Expanded Rough Approximation Spaces Using Grill and Maximal Rough Neighborhoods for Medical Applications

Abstract

An important mathematical way to deal with ambiguity and uncertainty in knowledge is rough set (RS) theory. It is believed that a grill is a necessary addition to this idea. Since it expands the approximate of RSs, it is a helpful technique for removing ambiguity and uncertainty. One of the key and important issues for developing rough sets, which subsequently aim to maximize the accuracy measure, is minimization of the boundary region (BR). One of the most practical and successful ways to accomplish this is with a grill. Thus, the goal of this work is to introduce novel grill-based approaches for rough sets (RSs). A few important aspects of these techniques are examined and illustrated to indicate that they produce accuracy measures that are higher and more significant than those of the previous methods. In the end, a medical application is shown to emphasize the need of using grills as instructed.

1. Introduction

Researchers have recently become interested in concerns regarding uncertainty, ambiguity, and inaccuracy of information in data analysis in a variety of sectors, including health, chemistry, engineering, computer science, and social sciences, due to their critical role in supporting decision-making. Scientists from a variety of disciplines, including mathematics, engineering, artificial intelligence, smart cities, and others, have attempted to address this issue in various ways. They have proposed a variety of methods, including RS theory, for achieving this goal [1,2]. The method uses two approximations (APPs), lower (LOW) and upper (UPP), to evaluate the accuracy measure and BR. A crisp set is defined as having equal LOW and UPP APPs, while an RS does not. The difference between the UPP and LOW APPs is used to compute the BR. Vagueness is typically linked to this boundary. Therefore, whether or not the BR is empty determines the set’s accuracy or ambiguity. A non-empty BR indicates that our knowledge of a set is insufficient to describe it precisely. This theory tries to reduce boundary regions (BRs) and improve set accuracy. Since an equivalency relation in a finite universe forms the basis of the traditional rough APPs, it can occasionally be challenging to derive this relation for real-world issues. Many methods consequently looked into extensions of this theory to address challenging real-world issues. One method replaced the equivalency relations with similarity or tolerance [3,4,5,6,7,8,9,10] or binary relations (BIRs) [11,12,13,14,15,16,17,18,19,20,21]. By defining APPs using a neighborhood instead of an equivalency class, RS theory is generalized through the use of neighborhood systems. The LOW and UPP APPs were defined using a variety of neighborhood types, including right and left neighborhoods [22,23,24], rough neighborhood ideal [25], neighborhoods with minimal left and minimal right [26,27], those including union and intersection [28,29], equal neighborhoods [30,31], minimal and maximal neighborhoods [3,32], cardinality neighborhoods [33,34], etc. Abo-Tabl [35], meanwhile, established the APPs through the use of minimal right neighborhoods, which are established through reflexive relations that constitute a topological space’s foundation. Three novel approximation (APP) types based on maximal right regions determined by similarity relations were introduced by Dai et al. [36] more recently. It has been thought to be an improvement over Abo-Tabl [35] APPs. A non-empty collection of sets that satisfies three requirements is called a grill. Because grills produce a new topology, their concept aims to generalize topology. Grill topological spaces are based on the notion of two operators, $\theta$ and $\eta$. This idea was first presented by Choquet [37]. It has been discovered that the Choquat notion and ideals, nets, and filters share certain similarities. Several theories and aspects have been discussed in [11,12,26,28]. The idea contributes to the enlargement of the topological structure, which is used to quantify qualities rather than quantities, such as love, intelligence, beauty, and educational attainment. Additionally, it offers up new boundaries in nano-topological spaces [38] by extending the topological structure through the use of the idea of grill modifications in the BR, UPP APP, and LOW APP. We are aware that grills are useful while studying raw sets, especially when it comes to eliminating ambiguity. The insufficiency of current models in conserving essential features, as well as the validity of particular applicable equations for measuring confirmed and potential knowledge, are the driving forces behind our idea, which is motivated by the ideas of grills and neighborhoods. Thus, the introduction of novel grill-based RS approaches is one of the main driving forces behind this study. Furthermore, unlike earlier studies [36] that relied on similarity relations, the current study focuses on describing the fundamental ideas of RSs using maximal neighborhoods inferred from BIRs. Although the application range of RSs is expanded using BIRs, many practical applications do not always follow the similarity relations. As a result, as this restriction makes clear at the end of the study, this set cannot be widely applied. As such, the current methodology is a continuation of the strategies used by Dai et al. [36]. This work is divided into seven sections, which are presented in the following order. The first part is an introduction. Section 2 discusses the principles and preliminary steps for following up on previous efforts. Section 3, Section 4 and Section 5 aim to develop three approaches for approximating the set by utilizing the concept of grills and maximal neighborhoods resulting from BIRs. We investigate the qualities of the current APPs. Theorems 3, 5, and 6 show that the grill with a rough neighborhood reduces the BR of a subset while improving its accuracy. Propositions 2, 3, 6, and 7 demonstrate that the UPP and LOW APPs for three types of APPs are all monotonic. The relationships between these APPs are analyzed and presented. Furthermore,

Table 1. Comparison between our approach and Dai’s.

H	$\underline{\mathit{AP}}\left(\mathit{H}\right)$	${\underline{\underline{\mathit{AP}}}}_{\mathbf{{\rm Y}}}^{\Delta }\left(\mathit{H}\right)$	$\overline{\mathit{AP}}\left(\mathit{H}\right)$	${\overline{\overline{\mathit{AP}}}}_{\mathbf{{\rm Y}}}^{\Delta }\left(\mathit{H}\right)$	$\mathit{BNDR}\left(\mathit{H}\right)$	$\underline{{\mathit{BND}}_{\mathbf{{\rm Y}}}^{\Delta }}\left(\mathit{H}\right)$
$\left\{f\right\}$	$\varphi$	$\left\{f\right\}$	$\left\{f\right\}$	$\left\{f\right\}$	$\left\{f\right\}$	$\varphi$
$\left\{l\right\}$	$\varphi$	$\left\{l\right\}$	$\{f,l,m\}$	$\left\{l\right\}$	$\{f,l,m\}$	$\varphi$
$\left\{m\right\}$	$\varphi$	$\left\{m\right\}$	$\{l,m\}$	$\left\{m\right\}$	$\{l,m\}$	$\varphi$
$\left\{n\right\}$	$\left\{n\right\}$	$\left\{n\right\}$	$\{f,n\}$	$\left\{n\right\}$	$\left\{f\right\}$	$\varphi$
$\{f,l\}$	$\varphi$	$\{f,l\}$	$\{f,l,m\}$	$\{f,l\}$	$\{f,l,m\}$	$\varphi$
$\{f,m\}$	$\varphi$	$\{f,m\}$	$\{f,l,m\}$	$\{f,m\}$	$\{f,l,m\}$	$\varphi$
$\{f,n\}$	$\varphi$	$\{f,n\}$	$\{f,n\}$	$\{f,n\}$	$\{f,n\}$	$\varphi$
$\{l,m\}$	$\{l,m\}$	$\{l,m\}$	$\{f,l,m\}$	$\{l,m\}$	$\left\{f\right\}$	$\varphi$
$\{l,n\}$	$\left\{n\right\}$	$\{l,n\}$	$\Omega$	$\{l,n\}$	$\{f,l,m\}$	$\varphi$
$\{m,n\}$	$\left\{n\right\}$	$\{m,n\}$	$\Omega$	$\{m,n\}$	$\{f,l,m\}$	$\varphi$
$\{f,l,m\}$	$\{l,m\}$	$\{f,l,m\}$	$\{f,l,m\}$	$\{f,l,m\}$	$\left\{f\right\}$	$\varphi$
$\{f,l,n\}$	$\{f,n\}$	$\{f,l,n\}$	$\Omega$	$\{f,l,n\}$	$\{l,n\}$	$\varphi$
$\{f,m,n\}$	$\left\{n\right\}$	$\{f,m,n\}$	$\Omega$	$\{f,m,n\}$	$\{f,l,m\}$	$\varphi$
$\{l,m,n\}$	$\{l,m,n\}$	$\{l,m,n\}$	$\Omega$	$\{l,m,n\}$	$\left\{f\right\}$	$\varphi$
$\varphi$	$\varphi$	$\varphi$	$\varphi$	$\varphi$	$\varphi$	$\varphi$
$\Omega$	$\Omega$	$\Omega$	$\Omega$	$\Omega$	$\varphi$	$\varphi$

,

Table 2. Comparison between our approach and Dai’s.

H	$\underline{\mathit{AP}}\left(\mathit{H}\right)$	${\underline{\underline{\mathit{AP}}}}_{\mathbf{{\rm Y}}}^{\Delta }\left(\mathit{H}\right)$	$\overline{\mathit{AP}}\left(\mathit{H}\right)$	${\overline{\overline{\mathit{AP}}}}_{\mathbf{{\rm Y}}}^{\Delta }\left(\mathit{H}\right)$	$\mathit{BNDR}\left(\mathit{H}\right)$	$\underline{{\mathit{BND}}_{\mathbf{{\rm Y}}}^{\Delta }}\left(\mathit{H}\right)$
$\left\{f\right\}$	$\varphi$	$\varphi$	$\left\{f\right\}$	$\left\{f\right\}$	$\left\{f\right\}$	$\left\{f\right\}$
$\left\{l\right\}$	$\varphi$	$\left\{l\right\}$	$\{f,l,m\}$	$\{l,m\}$	$\{f,l,m\}$	$\left\{l\right\}$
$\left\{m\right\}$	$\varphi$	$\varphi$	$\{l,m\}$	$\left\{m\right\}$	$\{l,m\}$	$\left\{m\right\}$
$\left\{n\right\}$	$\left\{n\right\}$	$\left\{n\right\}$	$\{f,n\}$	$\left\{n\right\}$	$\left\{f\right\}$	$\varphi$
$\{f,l\}$	$\varphi$	$\{f,l\}$	$\{f,l,m\}$	$\{f,l,m\}$	$\{f,l,m\}$	$\left\{m\right\}$
$\{f,m\}$	$\varphi$	$\varphi$	$\{f,l,m\}$	$\{f,m\}$	$\{f,l,m\}$	$\{f,m\}$
$\{f,n\}$	$\varphi$	$\left\{n\right\}$	$\{f,n\}$	$\{f,n\}$	$\{f,n\}$	$\left\{f\right\}$
$\{l,m\}$	$\{l,m\}$	$\{l,m\}$	$\{f,l,m\}$	$\{f,l,m\}$	$\left\{f\right\}$	$\left\{f\right\}$
$\{l,n\}$	$\left\{n\right\}$	$\{l,n\}$	$\Omega$	$\Omega$	$\{f,l,m\}$	$\{f,m\}$
$\{m,n\}$	$\left\{n\right\}$	$\left\{n\right\}$	$\Omega$	$\{m,n\}$	$\{f,l,m\}$	$\left\{m\right\}$
$\{f,l,m\}$	$\{l,m\}$	$\{f,l,m\}$	$\{f,l,m\}$	$\{f,l,m\}$	$\left\{f\right\}$	$\varphi$
$\{f,l,n\}$	$\{f,n\}$	$\{f,l,n\}$	$\Omega$	$\Omega$	$\{l,m\}$	$\left\{m\right\}$
$\{f,m,n\}$	$\left\{n\right\}$	$\left\{n\right\}$	$\Omega$	$\{f,m,n\}$	$\{f,l,m\}$	$\{f,m\}$
$\{l,m,n\}$	$\{f,m,n\}$	$\{f,m,n\}$	$\Omega$	$\Omega$	$\left\{f\right\}$	$\left\{f\right\}$
$\varphi$	$\varphi$	$\varphi$	$\varphi$	$\varphi$	$\varphi$	$\varphi$
$\Omega$	$\Omega$	$\Omega$	$\Omega$	$\Omega$	$\varphi$	$\varphi$

and

Table 3. Comparison between our approach and Dai’s.

H	$\underline{\mathit{AP}}\left(\mathit{H}\right)$	${\stackrel{´}{{\underline{\mathit{AP}}}_{\mathbf{{\rm Y}}}}}^{\Delta }\left(\mathit{H}\right)$	$\overline{\mathit{AP}}\left(\mathit{H}\right)$	${\overline{{\acute{\mathit{AP}}}_{\mathbf{{\rm Y}}}}}^{\Delta }\left(\mathit{H}\right)$	$\mathit{BNDR}\left(\mathit{H}\right)$	$\stackrel{´}{{\underline{\mathit{BND}}}_{\mathbf{{\rm Y}}}^{\Delta }}\left(\mathit{H}\right)$
$\left\{f\right\}$	$\varphi$	$\varphi$	$\left\{f\right\}$	$\varphi$	$\left\{f\right\}$	$\varphi$
$\left\{l\right\}$	$\varphi$	$\Omega$	$\{f,l,m\}$	$\Omega$	$\{f,l,m\}$	$\varphi$
$\left\{m\right\}$	$\varphi$	$\varphi$	$\{l,m\}$	$\varphi$	$\{l,m\}$	$\varphi$
$\left\{n\right\}$	$\left\{n\right\}$	$\varphi$	$\{f,n\}$	$\varphi$	$\left\{f\right\}$	$\varphi$
$\{f,l\}$	$\varphi$	$\Omega$	$\{f,l,m\}$	$\Omega$	$\{f,l,m\}$	$\varphi$
$\{f,m\}$	$\varphi$	$\varphi$	$\{f,l,m\}$	$\varphi$	$\{f,l,m\}$	$\varphi$
$\{f,n\}$	$\varphi$	$\varphi$	$\{f,n\}$	$\varphi$	$\{f,n\}$	$\varphi$
$\{l,m\}$	$\{l,m\}$	$\Omega$	$\{f,l,m\}$	$\Omega$	$\left\{f\right\}$	$\varphi$
$\{l,n\}$	$\left\{n\right\}$	$\Omega$	$\Omega$	$\Omega$	$\{f,l,m\}$	$\varphi$
$\{m,n\}$	$\left\{n\right\}$	$\varphi$	$\Omega$	$\varphi$	$\{f,l,m\}$	$\varphi$
$\{f,l,m\}$	$\{l,m\}$	$\Omega$	$\{f,l,m\}$	$\Omega$	$\left\{f\right\}$	$\varphi$
$\{f,l,n\}$	$\{f,n\}$	$\Omega$	$\Omega$	$\Omega$	$\{l,m\}$	$\varphi$
$\{f,m,n\}$	$\left\{n\right\}$	$\varphi$	$\varphi$	$\varphi$	$\{f,l,m\}$	$\varphi$
$\{l,m,n\}$	$\{f,m,n\}$	$\Omega$	$\Omega$	$\Omega$	$\left\{f\right\}$	$\varphi$
$\varphi$	$\varphi$	$\varphi$	$\varphi$	$\varphi$	$\varphi$	$\varphi$
$\Omega$	$\Omega$	$\Omega$	$\Omega$	$\Omega$	$\varphi$	$\varphi$

show how the current approaches decrease the UPP APPs and increase the LOW APPs in order to decrease the BR. These strategies reduce them more effectively than those employed by Dai et al. [36], and they additionally show that the accuracy surpassed that which was also attained. It is important to remember that the current APPs coincide with those of Dai et al. [36] if the BIR is a similarity relation and the grill is minimal. As a result, our work improves and generalizes the work of Dai et al. [36]. Medical applications in Section 6 demonstrate the importance of using grills in current methodologies. It is a key factor in real-world decision-making issues. It is demonstrated that existing methods enable the medical team to successfully categorize patients in dengue infection stages. A BIR is employed in the application. The approaches of Dai et al. [36] are not applicable to the application. This is due to the fact that they are predicated on similarity relations, which limits the applicability of RS theory. This indicates that there are additional applications that are possible with the current approaches. Finally, Section 7 discusses the paper’s conclusion as well as suggestions for further study.

2. Preliminaries

Several key concepts and results that will be included in this study are reviewed in this section. In what follows, by a space $\Omega$ we shall mean a space $\Omega$ that carries topology ℑ. For $H⊑\Omega$, we shall adopt the usual notations $C\left(H\right)$ and $I\left(H\right)$ to denote the closure and interior of H in $\Omega$, respectively. Also, the power set of $\Omega$ will be written as $\wp (\Omega )$.

Definition 1. 

A grill [37] refers to a nonempty subcollection Δ of a space Ω with topology ℑ that satisfies the following conditions:

• $\varphi \notin \Delta$.
• $H\in \Delta$, $H⊑J⊑\Omega$ leads to $J\in \Delta$.
• if $H\bigsqcup J\in \Delta$ for $H,J⊑\Omega$, then $H\in \Delta$ or $J\in \Delta$.

Remark 1. 

(a) $\Delta =\{\Omega \}$ is the smallest grill in any space Δ with topology ℑ. (b) $\Delta =\wp (\Omega )\setminus \left\{\varphi \right\}$ is the largest grill for each topological space $(\Omega ,\Im )$.

In order to provide a different topological space for the grill that is finer than the space $\Omega$, which is represented by ${\Im }_{\Delta }$ on $\Omega$, the grill depends on two mappings, $\theta$ and $\eta$. This has already been covered in [29].

Definition 2 

([39]). Let $(\Omega ,\Im ,\Delta )$ be a grill topological space. Consider the operator $\theta :\wp (\Omega )\to \wp (\Omega )$ denoted by $\theta \left(H\right)$ for $H\in \wp (\Omega )$, which is defined as $\theta \left(H\right)=\{\omega \in \Omega :H\sqcap \aleph \in \Delta$ for all open sets ℵ containing $\omega \}$.

Definition 3 

([39]). Let $(\Omega ,\Im ,\Delta )$ be a grill topological space. Consider the operator $\eta :\wp (\Omega )\to \wp (\Omega )$ denoted by $\eta \left(H\right)$ for $H\in \wp (\Omega )$, which is defined as $\eta \left(H\right)=H\bigsqcup \theta \left(H\right)$. The map η is a Kuratowski closure operator and hence induces a topology ${\Im }_{\Delta }=\{\aleph ⊑\Omega :\eta (\Omega \setminus \aleph )=\Omega \setminus \aleph \}$, where $H⊑\Omega ,\eta \left(H\right)={\Im }_{\Delta }-C\left(H\right)$, and we have $\Im ⊑{\Im }_{\Delta }$.

Definition 4 

([1,2]). Suppose that ${\rm Y}\left(\omega \right)$ is the equivalence class that contains ω, and that Υ is an equivalence relation on a universe Ω. For any subset H of Ω, the first LOW $\underline{AP}\left(H\right)$ and UPP APP $\overline{AP}\left(H\right)$ are denoted by

$$\underline{AP}\left(H\right)=\{\omega \in \Omega :{\rm Y}\left(\omega \right)⊑H\}$$

$$\overline{AP}\left(H\right)=\{\omega \in \Omega :{\rm Y}\left(\omega \right)\sqcap H\ne \varphi \}$$

Several interesting characteristics are now demonstrated; these are enumerated below.

Let $H,H_1,H_2⊑\Omega :$

(Ł1)

$\underline{AP}\left(H\right)={\left[\overline{AP}\left(H^c\right)\right]}^c$.

(Ł2)

$\underline{AP}(\Omega )=\Omega$.

(Ł3)

$\underline{AP}(H_1\sqcap H_2)=\underline{AP}\left(H_1\right)\sqcap \underline{AP}\left(H_2\right)$.

(Ł4)

$\underline{AP}\left(H_1\right)\bigsqcup \underline{AP}\left(H_2\right)⊑\underline{AP}(H_1\sqcap H_2)$.

(Ł5)

$\underline{AP}\left(H_1\right)⊑\underline{AP}\left(H_2\right)$ whenever $H_1⊑H_2$.

(Ł6)

$\underline{AP}\left(H\right)⊑H$.

(Ł7)

$\underline{AP}\left(\varphi \right)=\varphi$.

(Ł8)

$\underline{AP}\left(\underline{AP}\left(H\right)\right)=\underline{AP}\left(H\right)$.

(Ł9)

$\underline{AP}\left(\overline{AP}\left(H\right)\right)⊒\overline{AP}\left(H\right)$.

(U1)

$\overline{AP}\left(H\right)={\left[\underline{AP}\left(H^c\right)\right]}^c$.

(U2)

$\overline{AP}(\Omega )=\Omega$.

(U3)

$\overline{AP}\left(\varphi \right)=\varphi$.

(U4)

$\overline{AP}\left(H\right)⊒H$.

(U5)

$\overline{AP}(H_1\bigsqcup H_2)=\overline{AP}\left(H_1\right)\bigsqcup \overline{AP}\left(H_2\right)$.

(U6)

$\overline{AP}(H_1\sqcap \left(H_2\right)⊑\overline{AP}\left(H_1\right)\sqcap \overline{AP}\left(H_2\right)$.

(U7)

$H_1⊑H_2\Rightarrow \overline{AP}\left(H_1\right)⊑\overline{AP}\left(H_2\right)$.

(U8)

$\overline{AP}\left(\overline{AP}\left(H\right)\right)=\overline{AP}\left(H\right)$.

(U9)

$\overline{AP}\left(\underline{AP}\left(H\right)\right)⊑\underline{AP}\left(H\right)$.

Definition 5 

([14]). Let Υ be an arbitrary relation on a universe Ω. The maximal right neighborhood of an $\omega \in \Omega$ is defined as follows.

$$N_{{\rm Y}}\left(\omega \right)={\bigsqcup }_{\omega \in M_{{\rm Y}}\left(\xi \right)}{M}_{{\rm Y}}\left(\omega \right),\mathit{where} M_{\xi }\left(\omega \right)=\{\xi \in \Omega :\omega {\rm Y}\xi \}$$

Definition 6 

([1,2]). For a universe Ω, let Υ be a similarity relation. For any subset $H⊑\Omega$, the first LOW and UPP APPs, as well as the accuracy and roughness, of H corresponding to Υ are, respectively, defined by

$${\underline{AP}}_{{\rm Y}}\left(H\right)=\{\omega \in \Omega :N_{{\rm Y}}\left(\omega \right)⊑H\}.$$

$${\overline{AP}}_{{\rm Y}}\left(H\right)=\{\omega \in \Omega :N_{{\rm Y}}\left(\omega \right)\sqcap H\ne \varphi \}.$$

$$A{c}_{{\rm Y}}\left(H\right)=\left|{\displaystyle \frac{{\underline{AP}}_{{\rm Y}}\left(H\right)}{{\overline{AP}}_{{\rm Y}}\left(H\right)}}\right|, {\overline{AP}}_{{\rm Y}}\left(H\right)\ne \varphi$$

$${\acute{Rness}}_{{\rm Y}}\left(H\right)=1-A{c}_{{\rm Y}}\left(H\right).$$

Definition 7 

([1,2]). Consider the similarity relation Υ on a universe Ω. For any subset $H⊑\Omega$, the second LOW and UPP APPs, as well as the accuracy and roughness, of H corresponding to Υ are, respectively, defined by

$$\acute{{\underline{AP}}_{{\rm Y}}}\left(H\right)=\bigsqcup \{N_{{\rm Y}}\left(\omega \right):N_{{\rm Y}}\left(\omega \right)⊑H\}$$

$$\overline{{\acute{AP}}_{{\rm Y}}}\left(H\right)={\left[\acute{{\underline{AP}}_{{\rm Y}}}\left(H^c\right)\right]}^c$$

$${\acute{Ac}}_{{\rm Y}}\left(H\right)=|{\displaystyle \frac{\acute{{\underline{AP}}_{{\rm Y}}}\left(H\right)}{\overline{{\acute{AP}}_{{\rm Y}}}\left(H\right)}}|, \overline{{\acute{AP}}_{{\rm Y}}}\left(H\right)\ne \varphi$$

$${\acute{Rness}}_{{\rm Y}}\left(H\right)=1-{\acute{Ac}}_{{\rm Y}}\left(H\right).$$

Definition 8 

([1,2]). For a universe Ω, let Υ be a similarity relation. The third LOW and UPP APPs, and the accuracy and roughness, of H corresponding to Υ for every subset $H⊑\Omega$ are determined by

$${\overline{\acute{\stackrel{´}{AP}}}}_{{\rm Y}}\left(H\right)=\bigsqcup \{N_{{\rm Y}}\left(\omega \right):N_{{\rm Y}}\left(\omega \right)\sqcap H\ne \varphi \}$$

$${\underline{\acute{\stackrel{´}{AP}}}}_{{\rm Y}}\left(H\right)={({\overline{\acute{\stackrel{´}{AP}}}}_{{\rm Y}}\left(H^c\right))}^c$$

$${\acute{\stackrel{´}{Ac}}}_{{\rm Y}}\left(H\right)=|{\displaystyle \frac{{\underline{\acute{\stackrel{´}{AP}}}}_{{\rm Y}}\left(H\right)}{{\overline{\acute{\stackrel{´}{AP}}}}_{{\rm Y}}\left(H\right)}}|, {\overline{\acute{\stackrel{´}{AP}}}}_{{\rm Y}}\left(H\right)\ne \varphi$$

$${\acute{\stackrel{´}{Rness}}}_{{\rm Y}}\left(H\right)=1-{\acute{\stackrel{´}{Ac}}}_{{\rm Y}}\left(H\right).$$

3. First Method for Obtaining Extended Rough Sets Using Grill

Generalized rough APPs for the first type are discussed in this section. These approximations are explored and compared to prior ones, demonstrating their generality.

Definition 9. 

Let Δ be a grill and Υ be a BIR on the universe Ω. For any subset H of Ω, the first kind of generalized LOW ${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)$, UPP APP ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)$, accuracy $A{c}_{{\rm Y}}^{\Delta }\left(H\right)$, and roughness $Rnes{s}_{{\rm Y}}^{\Delta }\left(H\right)$ of H using a grill and according to Υ are defined as follows:

$${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)=\{\omega \in \Omega :N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin \Delta \}$$

$${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)=\{\omega \in \Omega :N_{{\rm Y}}\left(\omega \right)\sqcap H\in \Delta \}$$

$$A{c}_{{\rm Y}}^{\Delta }\left(H\right)=|{\displaystyle \frac{{\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)}{{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)}}|, {\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)\ne \varphi$$

$$Rnes{s}_{{\rm Y}}^{\Delta }\left(H\right)=1-A{c}_{{\rm Y}}^{\Delta }\left(H\right).$$

Proposition 1. 

Assume Υ is a BIR on the universe Ω and Δ is a grill; then the following condition holds:

(i) 

If $\Delta =\{\Omega \}$ then ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)=\varphi$,

(ii) 

If $\Delta =\wp (\Omega )\setminus \left\{\varphi \right\}$ then ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)={\overline{AP}}_{{\rm Y}}\left(H\right)$.

Proof. 

(i) If $\Delta =\{\Omega \}$, then $N_{{\rm Y}}\left(\omega \right)\sqcap H\ne \Omega$, and ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)=\{\omega \in \Omega :N_{{\rm Y}}\left(\omega \right)\sqcap H\notin \Delta \}$. So, ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)=\varphi$. (ii) If $\Delta =\wp (\Omega )\setminus \left\{\varphi \right\}$ then ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)={\overline{AP}}_{{\rm Y}}\left(H\right)$. □

Proposition 2. 

If Υ be a BIR on a universe Ω, $H_1,H_2⊑\Omega$, and $\Delta =\wp (\Omega )\setminus \left\{\varphi \right\}$. Then, the properties listed below are true:

(i) 

${\overline{AP}}_{{\rm Y}}^{\Delta }\left(\varphi \right)=\varphi$.

(ii) 

$H_1⊑H_2\Rightarrow {\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)⊑{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)$.

(iii) 

${\overline{AP}}_{{\rm Y}}^{\Delta }(H_1\bigsqcup H_2)={\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)\bigsqcup {\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)$.

(iv) 

${\overline{AP}}_{{\rm Y}}^{\Delta }(H_1\sqcap H_2)⊑{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)\sqcap {\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)$.

(v) 

${\overline{AP}}_{{\rm Y}}^{\Delta }\left({\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)\right)⊒{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)$.

(vi) 

${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)={\left[{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H^c\right)\right]}^c$.

(vii) 

If $H\notin \Delta$, then ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)=\varphi$.

Proof. 

(i) It is obvious from Definition 9. (ii) Let $\omega \in {\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)$, then $N_{{\rm Y}}\left(\omega \right)\sqcap H_1\in \Delta$. Since $N_{{\rm Y}}\left(\omega \right)\sqcap H_1⊑N_{{\rm Y}}\left(\omega \right)\sqcap H_2$, $\omega \in {\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)$. (iii) ${\overline{AP}}_{{\rm Y}}^{\Delta }(H_1\bigsqcup H_2)=\{\omega \in \Omega :N_{{\rm Y}}\left(\omega \right)\sqcap (H_1\bigsqcup H_2)\in \Delta \}$$=\{\omega :(N_{{\rm Y}}\left(\omega \right)\sqcap H_1)\bigsqcup (N_{{\rm Y}}\left(\omega \right)\sqcap H_2)$$\in \Delta =\{\omega :N_{{\rm Y}}\left(\omega \right)\sqcap H_1\in \Delta \}\bigsqcup \{\omega :N_{{\rm Y}}\left(\omega \right)\sqcap H_2\in \Delta \}$$={\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)\bigsqcup {\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)$. (iv) Let $\omega \in {\overline{AP}}_{{\rm Y}}^{\Delta }(H_1\sqcap H_2)$, which leads to $N_{{\rm Y}}\left(\omega \right)\sqcap (H_1\sqcap H_2)\in \Delta$, $(N_{{\rm Y}}\left(\omega \right)\sqcap H_1)\sqcap (N_{{\rm Y}}\left(\omega \right)\sqcap H_2)\in \Delta$, and $\omega \in {\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)\sqcap {\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)$. (v) Let $\omega \in {\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)$. Then $N_{{\rm Y}}\left(\omega \right)\sqcap H\ne \varphi$. So, there exists ${\omega }_1\in N_{{\rm Y}}\left(\omega \right)\sqcap H$—i.e., $N_{{\rm Y}}\left({\omega }_1\right)⊑N_{{\rm Y}}\left(\omega \right)$, and $N_{{\rm Y}}\left({\omega }_1\right)\sqcap H\in \Delta$. So, $\omega \in {\overline{AP}}_{{\rm Y}}^{\Delta }\left({\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)\right)$. (vi) ${\left[{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H^c\right)\right]}^c={\{\omega \in \Omega :N_{{\rm Y}}\left(\omega \right)\sqcap H^c\in \Delta \}}^c=\{\omega \in \Omega :N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin \Delta \}={\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)$. (vii) If $H\notin \Delta$, then ${\overline{\varphi }}_{{\rm Y}}^{\Delta }\left(H\right)=\varphi$. So, $N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin \Delta$ and ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)=\varphi$. □

Remark 2. 

The example that follows demonstrates that

(1) 

In general, $H⊈{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)$.

(2) 

The converse of (ii) in Proposition 2 does not always hold true.

Example 1. 

Let $\Omega =\{f,l,m,n\}$, $\Delta =\left\{\right\{l\},\{f,l\},\{l,m\},\{l,n\},\{f,l,m\},$$\{f,l,n\},\{l,m,n\},\Omega \}$, and ${\rm Y}=\left\{\right(f,f),(l,l),(m,m),(n,n),(f,l),(f,m),(l,m),(l,n),(m,n),(m,f\left)\right\}$ be relation on Ω. Then, $N_{{\rm Y}}\left(f\right)=\Omega$, $N_{{\rm Y}}\left(l\right)=\Omega$, $N_{{\rm Y}}\left(m\right)=\Omega$, and $N_{{\rm Y}}\left(n\right)=\Omega$. Consider

(1) 

If $H=\left\{f\right\}$ then ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)=\varphi$—i.e., $H⊈{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)$.

(2) 

If $H_1=\left\{f\right\}$ and $H_2=\left\{l\right\}$, then ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)=\varphi$, ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)=\Omega$. Therefore, ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)⊑{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)$, but $H_1⊈H_2$.

(3) 

If $H_1=\left\{f\right\}$ and $H_2=\left\{l\right\}$, then ${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)=\varphi$, ${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)=\Omega ,$ and $A{c}_{{\rm Y}}^{\Delta }\left(H_2\right)=1$.

Example 2. 

Let $\Omega =\{f,l,m,n\}$, $\Delta =2^{\Omega }\setminus \varphi$, and ${\rm Y}=\left\{\right(f,l),(l,l),(f,m),(l,m),(l,n),(m,f),(m,n\left)\right\}$ be a relation on Ω. Then, $N_{{\rm Y}}\left(f\right)=\{f,n\}$, $N_{{\rm Y}}\left(l\right)=\{l,m\}$, and $N_{{\rm Y}}\left(m\right)=\{l,m,n\}$, $N_{{\rm Y}}\left(n\right)=\{f,m,n\}$. Consider

(1) 

If $H_1=\{l,n\}$, $H_2=\{m,n\}$, then ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)=\Omega$, ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)=\Omega$, and ${\overline{AP}}_{{\rm Y}}^{\Delta }(H_1\sqcap H_2)=\{f,m,n\}$—i.e., ${\overline{AP}}_{{\rm Y}}^{\Delta }(H_1\sqcap H_2)⊑{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)\sqcap {\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)$.

(2) 

If $H=\left\{l\right\}$, then ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)=\{l,m\}$ and ${\overline{AP}}_{{\rm Y}}^{\Delta }\left({\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)\right)=\{l,m,n\}$.

Remark 3. 

Example 2 illustrates that the equality is not true in general for Items (iv) and (v).

Theorem 1. 

Given a universe Ω, $H_1,H_2⊑\Omega$, Υ as a BIR, and ${\Delta }_1$ and ${\Delta }_2$ as two grills on Ω, we obtain

(i) 

${\Delta }_1⊑{\Delta }_2\Rightarrow {\overline{AP}}_{{\rm Y}}^{{\Delta }_1}⊑{\overline{AP}}_{{\rm Y}}^{{\Delta }_2}$, and ${\underline{AP}}_{{\rm Y}}^{{\Delta }_2}⊑{\underline{AP}}_{{\rm Y}}^{{\Delta }_1}$;

(ii) 

${\underline{AP}}_{{\rm Y}}^{{\Delta }_1\sqcap {\Delta }_2}\left(H\right)={\underline{AP}}_{{\rm Y}}^{{\Delta }_1}\left(H\right)\bigsqcup {\underline{AP}}_{{\rm Y}}^{{\Delta }_2}\left(H\right)$;

(iii) 

${\overline{AP}}_{{\rm Y}}^{{\Delta }_1\sqcap {\Delta }_2}\left(H\right)={\overline{AP}}_{{\rm Y}}^{{\Delta }_1}\left(H\right)\sqcap {\overline{AP}}_{{\rm Y}}^{{\Delta }_2}\left(H\right)$.

Proof. 

(i) Let $\omega \in {\overline{AP}}_{{\rm Y}}^{{\Delta }_1}$; hence $N_{{\rm Y}}\left(\omega \right)\sqcap H\in {\Delta }_1$. Since ${\Delta }_1⊑{\Delta }_2$, $N_{{\rm Y}}\left(\omega \right)\sqcap H\in {\Delta }_2$ and $\omega \in {\overline{AP}}_{{\rm Y}}^{{\Delta }_2}$. Let $\omega \in {\underline{AP}}_{{\rm Y}}^{{\Delta }_2}$, and hence $N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin {\Delta }_2$. So, $N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin {\Delta }_1$, and $\omega \in {\underline{AP}}_{{\rm Y}}^{{\Delta }_1}$—i.e., ${\underline{AP}}_{{\rm Y}}^{{\Delta }_2}⊑{\underline{AP}}_{{\rm Y}}^{{\Delta }_1}$. (ii) ${\underline{AP}}_{{\rm Y}}^{{\Delta }_1\sqcap {\Delta }_2}\left(H\right)=\{\omega \in \Omega :N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin {\Delta }_1\sqcap {\Delta }_2\}$$=\{\omega \in \Omega :N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin {\Delta }_1\}\bigsqcup \{\omega \in \Omega :N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin {\Delta }_2\}$$={\underline{AP}}_{{\rm Y}}^{{\Delta }_1}\left(H\right)\bigsqcup {\underline{AP}}_{{\rm Y}}^{{\Delta }_2}\left(H\right)$. (iii) ${\overline{AP}}_{{\rm Y}}^{{\Delta }_1\sqcap {\Delta }_2}\left(H\right)=\{\omega \in \Omega :N_{{\rm Y}}\left(\omega \right)\sqcap H\in {\Delta }_1\sqcap {\Delta }_2\}$$=\{\omega \in \Omega :N_{{\rm Y}}\left(\omega \right)\sqcap H\in {\Delta }_1\}\sqcap \{\omega \in \Omega :N_{{\rm Y}}\left(\omega \right)\sqcap H\in {\Delta }_2\}$$={\overline{AP}}_{{\rm Y}}^{{\Delta }_1}\left(H\right)\sqcap {\overline{AP}}_{{\rm Y}}^{{\Delta }_2}\left(H\right)$. □

Remark 4. 

The example that follows shows that the converse of (i) in Theorem 1 does not always hold true.

Example 3. 

Let $\Omega =\{f,l,m,n\}$, ${\Delta }_1=\{\{f,n\},\{f,l,n\},\{f,m,n\},\Omega \}\}$, ${\Delta }_2=\{\left\{l\right\},\{f,l\},\{l,m\},\{l,n\},\{f,l,m\},$$\{f,l,n\},\{l,m,n\},\Omega \}$, and ${\rm Y}=\left\{\right(f,f),(l,l),(m,m),(n,n),(f,l),(f,m),(l,m),(l,n),(m,n),(m,f\left)\right\}$ be a relation on Ω. Then, $N_{{\rm Y}}\left({\omega }_1\right)=\Omega$, $N_{{\rm Y}}\left(l\right)=\Omega$, and $N_{{\rm Y}}\left(m\right)=\Omega$, $N_{{\rm Y}}\left(n\right)=\Omega$. Consider the following: if $H=\{l,m\}$, then ${\overline{AP}}_{{\rm Y}}^{{\Delta }_1}\left(H\right)=\varphi$, ${\overline{AP}}_{{\rm Y}}^{{\Delta }_2}\left(H\right)=\Omega$, but ${\Delta }_1⊈{\Delta }_2$.

Theorem 2. 

Let Δ be a maximal grill on a universe Ω and Υ a BIR; then the following properties hold:

(i) 

${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)⊑H$

(ii) 

If $H_1⊑H_2\Rightarrow {\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)⊑{\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)$;

(iii) 

${\underline{AP}}_{{\rm Y}}^{\Delta }(H_1\sqcap H_2)⊑{\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)\bigsqcup {\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)$;

(iv) 

${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)\bigsqcup {\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)⊑{\underline{AP}}_{{\rm Y}}^{\Delta }(H_1\bigsqcup H_2)$;

(v) 

${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)⊒{\underline{AP}}_{{\rm Y}}^{\Delta }\left({\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)\right)$;

(vi) 

${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)={\left[{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H^c\right)\right]}^c$;

(vii) 

If $H^c\notin \Delta \Rightarrow {\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)=\Omega$.

Proof. 

(i) From ${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)=\{\omega \in \Omega :N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin \Delta \}$, we have ${\underline{AP}}_{{\rm Y}}^{\Delta }(\Omega )=\{\omega \in \Omega :N_{{\rm Y}}\left(\omega \right)\sqcap \varphi =\varphi \notin \Delta \}=\Omega$. (ii) If $H_1⊑H_2$ and $\omega \in {\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)$, then $N_{{\rm Y}}\left(\omega \right)\sqcap H_1^c\notin \Delta$ and $N_{{\rm Y}}\left(\omega \right)\sqcap H_2^c\notin \Delta$—i.e., $\omega \in {\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)$. (iii) Let $\omega \in {\underline{AP}}_{{\rm Y}}^{\Delta }(H_1\sqcap H_2)$. Then, $N_{{\rm Y}}\left(\omega \right)\sqcap {(H_1\sqcap H_2)}^c\notin \Delta$—i.e., $\omega \in {\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)\bigsqcup {\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)$. (iv) Let $\omega \in {\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)\bigsqcup {\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)$. Then, $\omega \in \{N_{{\rm Y}}\left(\omega \right)\sqcap H_1^c\notin \Delta \}\bigsqcup \{N_{{\rm Y}}\left(\omega \right)\sqcap H_2^c\notin \Delta \}$, and $\omega \in {\underline{AP}}_{{\rm Y}}^{\Delta }(H_1\bigsqcup H_2)$. The proofs for (v), (vi), and (vii) are simple. □

Remark 5. 

The example that follows shows that the converse of (ii) in Theorem 2 does not always hold true.

Example 4. 

Let $\Omega =\{f,l,m,n\}$, $\Delta =\left\{\right\{l\},\{f,l\},\{l,m\},\{l,n\},\{f,l,m\},$$\{f,l,n\},\{l,m,n\},\Omega \}$ and ${\rm Y}=\left\{\right(f,f),(l,l),(m,m),(n,n),(f,l),(f,m),(l,m),(l,n),(m,n),(m,f\left)\right\}$ be a relation on Ω. Then, $N_{{\rm Y}}\left(f\right)=\Omega$, $N_{{\rm Y}}\left(l\right)=\Omega$, $N_{{\rm Y}}\left(m\right)=\Omega$, and $N_{{\rm Y}}\left(n\right)=\Omega$. Consider

(1) 

If $H=\left\{f\right\}$, then ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)=\varphi$, i.e., $H⊈{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)$.

(2) 

If $H_1=\left\{f\right\}$ and $H_2=\left\{l\right\}$, then ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)=\varphi$, ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)=\Omega$. Therefore, ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)⊑{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)$, but $H_1⊈H_2$.

Example 5. 

Continued in Example 2. Consider the following:

(1) 

If $H_1=\{l,n\}$, $H_2=\{m,n\}$, then ${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)=\varphi$, ${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)=\varphi$, and ${\underline{AP}}_{{\rm Y}}^{\Delta }(H_1\bigsqcup H_2)=\{l,m\}$—i.e., ${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)\bigsqcup {\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)⊑{\underline{AP}}_{{\rm Y}}^{\Delta }(H_1\bigsqcup H_2)$

(2) 

If $H=\{n,f\}$, then ${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)=\left\{f\right\}$ and ${\underline{AP}}_{{\rm Y}}^{\Delta }\left({\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)\right)=\varphi$; this shows that ${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)⊒{\underline{AP}}_{{\rm Y}}^{\Delta }\left({\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)\right)$.

Theorem 3. 

If Υ is a BIR on a universe Ω and ${\Delta }_1$ and ${\Delta }_2$ are two grills on Ω, and if ${\Delta }_1⊑{\Delta }_2$, then the following condition holds:

(i) 

$Boun{d}_{{\rm Y}}^{{\Delta }_2}\left(H\right)⊑Boun{d}_{{\rm Y}}^{{\Delta }_1}\left(H\right)$.

(ii) 

$A{c}_{{\rm Y}}^{{\Delta }_1}\left(H\right)\le A{c}_{{\rm Y}}^{{\Delta }_2}\left(H\right)$.

Proof. 

(i) Let ${\omega }_1\in Boun{d}_{{\rm Y}}^{{\Delta }_2}\left(H\right)$; then ${\omega }_1\in {\overline{AP}}_{{\rm Y}}^{{\Delta }_2}\left(H\right)-{\underline{AP}}_{{\rm Y}}^{{\Delta }_2}\left(H\right)$. So, ${\omega }_1\in {\overline{AP}}_{{\rm Y}}^{{\Delta }_2}\left(H\right)$ and ${\omega }_1\in {\left({\underline{AP}}_{{\rm Y}}^{{\Delta }_2}\left(H\right)\right)}^c$. Then, ${\omega }_1\in {\overline{AP}}_{{\rm Y}}^{{\Delta }_1}\left(H\right)$ and ${\omega }_1\in {\left({\underline{AP}}_{{\rm Y}}^{{\Delta }_1}\left(H\right)\right)}^c$ according to (i) of Theorem 1. It follows that ${\omega }_1\in Boun{d}_{{\rm Y}}^{{\Delta }_1}\left(H\right)$. Therefore, $Boun{d}_{{\rm Y}}^{{\Delta }_2}\left(H\right)⊑Boun{d}_{{\rm Y}}^{{\Delta }_2}\left(H\right)$. (ii) $A{c}_{{\rm Y}}^{{\Delta }_1}\left(H\right)=|{\displaystyle \frac{{\underline{AP}}_{{\rm Y}}^{{\Delta }_1}\left(H\right)}{{\overline{AP}}_{{\rm Y}}^{{\Delta }_1}\left(H\right)}}|\le |{\displaystyle \frac{{\underline{AP}}_{{\rm Y}}^{{\Delta }_2}\left(H\right)}{{\overline{AP}}_{{\rm Y}}^{{\Delta }_2}\left(H\right)}}|=A{c}_{{\rm Y}}^{{\Delta }_2}\left(H\right)$. □

Remark 6. 

The converse of (i) in Theorem 3 is not always true, as demonstrated in Example 5. Considering $H=\{f,n\}$, we find that ${\Delta }_1⊈{\Delta }_2$, but $Bou{n}_{{\rm Y}}^{{\Delta }_2}\left(H\right)=\varphi ⊑\Omega =Boun{d}_{{\rm Y}}^{{\Delta }_1}\left(H\right)$.

4. Second Method for Obtaining Extended Rough Sets Using Grill

This section’s goal is to suggest a second kind of rough approximation extension. This approximation’s properties are proposed. Furthermore, with the aid of some clarifying examples, certain connections between these estimates and the first category of approximations in the preceding section are revealed. We provide comparisons with the approximations in [36].

Definition 10. 

Assume Υ is a BIR on the universe Ω, and let Δ be a grill. For any subset H of Ω, the second kind of generalized LOW $\underline{\underline{AP}}\left(H\right)$, UPP APP $\overline{\overline{AP}}\left(H\right)$, BRs, accuracy, and roughness of H using a grill and according to Υ are defined by

$${\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)=\{\omega \in H:N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin \Delta \}$$

$${\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)=H\bigsqcup {\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)$$

$${\underline{Ac}}_{{\rm Y}}^{\Delta }\left(H\right)=|{\displaystyle \frac{{\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)}{{\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)}}|, {\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)\ne \varphi$$

$${\underline{Rness}}_{{\rm Y}}^{\Delta }\left(H\right)=1-{\underline{Ac}}_{{\rm Y}}^{\Delta }\left(H\right).$$

Proposition 3. 

Assume Υ is a BIR on the universe Ω, and let Δ be a grill, then ${\overline{\overline{AP}}}_{{\rm Y}}\left(H\right)⊑{\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)$

Proof. 

It is obvious. □

Proposition 4. 

Assume Υ is a BIR on the universe Ω, and let Δ be a grill; then the following properties hold:

(i) 

${\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }(\Omega )=\Omega$.

(ii) 

$H⊑{\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)$.

(iii) 

If $H_1⊑H_2\Rightarrow {\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H_1\right)⊑{\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H_2\right)$.

(iv) 

${\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }(H_1\bigsqcup H_2)={\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H_1\right)\bigsqcup {\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H_2\right)$.

(v) 

${\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }(H_1\sqcap H_2)⊑{\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H_1\right)\sqcap {\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H_2\right)$.

(vi) 

${\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left({\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)\right)⊒{\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)$.

(vii) 

${\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)={\left[{\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H^c\right)\right]}^c$.

(viii) 

$H\notin \Delta \Rightarrow H⊑{\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)$.

(ix) 

${\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }(\Omega )=\Omega$.

(x) 

${\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)⊑H$.

(xi) 

$H_1⊑H_2\Rightarrow {\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H_1\right)⊑{\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H_2\right)$.

(xii) 

${\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H_1\right)\bigsqcup {\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H_2\right)⊑{\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }(H_1\bigsqcup H_2)$.

(xiii) 

${\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }(H_1\sqcap H_2)={\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H_1\right)\sqcap {\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H_2\right)$.

(xiv) 

${\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left({\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)\right)={\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)$.

(xv) 

${\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)={\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }{\left[\left(H^c\right)\right]}^c$.

(xvi) 

If $H^c\notin \Delta$ then, ${\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)=H$.

Proof. 

In line with Proposition 2. □

Example 6. 

Continued in Example 2. Consider the following:

(1) 

If $H_1=\left\{l\right\}$, $H_2=\{m,n\}$, then ${\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H_1\right)=\left\{l\right\}$, ${\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H_2\right)=\left\{n\right\}$, and ${\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }(H_1\bigsqcup H_2)=\{l,m,n\}$—i.e., ${\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H_1\right)\bigsqcup {\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H_2\right)⊑{\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }(H_1\bigsqcup H_2)$.

(2) 

$H_1=\{l,n\}$, $H_2=\{f,m\}$; then ${\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }(H_1\sqcap H_2)=\varphi$ and ${\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H_1\right)\sqcap {\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H_2\right)=\Omega$, and then we have ${\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }(H_1\sqcap H_2)⊑{\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H_1\right)\sqcap {\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H_2\right)$.

Theorem 4. 

Suppose Υ is a BIR on a universe Ω and ${\Delta }_1$ and ${\Delta }_2$ are two grills on Ω. If ${\Delta }_1⊑{\Delta }_2$, then ${\underline{boundary}}_{{\rm Y}}^{{\Delta }_1}\left(H\right)⊑{\underline{boundary}}_{{\rm Y}}^{{\Delta }_2}\left(H\right)$, where ${\underline{boundary}}_{{\rm Y}}^{\Delta }\left(H\right)={\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)-{\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)$.

Proof. 

Let $\omega \in {\underline{boundary}}_{{\rm Y}}^{{\Delta }_1}\left(H\right)$. Then, $\omega \in {\overline{\overline{AP}}}_{{\rm Y}}^{{\Delta }_1}\left(H\right)$, and $\omega \in ({\underline{\underline{AP}}}_{{\rm Y}}^{{\Delta }_1}{\left(H\right)}^c$. So, $\omega \in {\overline{\overline{AP}}}_{{\rm Y}}^{{\Delta }_2}\left(H\right)$, and $\omega \in {\left({\underline{\underline{AP}}}_{{\rm Y}}^{{\Delta }_2}\left(H\right)\right)}^c$. Hence, $\omega \in {\underline{boundary}}_{{\rm Y}}^{{\Delta }_2}\left(H\right)$. □

In contrast to Dai’s technique [36], we demonstrate in Example 7 that the boundary region is reduced by the maximal rough sets utilizing a grill.

Example 7. 

Let $\Omega =\{f,l,m,n\},\Delta =\{\Omega \}$, and ${\rm Y}=\left\{\right(f,f),(l,l),(m,m),(n,n),(f,l),(f,n),(l,m),(m,l\left)\right\}$ be a relation on Ω, as given in Table 1.

Example 8. 

Let $\Omega =\{f,l,m,n\}$, $\Delta =\left\{\right\{l\},\{f,l\},\{l,m\},\{l,n\},\{f,l,m\},\{f,l,n\},\{\Omega \left\}\right\}$, and ${\rm Y}=\left\{\right(f,f),(l,l),(m,m),(n,n),(f,l),(f,n),(l,m),(m,l\left)\right\}$ be a relation on Ω. As in Table 2 with a different grill, our method is the best compared to Dai’s method.

5. Third Method for Obtaining Extended Rough Sets Using Grill

This section’s goal is to define the generalized rough approximation’s third kind. We create and establish the fundamental properties of these approximations. Additionally, there are examples of how these approximations compare to the approximations in Section 3, Section 4 and Section 5. The connections between these approximations and the approximations in [36] are shown at the conclusion of this section.

Definition 11. 

Let Υ be a BIR on a universe Ω. For any subset $H⊑\Omega$, the third LOW and UPP APPs, accuracy, and roughness of H corresponding to Υ are, respectively, defined by

$${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)=\bigsqcup \{N_{{\rm Y}}\left(\omega \right):N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin \Delta \}$$

$${\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)={\left[{\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H^c\right)\right]}^c$$

$${\acute{Ac}}_{{\rm Y}}^{\Delta }\left(H\right)=|{\displaystyle \frac{{\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)}{{\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)}}|, {\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)\ne \varphi$$

$${\acute{Rness}}_{{\rm Y}}^{\Delta }\left(H\right)=1-{\acute{Ac}}_{{\rm Y}}^{\Delta }\left(H\right).$$

Proposition 5. 

Given a universe Ω, $H_1,H_2⊑\Omega$, Υ as a BIR, and ${\Delta }_1$ and ${\Delta }_2$ as two grills on Ω, we obtain

(i) 

${\Delta }_1⊑{\Delta }_2\Rightarrow {\overline{{\acute{AP}}_{{\rm Y}}}}^{{\Delta }_1}\left(H\right)⊑{\overline{{\acute{AP}}_{{\rm Y}}}}^{{\Delta }_2}\left(H\right)$, and ${\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_2}\left(H\right)⊑{\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_1}\left(H\right)$;

(ii) 

${\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_1\sqcap {\Delta }_2}\left(H\right)={\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_1}\left(H\right)\bigsqcup {\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_2}\left(H\right)$;

(iii) 

${\overline{{\acute{AP}}_{{\rm Y}}}}^{{\Delta }_1\sqcap {\Delta }_2}\left(H\right)={\overline{{\acute{AP}}_{{\rm Y}}}}^{{\Delta }_1}\left(H\right)\sqcap {\overline{{\acute{AP}}_{{\rm Y}}}}^{{\Delta }_2}\left(H\right)$.

Proof. 

(i) Let $\omega \in {\overline{{\acute{AP}}_{{\rm Y}}}}^{{\Delta }_1}\left(H\right)$; hence, $\omega \in {\left[{\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_1}\left(H^c\right)\right]}^c$, and $\omega \in \left[{\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_2}\left(H^c\right)\right]$. So, $\omega \in {\overline{{\acute{AP}}_{{\rm Y}}}}^{{\Delta }_2}\left(H\right)$. Let $\omega \in {\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_2}\left(H\right)$, and hence $N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin {\Delta }_2$. So, $N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin {\Delta }_1$, and $\omega \in {\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_1}\left(H\right)$—i.e., ${\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_2}\left(H\right)⊑{\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_1}\left(H\right)$. (ii) ${\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_1\sqcap {\Delta }_2}\left(H\right)=\bigsqcup \{N_{{\rm Y}}\left(\omega \right):N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin {\Delta }_1\sqcap {\Delta }_2\}=\bigsqcup \{N_{{\rm Y}}\left(\omega \right):N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin {\Delta }_1\}\bigsqcup (\bigsqcup \{N_{{\rm Y}}\left(\omega \right):N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin {\Delta }_2\})={\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_1}\left(H\right)\bigsqcup {\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_2}\left(H\right)$. (iii) ${\overline{{\acute{AP}}_{{\rm Y}}}}^{{\Delta }_1\sqcap {\Delta }_2}\left(H\right)={\left[{\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_1\sqcap {\Delta }_2}\left(H^c\right)\right]}^c={[{\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_1}\left(H^c\right)\bigsqcup {\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_2}\left(H^c\right)]}^c={\left[{\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_1}\left(H^c\right)\right]}^c\sqcap {\left[{\acute{{\underline{AP}}_{{\rm Y}}}}^{{\Delta }_2}\left(H^c\right)\right]}^c{]}^c={\overline{{\acute{AP}}_{{\rm Y}}}}^{{\Delta }_1}\left(H\right)\sqcap {\overline{{\acute{AP}}_{{\rm Y}}}}^{{\Delta }_2}\left(H\right)$. □

Remark 7. 

The example that follows shows that in Proposition 5, the opposite of (i) is not always true.

Example 9. 

Let $\Omega =\{f,l,m,n\}$, ${\Delta }_1=\{\{f,n\},\{f,l,n\},\{f,m,n\},\Omega \}$, ${\Delta }_2=\{\left\{l\right\},\{f,l\},\{l,m\},\{l,n\},\{f,l,m\},\{f,l,n\},\{l,m,n\},\Omega \}$, and ${\rm Y}=\left\{\right(f,f),(l,l),(m,m),(n,n),(f,l),(f,m),(l,m),(l,n),(m,n),(m,f\left)\right\}$ be a relation on Ω. Then, $N_{{\rm Y}}\left(f\right)=\Omega$, $N_{{\rm Y}}\left(l\right)=\Omega$, and $N_{{\rm Y}}\left(m\right)=\Omega$, $N_{{\rm Y}}\left(n\right)=\Omega$. Consider $H=\{l,m\}$; then ${\overline{{\acute{AP}}_{{\rm Y}}}}^{{\Delta }_1}\left(H\right)⊑{\overline{{\acute{AP}}_{{\rm Y}}}}^{{\Delta }_2}\left(H\right)$, but ${\Delta }_1⊈{\Delta }_2$.

Also, as in Table 3 with grill ${\Delta }_2$, our method is the best compared to Dai’s method.

Proposition 6. 

Let Υ be a BIR on a universe Ω and Δ be an grill on Ω; then the following properties hold:

(i) 

${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }(\Omega )=\Omega$.

(ii) 

$H_1⊑H_2\Rightarrow {\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_1\right)⊑{\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_2\right)$.

(iii) 

${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_1\right)\bigsqcup {\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_2\right)⊑{\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }(H_1\bigsqcup H_2)$.

(iv) 

${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }(H_1\sqcap H_2)={\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_1\right)\bigsqcup {\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_2\right)$.

(v) 

${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left({\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)\right)={\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)$.

(vi) 

${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)={\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }{\left[\left(H^c\right)\right]}^c$.

(vii) 

If $H^c\notin \Delta$, then ${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)=H$.

Proof. 

(i) ${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }(\Omega )=\bigsqcup \{N_{{\rm Y}}\left(\omega \right):N_{{\rm Y}}\left(\omega \right)\sqcap {\Omega }^c\notin \Delta \}=\bigsqcup \{N_{{\rm Y}}\left(\omega \right):N_{{\rm Y}}\left(\omega \right)\sqcap \varphi \notin \Delta \}=\Omega$. (ii) Let $H_1⊑H_2$, and $\omega \in {\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_1\right)$. Then, $\omega \in N_{{\rm Y}}\left(\omega \right):N_{{\rm Y}}\left(\omega \right)\sqcap H_1^c\notin \Delta$. So, $\omega \in N_{{\rm Y}}\left(\omega \right):N_{{\rm Y}}\left(\omega \right)\sqcap H_2^c\notin \Delta$, since $H_2^c⊑H_1^c$. Thus, $\omega \in {\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_2\right)$. (iii) The evidence follows immediately from (ii). (iv) ${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }(H_1\sqcap H_2)=\bigsqcup \{N_{{\rm Y}}\left(\omega \right):N_{{\rm Y}}\left(\omega \right)\sqcap {(H_1\sqcap H_2)}^c\notin \Delta \}=\bigsqcup \{N_{{\rm Y}}\left(\omega \right):N_{{\rm Y}}\left(\omega \right)\sqcap (H_1^c\bigsqcup H_2^c)\notin \Delta \}=\bigsqcup \{N_{{\rm Y}}\left(\omega \right):N_{{\rm Y}}\left(\omega \right)\sqcap \left(H_1^c\right)\notin \Delta \}$ or $\bigsqcup \{N_{{\rm Y}}\left(\omega \right):N_{{\rm Y}}\left(\omega \right)\sqcap \left(H_2^c\right)\notin \Delta \}=$${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_1\right)\bigsqcup {\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_2\right)$. (v) The proof comes straight out of (i). (vi) The proof follows immediately from Definition 11. (vii) Definition 11 immediately leads to the proof. □

Proposition 7. 

Let Υ be a BIR on a universe Ω and Δ be an grill on Ω; then the following properties hold:

(i) 

${\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }(\Omega )=\Omega$.

(ii) 

If $H_1⊑H_2\Rightarrow {\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }{\left(H\right)}_1⊑{\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }{\left(H\right)}_2$.

(iii) 

${\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }(H_1\bigsqcup H_2)⊑{\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H_1\right)\bigsqcup {\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H_2\right)$.

(iv) 

${\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }(H_1\sqcap H_2)={\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }(H_1)\sqcap {\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta })\left(H_2\right)$.

(v) 

${\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)={\left[{\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H^c\right)\right]}^c$.

(vi) 

$H\notin \Delta \Rightarrow H⊑{\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)$.

Proof. 

(i) ${\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }(\Omega )={\left[{\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left({\Omega }^c\right)\right]}^c={\left[\varphi \right]}^c=\Omega$. (ii) Let $H_1⊑H_2$. Thus, $H_2^c⊑H_1^c$ and ${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_2^c\right)⊑{\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_1^c\right)$ (from (ii) Proposition 6). So, ${\left({\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_1^c\right)\right)}^c⊑{\left({\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_2^c\right)\right)}^c$. Then, ${\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }{\left(H\right)}_1⊑{\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }{\left(H\right)}_2$. (iii) The proof follows immediately from (ii). (iv) ${\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }(H_1\sqcap H_2)={\left[{\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }(H_1\sqcap H_2)\right]}^c={\left[{\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }(H_1^c\bigsqcup H_2^c)\right]}^c={\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H_1\right)\sqcap {\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H_2\right)$. (v) The proof follows immediately from Definition 11. (vi) Definition 11 immediately leads to the proof. □

Remark 8. 

The converse of (iii) Propositions 6 and 7 is not always true, as demonstrated by Remark 2. Consider the following:

(a) 

$H_1=\left\{r\right\}$, $H_2=\{s,t\}$; then ${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_1\right)=\{r,t,u\}$, and ${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_2\right)=\{{\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4\}$. Therefore ${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_1\right)⊑{\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_2\right)$, but $H_1⊈H_2$.

(b) 

$H_1=\left\{r\right\}$, $H_2=\{s,t\}$; then ${\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H_1\right)=\varphi$, and ${\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H_2\right)=\left\{s\right\}$. Therefore ${\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H_1\right)⊑{\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H_2\right)$, but $H_1⊈H_2$.

Theorem 5. 

Let Υ be a reflexive relation on a universe Ω,$H⊑\Omega$ and $\Delta \ne \Omega$ be a grill on Ω; then the following properties hold:

(i) 

${\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)⊑{\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)⊑{\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)⊑H⊑{\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)⊑{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)⊑{\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)$.

(ii) 

${{bound}^{\prime }}_{{\rm Y}}^{\Delta }\left(H\right)⊑boun{d}_{{\rm Y}}^{\Delta }\left(H\right)⊑{\underline{bound}}_{{\rm Y}}^{\Delta }\left(H\right)$.

(iii) 

${\underline{Ac}}_{{\rm Y}}^{\Delta }\left(H\right)\le A{c}_{{\rm Y}}^{\Delta }\left(H\right)\le {\acute{Ac}}_{{\rm Y}}^{\Delta }\left(H\right)$.

Proof. 

We only prove (i), whereas the others are obvious forms of (i). From Definitions 9 and 10, we have ${\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)⊑{\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)$. To prove ${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)⊑{\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)$, let $\omega \in {\underline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)$; then $N_{{\rm Y}}\left(\omega \right)\sqcap H^c\notin \Delta$. Hence, $N_{{\rm Y}}\left(\omega \right)⊑{\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)$. It follows that $\omega \in N_{{\rm Y}}\left(\omega \right)⊑{\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)$, since ${\rm Y}$ is reflexive relation. Therefore, $\omega \in {\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)$, since ${\rm Y}$ is reflexive relation. It follows that ${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)⊑H⊑{\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)$, which is straightforward from Definition 11. To prove ${\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)⊑{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)$, let $\omega \in {\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H\right)={\left({\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H^c\right)\right)}^c$; then $\omega \notin \acute{{\underline{AP}}_{{\rm Y}}}\left(H^c\right)$. Hence, by Definition 5.1, we get $N_{{\rm Y}}\left(\omega \right)\sqcap H\in \Delta$. It follows that $\omega \in {\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)$. By Definitions 9 and 10, we have ${\overline{AP}}_{{\rm Y}}^{\Delta }\left(H\right)⊑{\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H\right)$. □

Remark 9. 

The maximal right neighborhood and grills are the foundation of the three previously suggested approaches. Finding estimates that rely on the right minimal neighborhood with grills is demonstrated in Example 10.

Example 10. 

Continued in Example 2. Let $H_1=\{l,m\}$, $H_2=\{m,n\}$. Then: First method: ${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)=\{l,m\},{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_1\right)=\{l,m\}$, ${\underline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)=\{m,n\},{\overline{AP}}_{{\rm Y}}^{\Delta }\left(H_2\right)=\Omega$, $A{c}_{{\rm Y}}^{\Delta }\left(H_1\right)=1,$ and $A{c}_{{\rm Y}}^{\Delta }\left(H_2\right)=0.5$. Second method: ${\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H_1\right)=\{l,m\},{\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H_1\right)=\{l,m\}$, ${\underline{\underline{AP}}}_{{\rm Y}}^{\Delta }\left(H_2\right)=\{m,n\},{\overline{\overline{AP}}}_{{\rm Y}}^{\Delta }\left(H_2\right)=\Omega$, ${\underline{Ac}}_{{\rm Y}}^{\Delta }\left(H_1\right)=1,$ and ${\underline{Ac}}_{{\rm Y}}^{\Delta }\left(H_2\right)=0.5$. Third method: ${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_1\right)=\{l,m\},{\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H_1\right)=\{l,m\}$, ${\acute{{\underline{AP}}_{{\rm Y}}}}^{\Delta }\left(H_2\right)=\{m,n\},{\overline{{\acute{AP}}_{{\rm Y}}}}^{\Delta }\left(H_1\right)=\Omega$, ${\acute{Ac}}_{{\rm Y}}^{\Delta }\left(H_1\right)=1,$ and ${\acute{Ac}}_{{\rm Y}}^{\Delta }\left(H_2\right)=0.5$.

6. Medical Application

The main purpose of this section is to apply the provided approaches to problems in the real world, notably in the field of medical diagnostics, where more precise decisions are necessary. As a result, the suggested APP grill properties are carefully examined for potential medical uses. These applications demonstrate how well rough sets may be generalized utilizing grills to address and simulate a wide range of real-world problems. It is demonstrated that using grills in RS theory reduces ambiguity and vagueness in data.

Example 11 

(Dengue fever decision-making). This example aims to illustrate the importance of the existing APPs in acquiring the most effective instruments to determine the essential elements of human dengue fever infections. The World Health Organization and medical organizations that specialize in dengue fever gathered the data that is given in

Table 4. Original dengue fever information system.

${\mathit{b}}_{\mathit{n}}$	$\mathit{mj}$	ℏ	$\mathit{sk}$	t	D
$b_{n1}$	⧫	⧫	⧫	||	⧫
$b_{n2}$	⧫	◊	◊	||	◊
$b_{n3}$	⧫	◊	◊	||	⧫
$b_{n4}$	◊	◊	◊	|||	◊
$b_{n5}$	◊	⧫	⧫	||	◊
$b_{n6}$	⧫	⧫	◊	|||	⧫
$b_{n7}$	⧫	⧫	◊	|	◊
$b_{n8}$	⧫	⧫	◊	|||	⧫

[40,41]. A temperature $\left(t\right)$ (very high _|||, high _||, or normal _|), headache with vomiting $(\hslash )$, muscular and joint pain $\left(mj\right)$, and a characteristic skin rash $\left(sk\right)$ are among the symptoms of dengue fever that are displayed in the columns of Table 4. The rows of patients $b_n=\{b_{n1},b_{n2},b_{n3},b_{n4},b_{n5},b_{n6},b_{n7},b_{n8}\}$ are the patients, and attribute D is the problem decision. As for symbols, ⧫ indicates that the patient exhibits symptoms, while ◊ signifies the absence of symptoms. The decision is also represented using the same two symbols, indicating whether the patient has dengue fever or not.

The following symptoms are taken from Table 4:

$b_{n1}{\rm Y}=\{b_{n1},b_{n2},b_{n3},b_{n5},b_{n6},b_{n7},b_{n8}\}$, $b_{n2}{\rm Y}=\{b_{n1},b_{n2},b_{n3}\}$, $b_{n3}{\rm Y}=\{b_{n1},b_{n2},b_{n3}\}$, $b_{n5}{\rm Y}=\{b_{n1},b_{n5}\}$, $b_{n6}{\rm Y}=\{b_{n1},b_{n6},b_{n7},b_{n8}\}$, $b_{n7}{\rm Y}=\{b_{n1},b_{n6},b_{n7},b_{n8}\}$, $b_{n8}{\rm Y}=\{b_{n1},b_{n2},b_{n6},b_{n7},b_{n8}\}$.

Here, $b_{ni}{\rm Y}{b}_{nj}$ if and only if $b_{ni}$ and $b_{nj}$ have at least two positive symptoms.

Let $\Delta =\{\{b_{n1},b_{n3},b_{n6},b_{n8}\},$  $\{b_{n1},b_{n2},b_{n3},b_{n6},b_{n8}\},$  $\{b_{n1},b_{n3},b_{n6},b_{n7},b_{n8}\},$  $\{b_{n1},b_{n3},b_{n4},b_{n6},b_{n8}\},$  $\{b_{n1},b_{n3},b_{n5},b_{n6},b_{n8}\},$  $\{b_{n1},b_{n2},b_{n3},b_{n6},b_{n7},b_{n8}\},$  $\{b_{n1},b_{n2},b_{n3},b_{n4},b_{n6},b_{n8}\},$  $\{b_{n1},b_{n2},b_{n3},b_{n5},b_{n6},b_{n8}\},$  $\{b_{n1},b_{n3},b_{n4},b_{n5},b_{n6},b_{n8}\},$  $\{b_{n1},b_{n3},b_{n4},b_{n6},b_{n7},b_{n8}\},$  $\{b_{n1},b_{n3},b_{n5},b_{n6},b_{n7},b_{n8}\},$  $\{b_{n1},b_{n2},b_{n3},b_{n4},b_{n5},b_{n6},b_{n8}\},$  $\{b_{n1},b_{n2},b_{n3},b_{n4},b_{n6},b_{n7},b_{n8}\},$  $\{b_{n1},b_{n2},b_{n3},b_{n5},b_{n6},b_{n7},b_{n8}\},$  $\{b_{n1},b_{n3},b_{n4},b_{n5},b_{n6},b_{n7},b_{n8}\},\Omega \}$.

Table 4 displays a decision system, indicating that individuals with confirmed dengue fever infections are $H=\{b_{n1},b_{n3},b_{n6},b_{n8}\}$. Then,

(i) 

Based on earlier APPs [36], the first (second/third) kind of LOW and UPP APPs, BRs, and accuracy of H are $\varphi ,\Omega ,\Omega$, and 0. This means Patients $b_{n1},b_{n3},b_{n6},b_{n8}$ do not have dengue fever, contradicting the decision system in Table 4. As a result, we are unable to determine if the patient has dengue fever, which creates uncertainty in the process of reaching a medical diagnosis. As a result, the methodologies used by Dai et al. [36] are not appropriate for reaching a precise conclusion.

(ii) 

In the suggested second type, the LOW and UPP APPs, BRs, and accuracy of $H=\{b_{n1},b_{n3},b_{n6},b_{n8}\}$ are $H,\{b_{n1},b_{n2},b_{n3},b_{n5},b_{n6},b_{n7},b_{n8}\},\{b_{n2},b_{n5},b_{n7}\},$ and ${\displaystyle \frac{4}{7}}$. According to the current method, this indicates that Patients $b_{n1},b_{n3},b_{n6}$, and $b_{n8}$ are unquestionably infected with dengue fever, which is in line with Table 4. As a result, the accuracy measure rises and the vagueness of the data decreases.

7. Conclusions

This work integrates two fields: grills and RS theory. While RS theory uses the LOW (UPP) APPs to deal with ambiguity and poor knowledge, the study of RS generalization heavily relies on the concept of grills, which is fundamental to topological spaces. The use of grills in RS theory suggests a natural extension of the notion, as demonstrated here. This article proposes various strategies for approximating sets using grills and the maximal right neighborhood created by BINs. Numerous types of neighborhood systems have been defined with grills for various reasons, including preserving the characteristics of Pawlak’s LOW (UPP) APPs, raising the accuracy measures, and rescinding the condition of an equivalence relation, among others. This is due to the inadequacy of previous approximations in preserving the main characteristics and their failure to analyze some data systems. The present technique was a generalization of the method used by Dai et al. [36]. The fundamental characteristics of the existing techniques were examined. More significantly, it was demonstrated that the corresponding roughness, accuracy, and LOW APPs for three of the existing approaches were monotonic. Additionally, a medical application was introduced to highlight the key points of current discoveries and to make way for further research efforts. The presented strategies effectively reduced BNs and improved accuracy measures.

We plan to investigate other forms of estimates from neighborhoods in future work, as well as to determine how these APPs can be used to model real-world problems and apply the outlined principles and results to generalize rough multisets using multiset grills.