n ® = {y ∈ ℵ : n®y}, ® n = {y ∈ ℵ : y ® n}, and defined new approximations for E ⊑ ℵ as ®(E) = {n ∈ ℵ : n ® ⊑ E} and ®(E) = {n ∈ ℵ : n ®⊓ E ≠ ∅} or ® (E) = {n ∈ ℵ : ®nE} and ®(E) = {n ∈ ℵ : ®nE ≠ ∅}.

### Definition 1.2

Abo-Table [17] generalized this relation by using the intersection of after set < n >®= ⊓{p ® : np ®}. ® is reflexive relation and defined a new approximation ®(E) = {n ∈ ℵ : < n >®E} and ®(E) = {n ∈ ℵ : < n >®E ≠ ∅}.

### Definition 1.3

Kandil et al. [18] used the ideal concept to generalize the rough approximation as ®(E) = {n ∈ ℵ : < n >®EcI} and ®(E) = {n ∈ ℵ : < n >®EI}, ® is reflexive relation and I is an ideal.

However, with this generalization, some problems arise in this relationship, where the Pawlak properties are not satisfied as: ®(E) ⊑ E®(E). And if I = (ℵ), then ®(E) = ∅, which is demonstrated in the following instance.

### Example 1.4

Let ℵ = {n, m, o, p} and ® = {(n,m), (n, p), (o,m), (o, p), (m, o), (m, n), (p, n), (p, o)} then n ® = {m, p} = o ®, m ® = {n, o} = p ®. Then from Yao’s definition [7], we have ®{n, o} = {m, p} and ®{m, p} = {n, o}.

Therefore, to generalize the rough approximation by an ideal induced neighborhood, we offer another method in Section 2 of this study.

### Definition 1.5 ([19])

A nonempty collection I(ℵ) ((ℵ) is the power set of ℵ) of space ℵ that is called ideal in this space if the following criteria are satisfied:

• (1) EI, gives (E) ⊑ I.

• (2) EI and CI; subsequently, ECI.

Given that ℵ carries topology τ with an ideal I on ℵ, the set operator()* : (ℵ) → (ℵ), which is a local function [20] of E. with respect to τ and I is defined as follows: E ⊑ ℵ, E*(I, τ ) = {y ∈ ℵ : OyEI for every Oyτ (y)} where τ (y) = {Oτ : yO}. Cl*() an operator for the Kuratowski closure in the τ*(I, τ ) topology known as the τ*topology, which is larger than the τ is defined as Cl*(E) = EE*(I, τ ) [17]. When there is no potential for confusion, we simply write as follows: E* for E*(I, τ ) and τ* for τ*(I, τ ). Space (ℵ, τ, I) is an ideal topological space if I is ideal for ℵ.

The ideal depends on two mappings, ()* and Cl*, which generate a unique ideal topological space finer than τ in the space ℵ denoted by τ* on ℵ, as discussed in [20, 21].

### Definition 1.6 ([19])

There exists a distinct topology on ℵ provided by τ* = {U ⊑ ℵ : Cl*(ℵ – U) = ℵ – U}, which corresponds to an ideal I in topological space (ℵ, τ ).

### Remark 1.7 ([20])

Let (ℵ, τ ) represent a topological space and I represent the ideal ℵ. Then and EI} is the open base for τ*.

### Definition 1.8 ([17])

All closed subsets of ℵ that contain E intersects at this point is defined as the closure of a subset E of a topological space (ℵ, τ ), represented by Cl(E).

### Definition 1.9 ([19])

Let (ℵ, τ ) be a space and E ⊑ ℵ. Int(E) represents the union of all the open sets included in E, which is the interior of E. It is also the largest open subset of ℵ in E.

### Definition 1.10 ([22, 23])

COVID-19 is a coronavirus disease that spread in 2019 and caused a global pandemic. This ailment is passed on to humans through animals. On December 31, 2019, pneumonia of unknown cause was discovered in Wuhan, China, and reported to the World Health Organization Country Office in China for the first time. Several fundamental public health instruments that were effectively employed in the fight against Ebola and COVID-19 are still in use today. Symptoms may appear 3 to 14 days after viral exposure. COVID-19 symptoms include high fever (T), cough (C), breathing difficulties (D), muscle and joint pain (P), and sore throat (S).

As of May 12, 2020, it had caused approximately 5 million symptomatic infections worldwide and 300,000 deaths. Consequently, we attempted to analyze this problem through neighborhoods, which reduced the condition attributes but increased the decision attribute accuracy.

### Definition 1.11

For a topological space (ℵ, τ ), it is natural to define the accuracy of subset E of ℵ as the accuracy Γ(E) = |Int(E)| / |Cl(E)|, where |Int(E)| is the cardinality of Int(E).

### Definition 1.12 (Variable precision)

• (1) It is clear for all N,E ⊑ ℵ, where ℵ is a finite universe that is not empty, that NE if E contains all of N’s elements.

• (2) In [1, 24], the authors show that NE with a ratio of error γ based on the measurement value M(N,E). In other words, Nγ E if and only if M(N,E) ≤ γ, where $M(N,E)=1-card(N⊓E)card(N)$ if card(N) > 0. M(N,E) = 0 if card(N) = 0, where 0 < γ ≤ 0.5 is the allowable classification error.

### 2. New Rough Set Approximations

In this section, the ideal concept is used to generalize the rough approximation. We demonstrate the results and compare the two approaches.

### Definition 2.1

Consider an approximation space (ℵ,®). ℵ is a universal set, ® is a binary relation on ℵ, and ℏ(n) = {y : n®y for all n, y ∈ ℵ}, where ℏ(n) is called a right neighbourhood of n for all n ∈ ℵ,

### Proposition 2.2

Let ℵ be a universe that is not empty and ® ⊑ ℵ × ℵ be a preordering relation on ℵ. Then, Ω = {ℏ(n) : n ∈ ℵ} is the base for topology on ℵ.

Proof

Because n ∈ ℏ(n) ∀ n ∈ ℵ, ® is reflexive. Then ℵ = ⊔n∈ℵ;ℏ(n), Let ℏ(n), ℏ(y) ∈ Ω such that ℏ(n) ⊓ ℏ(y) ≠ ∅ ∀ n, y ∈ ℵ that is, ∃ n ∈ ℵ such that n ∈ ℏ(n) and n ∈ ℏ(y). However, z ∈ ℏ(n) ⇒, ℏ(z) ⊑ ℏ(n) and z ∈ ℏ(y) ⇒ ℏ(z) ⊑ ℏ(y) because ® is transitive, ℏ(z) ⊑ ℏ(n) ⊓ ℏ(y). i.e., ∀ ℏ(n), ℏ(y) ∈ Ω such that ℏ(x) ⊓ ℏ(y) ∈ Ω. Then, Ω is the basis for the topology of ℵ.

### Definition 2.3

We consider an ideal on ℵ using ℏ(n) as follows: I® = {ℏ(n) ⊔ E : E is a subset of ℵ that satisfies the ideal alongside ℏ(n)}, and the lower and upper approximations can then be defined as follows using the ideal I®(E) = {II® : IE} and I®(E) = {II® : IE ≠ ∅}.

### Example 2.4

Let ℵ = {n, m, o, p} and any relation ® = {(n, n), (m, n), (m,m), (o, p), (p,m), (p, o)} and ℏ(n) = {n}, ℏ(m) = {n, m}, ℏ(o) = {p}, ℏ(p) = {m, o}, then we have I® = {{n}, {m}, {o}, {p}, {n, m}, {n, o}, {n, p}, {m, o}, {m, p}, {o, p}, {n, m, o}, {n, m, p}, {m, o, p}, {n, o, p}, ℵ, ∅}.

For E = {n, o}, we have I®(E) = {n, o} and I®(E) = ℵ. In addition, bI (E) = {m, p} and ΓI (E) = 0.5. For C = {n, p}, we have I®(C) = {n, p}, I®(C) = ℵ. Additionally, bI (C) = {m, o} and ΓI (C) = 0.5.

### Remark 2.5

By comparing Examples 2.4 and 1.1, we note that the problem in 1.4, has been solved, i.e., I®(E) ⊑ EI®(E).

### Example 2.6

Let ℵ = {n, m, o, p} and any relation ® = {(n, n), (m, n), (m,m), (o, p), (p,m), (p, o)} and ℏ(n) = {n}, ℏ(m) = {n, m}, ℏ(o) = {p}, ℏ(p) = {m, o}, then we have I® = {{n}, {m}, {o}, {p}, {n, m}, {n, o}, {n, p}, {m, o}, {m, p}, {o, p}, {n, m, o}, {n, m, p}, {m, o, p}, {n, o, p}, ℵ, ∅}.

For E = {m, p}, ®(E) = {m, p} and ®(E) = {m}, However, I®(E) = {o} and $Γ(B)=23$. But, I®(E) = {m, p} and I®(E) = ℵ for E = {m, p}; thus, we have bI (E) = {n, o} and $ΓI(E)=12$.

### Proposition 2.7

Let E be any subset of ℵ and let Ec be the set’s complement. The pair of ideal approximation operators satisfies certain useful properties with respect to the UI-upper approximation. Let E, CU.

• (1) I®[(Ec)]c = I®(E).

• (2) EI®(E).

• (3) I®(∅) = ∅.

• (4) ECI®(E) ⊑ I®(C).

• (5) I®(EC) = I®(E) ⊔ I®(C).

• (6) I®(EC) ⊑ I®(E) ⊓ I®(C).

• (7) I®(I®(E)) ⊑ I®(E).

Proof

(1) I®[(Ec)]c = {II® : IEc ≠ ∅}c = {II® : IEc = ∅} =, {II® : IE} = I®(E).

From Definition 2.3, (2), (3), and (4) are obvious.

(5) I®(EC) = {II® : I ⊓ (EC) ≠ ∅} = {II® : (IE)⊔(IC) ≠ ∅} = {II® : IE ≠ ∅}⊔{II® : IC ≠ ∅} = I®(E) ⊔ I®(C).

(6) Proof of (6) is clear from the fourth part.

(7) Let nI®(I®(E)). Then, nII®(E) ≠ ∅. Thus, nI®(E) and I®(I®(E)) ⊑ I®(E).

### Proposition 2.8

Let E be any subset of ℵ and Ec denote the complement of the set E. Some useful properties are satisfied with respect to I®-lower approximation using the pair of ideal the approximation operators. Let E,C ⊑ ℵ.

• (1) [I®(Ec)]c = I®(E).

• (2) I®(E) ⊑ E.

• (3) ECI®(E) ⊑ I®(C).

• (4) I®(EC) ⊒, I(E) ⊔ I®(C).

• (5) I®(EC) = I®(E) ⊓ .I®.(C).

• (6) I®(I®(E)) = I®(E).

• (7) I®(ℵ) = ℵ.

Proof

Straightforward.

### Definition 2.9

Let (ℵ,®, I®) be IAS and E ⊑ ℵ. Then E is called I®-definable if and only if I®(E) = I®(E).

### Theorem 2.10

Let (ℵ,®, I®) be IAS on ℵ and E ⊑ ℵ. Then nI®(E) ⇔ ∃II® : IE ≠ ∅, for every nE.

Proof

It is clear.

### Definition 2.11

Let (ℵ,®, I®) be IAS on ℵ and E ⊑ ℵ. Then:

• 1. The ideal boundary of E (I®-bnd(E)) is I-bnd(E) = I®(E)-I®(E);

• 2. The ideal internal edge of E(I®edg(E)) is I®edg(E) = EI®(E);

• 3. The ideal external edge of E(I®edg(E)) is I®edg(E) = I®(E) – E.

### Proposition 2.12

Let (ℵ,®, I®) be IAS, and E ⊑ ℵ. Then E is called I®-definable (exact) if and only if I®(E) = I®(E).

Proof

It is clear.

### Example 2.13

Let ℵ = {n, m, o, p} and ® be a binary relation of ℵ and ℏ(n) = {n,m}, ℏ(m) = {m, o}, ℏ(o) = {p}, ℏ(p) = {o}. Then, I® = {{n}, {m}, {o}, {p}, {n, m}, {n, o}, {n, p}, {m, o}, {m, p}, {o, p}, {n, m, p}, {n, m, o}, {m, o, p}, {n, o, p}, ℵ, ∅}.

Let E = {n, m}, C = {m, o}, D = {m, o, p}. Subsequently, I®(E) = {n, m}, I®(E) = ℵ, hence Ibnd(E) = {o, p}. For set C, we have I®(C) = C, I®(C) = ℵ. Hence, Ibnd(C) = {n, p}. Finally, for the set D, we have I®(D) = D and I®(D) = ℵ. Hence, Ibnd(D) = {n}.

The link between the ideal boundary edge, ideal internal edge, and ideal external edge is described as follows:

### Remark 2.14

Let (ℵ,®, I®) be the IAS. Then the universal set ℵ can be separated into three disjoint regions relative to any E ⊑ ℵ

• 1- positive region by I®(E);

• 2- boundary region by I®(E) – I®(E); and

• 3- negative region by EI®(E).

### Definition 2.15

Let (ℵ,®, I®) be an IAS and E ⊑ ℵ. Then E is called

• 1- I® internally definable if and only if I®(E) = E,

• 2- I is externally definable if and only if I®(E) = E.

### Proposition 2.16

Let (ℵ,®, I®) be an IAS and E ⊑ ℵ. Then E is called I®-definable if I® is internally definable, and I® is defined externally.

Proof

It is clear.

### Definition 2.17

Let (ℵ,®, I®) be an ideal topological space. Then for every E ⊑ ℵ we have:

• 1- E is called I®-internally (I®-externally, I®-totally) definable if and only if E is I®-open (I®-closed, I®-clopen) set in ideal topological space.

• 2- E is called I®-undefinable (rough) set if and only if E neither open ideal nor closed ideal in an ideal topological space.

Thus we concluded that the ideal structure of topology show that the members of the I® sets, (I®)c sets is I®-open sets, I®-closed sets are the basic tools to measure exactness and the roughness of sets.

### Proposition 2.18

Let (ℵ,®, I®) be an IAS and E ⊑ ℵ. Then

• 1- E is called I®-internally definable if and only if Ec is I® is externally definable; that is, I®(E) = E if and only if I®(Ec) = Ec.

• 2- E is called I®-definable if and only if: Ec is I®-definable; that is, I®(E) = I®(E) if and only if I®(Ec) = I®(Ec).

• 3- E is called I®-undefinable if and only if Ec is I®-undefinable; that is, I®(E) ≠ E and I®(E) ≠ E if and only if I®(Ec) ≠ Ec and I®(Ec) ≠ Ec.

Proof

It is evident.

### 2.1 Applications of Rough Set Models

Before presenting the experimental results, this subsection presents a preliminary investigation of the five symptoms of COVID-19. The investigation was carried out on August 5, 2020, at Kharga General Hospital in Kharga City, New Valley Governorate, Egypt. Due to comparable patients, the COVID-19 problem was explained using the information system data for only 14 occurrences, as shown in Table 1. The columns show the symptoms of COVID-19; yes (1) denotes the presence of symptoms, and no (0) denotes the absence of symptoms in the patient (condition attributes), where T is the temperature. C represents cough, D represents difficulty in breathing, P represents muscle and joint pain, and S represents sore throat. Attribute d denotes the decision of COVID-19. In Table 1, O = {o1, o2, o3, o4, o5}, represent the patients.

We converted Tables 1 and 2 based on the similarity of attributes concerning patients.

Now, we construct an ideal via the neighborhood relation, which is related to the problem studied. Notably, this relationship was determined by the opinions of the problem experts. In this case, ab ⇔ ℏ(a, b) > 0.8, where ℏ(a, b) denotes the degree of similarity between a and b. Table 2 shows that ℵ(o1) = {o1}, ℵ(o2) = {o2}, ℵ(o3) = {o3}, ℵ(o4) = {o4}, ℵ(o5) = {o5}.

The ideal on O is I® = {{o1}, {o2}, {o3}, {o4}, {o5}, {o1, o2}, {o1, o3}, {o1, o4}, {o1, o5}, {o2, o3}, {o2, o4}, {o2, o5}, {o3, o4}, {o3, o5}, {o4, o5}, {o1, o2, o3}, {o1, o2, o4}, {o1, o2, o5}, {o2, o3, o4}, {o2, o3, o5}, {o3, o4, o5}, {o1, o3, o4}, {o1, o3, o5}, {o1, o4, o5}, {o2, o4, o5}, {o1, o2, o3, o4}, {o1, o2, o3, o5}, {o2, o3, o4, o5}, {o1, o3, o4, o5}, {o1, o2, o4, o5}, ∅, O}.

In case 1: Let E1 = {o1, o3} be the set of patients with COVID-19. We can measure the accuracy of E1 using rough approximations (lower and upper)and Definition 2.3 as I®(E1) = {o1, o3}, I®(E1) = {o1, o2, o3, o4, o5} and $Γ(E1)=25$, bI (E1) = {o2, o4, o5}.

In case 2: Let E2 = {o2, o4, o5} be the set of patients without COVID-19. By rough approximations (lower and upper) and by Definition 2.3, we can measure the accuracy of E2 as I®(E2) = {o2, o4, o5}, I®(E2) = {o1, o2, o3, o4, o5} and $Γ(E2)=35$, bI (E2) = {o1, o3}.

### 2.2 Improving Accuracy Via New Approaches

All sectors, particularly in the medical field, must be considered when making decisions. As a result, we aim to increase decision-making precision. A comparison of these two approaches was presented. Consequently, the following approximation pairs were used:

Definition 2.19

Let ® be a binary relationship between ℵ and $E⊑ℵI®γ(E)=⊔{G⊓E:M(G,E)≤γ,G∈I®}$ and $Iγ®(E)=⊓{F⊑E:C(F,E)<1-γ}$.

By applying Definition 2.19 to improve the accuracy of the decision-making, we resolve the problem in Table 2 and obtain.

In case 1: Let E1 = {o1, o3} be the set of patients with COVID-19. We can measure the accuracy of E1 using rough approximations (lower and upper) and Definition 2.3 as $I®0.5(E1)={o1,o3},I0.5®(E1)={o1,o3}$ and Γ0.5(E1) = 1, $bI0.5(E1)=∅$.

In case 2: Let E2 = {o2, o4, o5} be the set of patients without COVID-19. By rough approximations (lower and upper) and by Definition 2.3, we can measure the accuracy of E2 as $I®0.5(E2)={o2,o4,o5},I0.5®(E2)={o2,o4,o5}$ and Γ0.5(E2) = 1, $bI0.5(E2)=∅$.

Notably, there is a clear improvement after using variable precision, as shown in Table 3.

Theorem 2.20

Let (ℵ,®, I®) be an IAS and E ⊑ ℵ. Then

• 1- $I®γ(ℵ)=ℵ$ and $Iγ®(∅)=∅$.

• 2- $I®γ(E)⊑E$ and $E⊑Iγ®(E)$.

• 3- If EC, then $Iγ®(E)⊑Iγ®(C)$, and $I®γ(E)⊑I®γ(C)$, for all E, C ∈ ℵ.

• 4- $I®γ(I®γ(E))=I®γ(E)$ and $Iγ®(Iγ®(E))=Iγ®(E)$.

• 5- $I®γ(ℵ\E)=ℵ\Iγ®(E)$ and $Iγ®(ℵ\E)=ℵ\I®γ(E)$.

Proof

1- By definition of value measure M(C,D) and Definition 4.1, $I®γ(E)=⊔{G⊓B:M(G,E)≤γ,G∈I®}$, GI®}, ℵ, ∅ ∈ I® and M(ℵ, ℵ) = 0, that is $I®γ(ℵ)=ℵ$. In addition, M(G, ∅) = 1; thus, $Iγ®(∅)=∅$.

Proving 2, 3, 4, and 5 directly from Definition 2.19.

### 3. Conclusion

In this study, new ideal notions for approximation spaces were introduced. We were able to address the issues in the Yao, Allam, and Kandil approximations because of these new notions. Furthermore, these new approximations based on variable precision assisted us in improving application decision-making. The current approximations generally assist in reducing the boundary region and obtaining the finest decision-making. These new approximations can also be used in the future in other fields such as engineering, artificial intelligence, and economics.

System for original medical information.

Oo1o2o3o4o5
T10101
C10100
D10100
P11111
S01101
d10100

Similarities in patients symptoms.

o1o2o3o4o5
o1115452525
o2151254545
o3452511535
o4254515135
o5254535351

Comparison between the method without variables and our improved method.

Without variable methodDefinition 2.19’s present technique

®®+Γ®γ=0.5-®γ=0.5+Γγ=0.5
1 = {o1, o3}{o1, o3}{o1, o2, o3, o4, o5}25{o1, o3}{o1, o3}1
2 = {o2, o4, o5}{o2, o4, o5}{o1, o2, o3, o4, o5}35{o2, o4, o5}{o2, o4, o5}1

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