EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6997 ISSN 1307-5543 – ejpam.com Published by New York Business Global Improving Cardinality Rough Neighborhoods via Grills and Their Applications A. A. Azzam1,∗, M. Aldawood1, B. Alreshidi1 1 Mathematics Department, Faculty of Science and Humanities, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia Abstract. In order to solve problems and provide practical solutions, researchers attempt to ac- curately describe societal difficulties and obstacles. An efficient method for handling complicated real-world data is rough set theory. The method finds confirmed and likely data from subsets using rough approximation operators. In order to increase accuracy while following Pawlak’s conven- tional approximation axioms, earlier research has created rough approximation models based on neighborhood systems. Based on cardinality rough neighborhoods and grills, we present new rough set notions in this study. These models are a suitable approach for a number of scenarios, including computational analysis, comparisons on medical datasets, real-world data analysis challenges, and classification accuracy metrics. We thoroughly examine the fundamental components of these ideas and clarify how they relate to each other and to earlier paradigms. Next, we describe the boundary regions and assess the accuracy of the data using a topological technique. Additionally, we look at how well our models handle heart failure disease in certain individuals and come to the conclusion that the suggested rough set concepts improve upon the characteristics of the earlier approach spaces. Finally, we identify the shortcomings of the current concepts and show their advantages in terms of extending the verified information gleaned from subsets of data while preserving the key elements of Pawlak’s original paradigms that were destroyed by the models that went before them. 2020 Mathematics Subject Classifications: 54D80, 54C55, 54A25, 54A05 Key Words and Phrases: Lower/upper approximation, grill, neighborhood, rough set 1. Introduction Rough sets theory (Rsst) applications to knowledge discovery entail gathering actual data and using it to create classification models according to the data [1, 2]. Each subset in this theory corresponds to two distinct sets that are derived from an equivalence relation (eqr) and are referred to as the lower (Lw) and upper (Up) approximations (Aps). Many researchers have substituted different kinds of relations for the eqr in order to expand the ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6997 Email addresses: azzam0911@yahoo.com (A. A. Azzam), m.aldawood@psau.edu.sa (M. Aldawood), b.alreshidi@psau.edu.sa (B. Alreshidi) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 2 of 31 applications of Rsst [3–11]. As a result, equivalence classes were replaced with various neighborhood (nbd) representations of blocks or granular computing to characterize sys- tems of information. These neighborhoods (nbds) include union, intersection, and right and left nbds [12–15], equal nbds [16, 17], rough nbd ideal [18]. The researchers demon- strated their usefulness in handling actual situations involving social difficulties, economics, and medical and in helping decision-makers make wise choices. In other words, they pro- vide a broad framework devoid of limitations on the kind of binary relationships. Notably, Rsst has demonstrated its usefulness as a crucial instrument for characterizing informa- tion content across numerous frameworks and applications in numerous fields [19–26]. In [27], the topological characteristics of Rss were examined. As a result, topological and rough set (Rs) theories were combined, and this became the main topic of several works [14, 28–32]. Additionally, this relationship involved topological generalizations, including minimal structures [33], supra topology [34], infra topology [35], nano-topology [36], and bitopology [37]. We direct readers to [38, 39] for a comprehensive summary of the works examining the connections between Rsst and topology. The symmetry between closure and interior topological operators with Lw and Up-Aps, as well, allows us to give abstract conceptions meaningful meanings and employ abstract methods to convey knowledge ex- tracted from systems of data. The concept of two operators, ξ and ζ, is the foundation of grill topological spaces. Choquet was the first to propose this concept [40]. Some similar- ities between the Choquat notion and ideals, nets, and filters have been found. [41] and [18] have addressed a number of hypotheses and aspects. It contributes to the expansion of the topological form, which is used to measure attributes like love, intelligence, beauty, and educational attainment rather than numbers. Furthermore, it extends the topological structure (tsr) by using the idea of grill alterations in the boundary region (Br), Up, and Lw-Aps, opening up new boundaries in nano topological spaces [42]. Grills are effective for analyzing raw datasets, especially for eliminating ambiguity. Consequently, a principal impetus for this endeavor is the implementation of novel grill-based Rs methodologies. To put it another way, it creates a special situation for its generic Rs model counterparts when the grill is the universal set. Because of this, a few intriguing studies examined the Rst that grills describe. This paper is laid out as follows. In order to explain why this study is necessary, we refer to the earlier types of R nbds and the key ideas associated with them in the following section. We then divide Sect. 3 into two subsections, each of which has two new Ap spaces based on cardinality rough nbds via grills. We present a new kind of Ap space and examine its primary characteristics in Sect. 3.1. To deal with irrational traits and ambiguous situations, In Sect. 3.2, we modified the prior method and demonstrate the advantages of the new one. The objective of Sect. 4 is to investigate the suggested Rs models from a topological perspective. In Sect. 5, we demonstrate the effectiveness of the provided models in handling a medical scenario involving the treatment of heart failure. We also highlight how our method enhances making choices and how we apply a topological approach motivated by this method to determine the most important characteristics or symptoms for making choices. Finally, in Sect. 6 we address the benefits and drawbacks of the current models and highlight the key findings of this work with a proposal for further research, respectively. A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 3 of 31 2. Preliminaries This part was devoted to reviewing the key terms and findings and explaining the advantages of hybridizing grills with cardinal rough nbds in order to improve accuracy. 2.1. Conventional approximation space Definition 1. [1] Let ℵ represent a universe, which is a finite, nonempty set. One way to describe a binary relation on ℵ is as a subcollection of ℵ×ℵ. A binary relation Γ on ℵ is a subclass Γ of ℵ×ℵ. To indicate that (b, n) is an element of Γ, we write bΓn. If a relation Γ on ℵ is reflexive (i.e., bΓb for all b ∈ ℵ), symmetric (bΓn ⇔ nΓb), and transitive (i.e.,bΓm when bΓn and nΓm), we call it an equivalence. Furthermore, a relation is referred to as comparable if it fulfills bΓn or nΓb for any b, n ∈ ℵ. Definition 2. [1] Let Γ represent an eqr on ℵ, and ∆ ⊑ ℵ so, the Lw and Up-Aps of ∆ will be represented as: Γ(∆) = ⊔{D ∈ ℵ Γ : D ⊑ ∆}. Γ(∆) = ⊔{D ∈ ℵ Γ : D ⊓ ∆ ̸= ϕ} Notation ℵ Γ represents the family that includes all equivalency classes brought to by the relation Γ. Henceforth referred to as an Ap space is the pair (ℵ,Γ). It is described as follows if Γ(∆) and Γ(∆) are not equal, ∆ is said to be rough. On the other hand, A subset ∆ is said to be exact if Γ(∆) = Γ(∆). The following proposition lists the main characteristics of the conventional Rs model. Proposition 1. [1, 2] Examine the definition of an eqr Γ on ℵ. The following properties are true for sets D and ∆: ( L1) Γ(D) ⊑ D. ( L2) Γ(ℵ) = ℵ. ( L3) Γ(ϕ) = ϕ. ( L4) Γ(D) ⊑ Γ(∆) whenever D ⊑ ∆. ( L5) Γ((D) ⊓ ∆) = Γ(D) ⊓ Γ(∆). ( L6) Γ(D) ⊔ Γ(∆) ⊑ Γ(D ⊔ ∆). A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 4 of 31 ( L7) Γ(Dc) = (Γ(D))c. ( L8) Γ(Γ(D)) = Γ(D). ( L9) Γ((Γ(D))c) = (Γ(D))c. ( L10) Γ(∆) = ∆,∀∆ ∈ ℵ Γ . (U1) D ⊑ Γ(D). (U2) Γ(ℵ) = ℵ. (U3) Γ(ϕ) = ϕ. (U4) If D ⊑ ∆, then Γ(D) ⊑ Γ(∆) . (U5) Γ(D ⊓ ∆) ⊑ Γ(D) ⊓AP (∆). (U6) Γ(D ⊔ ∆) = Γ(D) ⊔ Γ(∆). (U7) Γ(Dc) = (Γ(D))c. (U8) Γ(Γ(D)) = Γ(D). (U9) Γ((Γ(D))c) = (Γ(D))c. (U10) Γ(∆) = ∆,∀∆ ∈ ℵ Γ . Numerous methods have been used to expand traditional theory [26], and the validity of these traits has been well validated. Regretfully, it has been discovered that some qualities are questionable. However, it is seen to be beneficial in these approaches to acquire as many of these traits as is practical. Definition 3. [1, 2] The Σ-accuracy and ℜ-roughness criteria of ∆ are ascertained by taking an eqr Γ on ℵ into consideration: Σ(∆) = |Γ(∆)| |Γ(∆)| , |Γ(∆)| ̸= 0. ℜ(∆) = 1 − Σ(∆). The equivalency relations are not achievable in many situations. Consequently, weaker relationships than full equivalency have been used to augment the traditional approach. A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 5 of 31 2.2. ϱ-Neighborhood space types Definition 4. [13, 15, 43, 44] In an arbitrary relation Γ on ℵ. where ϱ ∈ {r, ⟨r⟩, l, ⟨l⟩, i, ⟨i⟩, u, ⟨u⟩}, the ϱ-nbds of an e ∈ ℵ (denoted by Mϱ(e)) are defined as follows: (1) Mr(e) = {π ∈ ℵ : eΓπ}. (2) Ml(e) = {π ∈ ℵ : πΓe}. (3) M⟨r⟩(e) = { ⊓e∈Mr(π)Mr(π) : ∃ Mr(π) containing e, ϕ : otherwise. (4) M⟨l⟩(e) = { ⊓e∈Ml(π)Ml(π) : ∃ Ml(π) containing e, ϕ : otherwise. . (5) Mi(e) = Mr(e) ⊓Ml(e). (6) Mu(e) = Mr(e) ⊔Ml(e). (7) M⟨i⟩(e) = M⟨r⟩(e) ⊓M⟨l⟩(e). (8) M⟨u⟩(e) = M⟨r⟩(e) ⊔M⟨l⟩(e). From now on, unless otherwise noted, we shall take into ϱ = {r, ⟨r⟩, l, ⟨l⟩, i, ⟨i⟩, u, ⟨u⟩}. Definition 5. [43] Let us examine a relation Γ on ℵ and demonstrate a mapping ξϱ from ℵ to 2ℵ that connects each member of ℵ to its ϱ-nbd in 2ℵ. The triple (ℵ,Γ, ξϱ), shortened to ϱ-NS, is hence referred to as a ϱ-nbd space. We used the aforementioned nbds to develop new Lw and Up-Ap, as well as accuracy (roughness) requirements. To improve Ap quality and accuracy, we compared several types of nbds. Definition 6. [13, 15, 43, 44] The Lw and Up-Ap of each subset ∆ in relative to Mϱ(e)- nbds are introduced as follows: MMϱ(∆) = {e ∈ ℵ : Mϱ(e) ⊑ ∆}. MMϱ(∆) = {e ∈ ℵ : Mϱ(e) ⊓ ∆ ̸= ϕ}. Definition 7. [12, 13, 15] The ΣMϱ-accuracy and ℜMϱ-roughness criteria of nonempty set ∆ in regarding to Γ are computed as: A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 6 of 31 ΣMϱ(∆) = |MMϱ(∆)⊓∆| |MMϱ(∆)⊔∆| , and ℜMϱ(∆) = 1 − ΣMϱ(∆). Definition 8. [1, 2] Let Γ1 and Γ2 be relations on ℵ such that Γ1 ⊑ Γ2. The monotonicity property in accuracy and roughness of any set is demonstrated by Ap obtained from M -nbds, ΣMϱ1(∆) ≥ ΣMϱ2(∆), and ℜMϱ1(∆) ≤ ℜMϱ2(∆), in that order. Definition 9. [16, 17] Let us examine a relation Γ on ℵ. The H-nbds of an element e ∈ ℵ for each ϱ are expressed as follows: (1) Hr(e) = {π ∈ ℵ : Mr(e) = Mr(π)}. (2) Hl(e) = {π ∈ ℵ : Ml(e) = Ml(π)}. (3) Hi(e) = Hr(e) ⊓Hl(e). (4) Hu(e) = Hr(e) ⊔Hl(e). (5) H⟨r⟩(e) = {π ∈ ℵ : M⟨r⟩(e) = M⟨r⟩(π)}. (6) H⟨l⟩(e) = {π ∈ ℵ : M⟨l⟩(e) = M⟨l⟩(π)}. (7) H⟨i⟩(e) = H⟨r⟩(e) ⊓H⟨l⟩(e). (8) H⟨u⟩(e) = H⟨r⟩(e) ⊔H⟨l⟩(e). Definition 10. [29] Let us examine a relation Γ on ℵ. The k-nbds of an element e ∈ ℵ for each ϱ are expressed as follows: (1) kr(e) ⊑ {π ∈ ℵ : Mr(π) ⊑ Mr(e)}. (2) kl(e) = {π ∈ ℵ : Ml(π) ⊑ Ml(e)}. (3) ki(e) = kr(e) ⊓ kl(e). (4) ku(e) = kr(e) ⊔ kl(e). (5) k⟨r⟩(e) = {π ∈ ℵ : M⟨r⟩(π) ⊑ M⟨r⟩(e)}. (6) k⟨l⟩(e) = {π ∈ ℵ : M⟨l⟩(π) ⊑ M⟨l⟩(e)}. (7) k⟨i⟩(e) = k⟨r⟩(e) ⊓ k⟨l⟩(e). A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 7 of 31 (8) k⟨u⟩(e) = k⟨r⟩(e) ⊔ k⟨l⟩(e). Definition 11. [45] Let us examine a relation Γ on ℵ. The ∦-nbds of an element e ∈ ℵ for each ϱ are expressed as follows: (1) ∦r(e) = {π ∈ ℵ : Mr(e) ⊑ Mr(π)}. (2) ∦l(e) = {π ∈ ℵ : Ml(e) ⊑ Ml(π)}. (3) ∦i(e) = ∦r(e) ⊓ ∦l(e). (4) ∦u(e) = ∦r(e) ⊔ ∦l(e). (5) ∦⟨r⟩(e) = {π ∈ ℵ : M⟨r⟩(e) ⊑ M⟨r⟩(π)}. (6) ∦⟨l⟩(e) = {π ∈ ℵ : M⟨l⟩(e) ⊑ M⟨l⟩(π)}. (7) ∦⟨i⟩(e) = ∦⟨r⟩(e) ⊓ ∦⟨l⟩(e). (8) ∦⟨u⟩(e) = ∦⟨r⟩(e) ⊔ ∦⟨l⟩(e). 2.3. Cardinality ϱ-Neighborhood system This section is dedicated to introducing the concept of cardinality nbds, in accordance with any binary relation. The goal of studying cardinality nbds is to address specific situations where the number of members in Mϱ-nbds affects the situation. We’ll examine their primary characteristics and identify the circumstances in which some of them are the same. Various examples are offered to bolster the links and outcomes obtained. The cardinality of Mϱ(.) is indicated by |Mϱ(.)| for each ϱ ∈ {r, ⟨r⟩, l, ⟨l⟩, i, ⟨i⟩, u, ⟨u⟩}. Definition 12. [46] For every ϱ, the ϱ-cardinality nbds of e ∈ ℵ, denoted by Cϱ(e), are found using an arbitrary relation Γ on ℵ as follows: (1) Cr(e) = {π ∈ ℵ : |Mr(e)| = |Mr(π)|}. (2) Cl(e) = {π ∈ ℵ : Ml(e) = |Ml(π)|}. (3) Ci(e) = Cr(e) ⊓ Cl(e). (4) Cu(e) = ∦r(e) ⊔ Cl(e). (5) C⟨r⟩(e) = {π ∈ ℵ : |M⟨r⟩(e)| = |M⟨r⟩(π)|}. A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 8 of 31 (6) C⟨l⟩(e) = {π ∈ ℵ : |M⟨l⟩(e)| = |M⟨l⟩(π)|}. (7) C⟨i⟩(e) = C⟨r⟩(e) ⊓ C⟨l⟩(e). (8) C⟨u⟩(e) = C⟨r⟩(e) ⊔ C⟨l⟩(e). Proposition 2. [46] (1) Ci ⊑ Cr ⊓ Cl ⊑ Cr ⊔ Cl ⊑ Cu, and C⟨i⟩ ⊑ C⟨r⟩ ⊓ C⟨l⟩ ⊑ C⟨r⟩ ⊔ C⟨l⟩ ⊑ C⟨u⟩. (2) All Cϱ are equal if Γ is a symmetric relation. Proposition 3. [46] (1) e ∈ Ci(b) iff |Mr(e)| = |Mr(b)| and |Ml(e)| = |Ml(b)|. (2) e ∈ Cu(b) iff |Mr(e)| = |Mr(b)| or |Ml(e)| = |Ml(b)|. (3) e ∈ C⟨i⟩(b) iff |M⟨r⟩(e)| = |M⟨r⟩(b)| and |M⟨l⟩(e)| = |M⟨l⟩(b)|. (4) e ∈ C⟨u⟩(b) iff |M⟨r⟩(e)| = |M⟨r⟩(b)| or |M⟨l⟩(e)| = |M⟨l⟩(b)|. Corollary 1. [46] If the relation Γ is symmetric, then: (1) Ci(b) = {π ∈ ℵ : |Mi(b)| = |Mr(π)|}. (2) C⟨i⟩(b) = {π ∈ ℵ : |M⟨i⟩(b)| = |M⟨i⟩(π)|}. (3) Cu(b) = {π ∈ ℵ : |Mu(b)| = |Mu(π)|}. (4) C⟨u⟩(b) = {π ∈ ℵ : |M⟨u⟩(b)| = |M⟨u⟩(π)|}. Proposition 4. [46] Let (ℵ,Γ, ξϱ) be a ϱ-NS. If e ∈ ℵ, then Cϱ(e) ̸= ϕ∀ϱ. Proposition 5. [46] Let (ℵ,Γ, ξϱ) be a ϱ-NS and e ∈ ℵ. Then, e ∈ Cϱ(b) iff b ∈ Cϱ(e)∀ϱ. Proposition 6. [46] Let (ℵ,Γ, ξϱ) be a ϱ-NS. If e ∈ Cϱ(b), b ∈ Cϱ(n), then e ∈ Cϱ(n), in the cases of ϱ ∈ {r, ⟨r⟩, l, ⟨l⟩, i, ⟨i⟩}. Corollary 2. [46] Let (ℵ,Γ, ξϱ) be a ϱ-NS and e ∈ ℵ. Then, e ∈ Cϱ(b) iff Cϱ(e) = Cϱ(b), in the cases of ϱ ∈ {r, ⟨r⟩, l, ⟨l⟩, i, ⟨i⟩}. Corollary 3. [46] In the instances of ϱ ∈ {r, ⟨r⟩, l, ⟨l⟩, i, ⟨i⟩}, the relation Γ where bΓe ⇔ b ∈ Cϱ(e) is an equivalence. A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 9 of 31 Corollary 4. [46] Under a symmetric relation, the cardinality nbds create a partition for ℵ for each ϱ. Proposition 7. [46] Cϱ = C⟨ϱ⟩ for ϱ belongs to {r, l, i, u}, if Γ is a preorder relation on ℵ. Proposition 8. [46] Let (ℵ,Γ, ξϱ) be a ϱ-NS. If e ∈ ℵ, then ∦ϱ(e) ⊑ Cϱ(e)∀ϱ. Definition 13. [46] Let (ℵ,Γ, ξϱ) be a ϱ-NS. Based on cardinality nbds, the Cϱ-Lw-Ap MCϱ(∆), and Cϱ-Up-Ap MCϱ(∆) of a set ∆, assigned as: MCϱ(∆) = {b ∈ ℵ : Cϱ(b) ⊑ ∆}. MCϱ(∆) = {b ∈ ℵ : Cϱ(b) ⊓ ∆ ̸= ϕ}. Definition 14. [46] The Cϱ-boundary, Cϱ-positive, Cϱ-negative regions of a subset ∆ within a ϱ-NS (ℵ,Γ, ξϱ) are identified respectively as: BCϱ(∆) = MCϱ(∆)\MCϱ(∆). PCϱ(∆) = MCϱ(∆). N Cϱ(∆) = ℵ\MCϱ(∆). Definition 15. [46] The Cϱ-accuracy and Cϱ-roughness criteria of ∆ ̸= ϕ of a ϱ-NS (ℵ,Γ, ξϱ) are identified respectively as: ΣCϱ(∆) = |MCϱ (∆)| |MCϱ (∆)| , |M Cϱ(∆)| ̸= 0. ℜCϱ(∆) = 1 − ΣCϱ(∆). Theorem 1. [46] Let (ℵ,Γ, ξϱ) be a ϱ-NS. Based on cardinality nbds, the family τCϱ = {∆ ⊑ ℵ : ∀e ∈ ∆, Cϱ(e) ⊑ ∆} is a topology on ℵ, for each ϱ. Lemma 1. [46] Let (ℵ,Γ, ξϱ) be a ϱ-NS and e ∈ ℵ. If ϱ ∈ {r, ⟨r⟩, l, ⟨l⟩, i, ⟨i⟩}, then Cϱ(e) is τCϱ-open set. Definition 16. [42] A nonempty subcollection G ⊑ 2ℵ is defined as a grill on ℵ if the following are satisfied: ϕ /∈ G H ∈ G, H ⊑ J ⊑ ℵ leads to J ∈ G, if H ⊔ J ∈ G for H, J ⊑ ℵ, then H ∈ G or J ∈ G. 3. Grills and cardinality create new rough-set paradigms in neighborhoods This section aims to delineate and examine novel R-Ap spaces generated by cardinality nbds and grills, with an emphasis on establishing new areas and standards for accuracy and roughness. A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 10 of 31 3.1. The initial class of Rs paradigms This section is devoted to presenting novel Rs paradigms that are influenced by the concepts of grills and cardinality nbds. We demonstrate that, in contrast to the earlier models of Rss, In these paradigms, the Lw-Ap is amplified and the Up-Ap is minified. On the other hand, we talk about the current models’ flaws. Definition 17. Let (ℵ,Γ, ξϱ) be a ϱ-NS and G be a grill on ℵ. The pair (GM̃Cϱ (∆),GM̃Cϱ (∆)) represents Lw, and Up-Ap of a subset ∆ with respect to grills and cardinality nbds, and they are calculated by: GM̃Cϱ (∆) = {b ∈ ℵ : Cϱ(b) ⊓ ∆c /∈ G}, GM̃Cϱ (∆) = {b ∈ ℵ : Cϱ(b) ⊓ ∆ ∈ G} Remark 1. If G = ℵ in Definition 3.1 then, GM̃Cϱ (∆) = ϕ. Theorem 2. Let (ℵ,Γ, ξϱ) be a ϱ-NS and G be a grill on ℵ. If ∆, D ⊑ ℵ, then ∀ϱ the next statement hold true. (1) GM̃Cϱ (ℵ) = ℵ and GM̃Cϱ (ϕ) = ϕ. (2) If D ⊑ ∆, then GM̃Cϱ (D) ⊑ GM̃Cϱ (∆) and GM̃Cϱ (D) ⊑ GM̃Cϱ (∆). (3) GM̃Cϱ (D ⊓ ∆) = GM̃Cϱ (D) ⊓ GM̃Cϱ (∆) and GM̃Cϱ (D ⊔ ∆) = GM̃Cϱ (D) ⊔ GM̃Cϱ (∆). (4) GM̃Cϱ (∆c) = (GM̃Cϱ (∆))c and GM̃Cϱ (∆c) = (GM̃Cϱ (∆))c. (5) If ∆c ∈ G, then GM̃Cϱ (∆) = ℵ and GM̃Cϱ (∆c) = ϕ. (6) GM̃Cϱ (GM̃Cϱ (∆)) ⊒ GM̃Cϱ (∆) and GM̃Cϱ (GM̃Cϱ (∆)) ⊑ GM̃Cϱ (∆), ∀ϱ ∈ {r, ⟨r⟩, l, ⟨l⟩, i, ⟨i⟩}. (7) GM̃Cϱ (Cϱ(n)) ⊒ Cϱ(n) ∀ϱ ∈ {r, ⟨r⟩, l, ⟨l⟩, i, ⟨i⟩}. Proof. (1) GM̃Cϱ (ℵ) = {b ∈ ℵ : Cϱ(b) \ ℵ = ϕ /∈ G} = ℵ and GM̃Cϱ (ϕ) = {b ∈ ℵ : Cϱ(b) ⊓ ϕ /∈ G} = ϕ. (2) Clear. (3) From (ii) we notice that GM̃Cϱ (D ⊓ ∆) ⊑ GM̃Cϱ (D) ⊓ GM̃Cϱ (∆). In contrast, sup- pose b ∈ GM̃Cϱ (D) ⊓ GM̃Cϱ (∆). Then b ∈ GM̃Cϱ (D) and b ∈ GM̃Cϱ (∆) it indicates that GCϱ(b)\D ∈ G and GCϱ(b)\∆ ∈ G. So, GCϱ(b)\(D ⊓ ∆) ∈ G. Thus, b ∈ GM̃Cϱ (D ⊓ ∆). Hence, GM̃Cϱ (D) ⊓ GM̃Cϱ (∆) ⊑ GM̃Cϱ (D ⊓ ∆). Similarly, it may be demonstrated that A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 11 of 31 GM̃Cϱ (D ⊔ ∆) = GM̃Cϱ (D) ⊔ GM̃Cϱ (∆). (4) b ∈ GM̃Cϱ (∆c) ⇔ Cϱ(b)\∆c ∈ G ⇔ Cϱ(b) ⊓ ∆ ∈ G ⇔ b ̸= GM̃Cϱ (∆) ⇔ b ∈ (GM̃Cϱ (∆))c. Likewise, it can be demonstrated GM̃Cϱ (∆c) = (GM̃Cϱ (∆))c. Let b ∈ GM̃Cϱ (∆c) ⇒ Cϱ(b)\∆c /∈ G ⇔ Cϱ(b) ⊓ ∆ /∈ G ⇔ b ∈ (GM̃Cϱ (∆))c. (5) Let ∆c ∈ G. For any b ∈ ℵ, Cϱ(b)\∆ = Cϱ(b) ⊓ ∆c ∈ G. Hence, GM̃Cϱ (∆) = ℵ. From (4), GM̃Cϱ (∆c) = ϕ. (6) Suppose ϱ ∈ {r, ⟨r⟩, l, ⟨l⟩, i, ⟨i⟩}. We will just demonstrate GM̃Cϱ (GM̃Cϱ (∆)) ⊑ GM̃Cϱ (∆). Let b ∈ GM̃Cϱ (GM̃Cϱ (∆)), then Cϱ(b)⊓GM̃Cϱ (∆) ∈ G. Hence, Cϱ(b)⊓GM̃Cϱ (∆) ̸= ϕ i.e. there exists m ∈ ℵ s.t. m ∈ Cϱ(b), and m ∈ GM̃Cϱ (∆). This results in that Cϱ(m)⊓∆ ∈ G. Con- sidering Corollary 2.20, Cϱ(m) = Cϱ(b). Consequently, Cϱ(b)⊓∆ ∈ G and so b ∈ GM̃Cϱ (∆). (7) Suppose ϱ ∈ {r, ⟨r⟩, l, ⟨l⟩, i, ⟨i⟩}. Let b ∈ Cϱ(n). In line with Corollary 2.20, Cϱ(n) = Cϱ(b). Then, Cϱ(b)\Cϱ(n) ̸= ϕ ∈ G, and so b ∈ GM̃Cϱ (Cϱ(n)), and Cϱ(n) ⊑ GM̃Cϱ (Cϱ(n)). It is clear that the following implication follows from points (2) of Theorem 3.3. Corollary 5. A grill on a ϱ-NS (ℵ,Γ, ξϱ) is denoted by G. For any ϱ, the following ideas are true if D,∆ ⊑ ℵ: (1) GM̃Cϱ (D) ⊔ GM̃Cϱ (∆) ⊑ GM̃Cϱ (D ⊔ ℵ). (2) GM̃Cϱ (D ⊓ ∆) ⊑ GM̃Cϱ (D) ⊓ GM̃Cϱ (∆). Proposition 9. Let G be a grill on a ϱ-NS (ℵ,Γ, ξϱ). If D,∆ ⊑ ℵ, then (1) GM̃Cu (∆) ⊑ GM̃Cr (∆) ⊓ GM̃Cl (∆) ⊑ GM̃Cr (∆) ⊔ GM̃Cl (∆) ⊑ GM̃Ci (∆). (2) GM̃Ci (∆) ⊑ GM̃Cr (∆) ⊓ GM̃Cl (∆) ⊑ GM̃Cr (∆) ⊔ GM̃Cl (∆) ⊑ GM̃Cu (∆). (3) GM̃C⟨u⟩ (∆) ⊑ GM̃C⟨r⟩ (∆) ⊓ GM̃C⟨l⟩ (∆) ⊑ GM̃C⟨r⟩ (∆) ⊔ GM̃C⟨l⟩ (∆) ⊑ GM̃C⟨i⟩ (∆). (4) GM̃C⟨i⟩ (∆) ⊑ GM̃C⟨r⟩ (∆) ⊓ GM̃C⟨l⟩ (∆) ⊑ GM̃C⟨r⟩ (∆) ⊔ GM̃C⟨l⟩ (∆) ⊑ GM̃C⟨u⟩ (∆). Proof. This derives from Proposition 2.14 (1). A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 12 of 31 Proposition 10. On a ϱ-NS (ℵ,Γ, ξϱ), let G be a grill, and let Γ be a symmetric relation. Then, ∀∆ ⊑ ℵ, all GM̃Cϱ (∆)(GM̃Cϱ (∆)) are equal. Proof. This derives from Proposition 2.14(2). To demonstrate this, we provide the following example: (1) In general, the opposite of items (2), (6), and (7) in Theorem 3.3 is not success- ful, (2) Corollary 3.4’s converse isn’t always true, (3) The subsets GM̃Cϱ (Cϱ(b)) and Cϱ(b) are independent of one another in the case of ϱ ∈ {u, ⟨i⟩}, (4) Proposition 3.5’s opposite need not be true, and (5) The current approach violates a few of Pawlak’s paradigm’s properties. Example 1. Consider the binary relation Γ = {(b, n), (n, n), ((n,m), (m, e)} on ℵ = {b, e, n,m}. The cardinality nbds for each element of ℵ in Table 1 are then calculated. Table 1: Cϱ-nbds for members of ℵ. b n m e Cr {b,m} {n} {b,m} {e} Cl {b} {n} {em} {e,m} Ci {b} {n} {m} {e} Cu {b,m} {n} {b, e,m} {e,m} C⟨r⟩ {b} {n, e} {m} {n, e} C⟨l⟩ {b} {n,m} {n,m} {e} C⟨i⟩ {b} {n} {m} {e} C⟨u⟩ {b} {e, n,m} {n,m} {n, e} If G = 2ℵ\ϕ, then for each ϱ, the GM̃Cϱ (∆), GM̃Cϱ (∆) are computed in Table 2 and Table 3. It can now be seen as follows: Currently, the following is visible: (1) GM̃Cϱ ({D}) ⊑ GM̃Cϱ ({∆}) for each ϱ, and GM̃Cϱ ({D}) ⊑ GM̃Cϱ ({∆}) whereas {D} ⊑ {∆}. (2) GM̃Cϱ ({n} ⊔ {e}) = {n, e} ⊈ GM̃Cϱ ({n}) ⊔ GM̃Cϱ ({e}) = ϕ. (3) GM̃C⟨r⟩{(n)} ⊓ GM̃C⟨r⟩{(e)} = {e} ⊈ GM̃C⟨r⟩{(n) ⊓ {e}} = ϕ. A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 13 of 31 Table 2: The Ap. for {r, l, i, u}. ∆ GM̃Cr (∆) GM̃Cr (∆) GM̃Cl (∆) GM̃Cl (∆) GM̃Ci (∆) GM̃Ci (∆) GM̃Cu (∆) GM̃Cu (∆) {b} ϕ {b, n} {b} {b} {b} {b} ϕ {b,m} {n} ϕ {n} {n} {n} {n} {n} {n} {n} {m} ϕ {b,m} ϕ {m, e} {m} {m} ϕ {b,m, e} {e} {e} {e} ϕ {m, e} {e} {e} ϕ {m, e} {b, n} {n} {b, n,m} {b, n} {b, n} {b, n} {b, n} {n} {b, n,m} {b,m} {b,m} {b,m} {b} {b,m, e} {b,m} {b,m} {b} {b,m, e} {b, e} {e} {b,m, e} {b} {b, e,m} {b, e} {b, e} ϕ {b,m, e} {n,m} {n} {b, n,m} {n} {n,m, e} {n,m} {n,m} {n} ℵ {n, e} {n, e} {n, e} {n} {n,m, e} {n, e} {n, e} {n, e} {n, e} {m, e} {e} {b,m, e} {m, e} {m, e} {m, e} {m, e} {e} {b,m, e} {b, n,m} {b, n,m} {b, n,m} {b, n} ℵ {b, n,m} {b, n,m} {b, n} ℵ {b, n, e} {n, e} ℵ {b, n} ℵ {b, n, e} {b, n, e} {n} ℵ {b,m, e} {b,m, e} {b,m, e} {b,m, e} {b,m, e} {b,m, e} {b,m, e} {b,m, e} {b,m, e} {n,m, e} {n, e} ℵ {n,m, e} {n,m, e} {n,m, e} {n,m, e} {n} ℵ ℵ ℵ ℵ ℵ ℵ ℵ ℵ ℵ ℵ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ The primary benefit of the current models over the earlier models presented in [24] is their ability to minimize the Br by enlarging the Lw-Ap and downsizing the Lw-Ap of subsets. This is demonstrated by the following outcome. Theorem 3. Let G be a grill on a ϱ-NS (ℵ,Γ, ξϱ) and let D ⊑ ℵ. We have the subsequent relations for each ϱ. (1) MCϱ(D) ⊑ GM̃Cϱ (D), (2) GM̃Cϱ ({D}) ⊑ MCϱ(D). Proof. Let π ∈ MCϱ(D). Then, Cϱ(π) ⊑ D, and Cϱ(π) ⊓Dc /∈ G. Currently, we have π ∈ GM̃Cϱ (D). Hence, MCϱ(D) ⊑ GM̃Cϱ (D). A similar argument can be used to prove the second statement. To demonstrate why the reverse of the preceding theorem fails in the first sentence, we shall provide an example. Example 2. Let D = {m}. Then, MCϱ(D) = ϕ, whereas GM̃Cϱ (D) = {m}. In the following, we show some of the models’ weaknesses. Remark 2. Let G ̸= 2ℵ\ϕ be a grill on a ϱ-NS (ℵ,Γ, ξϱ) and let D,∆ ⊑ ℵ. Some short- comings of the present preliminary set models are illustrated in the following assertions. (1) GM̃Cϱ (D) ⊈ D ⊈ GM̃Cϱ ({D}). A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 14 of 31 Table 3: The Ap. for {⟨r⟩, ⟨l⟩, ⟨i⟩, ⟨u⟩}. ∆ GM̃C⟨r⟩ (∆) GM̃C⟨r⟩ (∆) GM̃C⟨l⟩ (∆) GM̃C⟨l⟩ (∆) GM̃C⟨i⟩ (∆) GM̃C⟨i⟩ (∆) GM̃C⟨u⟩ (∆) GM̃C⟨u⟩ (∆) {b} {b} {b} {b} {b} {b} {b} {b} {b} {n} ϕ {n, e} ϕ {n,m} {n} {n} ϕ {n,m, e} {m} {m} {n,m} ϕ {n,m} {m} {m} ϕ {n,m} {e} ϕ {e} {e} {e} {e} {e} ϕ {n, e} {b, n} {b} {b, n, e} {b} {b, n,m} {b, n} {b, n} {b} ℵ {b,m} {b,m} {b, n,m} {b} {b, n,m} {b,m} {b,m} {b} {b, n,m} {b, e} {b} {b, e} {b, e} {b, e} {b, e} {b, e} {b} {b, n, e} {n,m} {n,m} {n,m, e} {n,m} {n,m} {n,m} {n,m} {m} {n,m, e} {n, e} {n, e} {n, e} {e} {n,m, e} {n, e} {n, e} {e} {n,m, e} {m, e} {m, e} {n,m, e} {m, e} {n,m, e} {m, e} {m, e} {e} {n,m, e} {b, n,m} {b, n,m} ℵ {b, n,m} {b, n,m} {b, n,m} {b, n,m} {b,m} ℵ {b, n, e} {b, e} {b, n, e} {b, e} ℵ {b, n, e} {b, n, e} {b, e} {b, n, e} {b,m, e} {b,m} ℵ {b, e} ℵ {b,m, e} {b,m, e} {b} ℵ {n,m, e} {n,m, e} {n,m, e} {n,m, e} {n,m, e} {n,m, e} {n,m, e} {n,m, e} {n,m, e} ℵ ℵ ℵ ℵ ℵ ℵ ℵ ℵ ℵ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ (2) GM̃Cϱ (ϕ) ̸= ϕ. (3) GM̃Cϱ (ℵ) ̸= ℵ. (4) GM̃Cϱ (GM̃Cϱ (∆)) ̸= GM̃Cϱ (∆) for ϱ ∈ {⟨r⟩, ⟨u⟩}. (5) Let π ∈ ℵ. Then, GM̃Cϱ (Cϱ(π)) ⊈ Cϱ(π)∀ϱ. Example 3.11 illustrates property (1 → 5) of Remark 3.10 Example 3. In Example 3.7. Let G = {{b}, {b, e}, {b, n}, {b,m}, {b, e, n}, {b, e,m}, {b, n,m},ℵ} be a grill on a ϱ-NS (ℵ,Γ, ξϱ) (1) If ∆ = {b, n,m}, then GM̃Cr (∆) = ℵ ⊈ ∆, (2) If ∆ = {b, e}, then ∆ ⊈ {b,m} = GM̃Cϱ ({∆}). (3) GM̃Cr (ϕ) = {n, e} ̸= ϕ. (4) GM̃Cr (ℵ) = {b,m} ̸= ℵ. (5) GM̃Cr ({n}) = {n, e} ⊈ {n},GM̃Ci ({n}) = {n,m, e} ⊈ {n},GM̃Cu ({e}) = {n, e} ⊈ {e}. Hence, ∀π ∈ ℵ,GM̃Cϱ ({π}) ⊈ Cϱ({π}) A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 15 of 31 (6) GM̃C⟨r⟩ (GM̃C⟨r⟩ ({e})) = {n, e} ̸= {n,m, e} = GM̃C⟨r⟩ ({e}). Proposition 11. Let G,J be grills on a ϱ-NS (ℵ,Γ, ξϱ). If D,∆ ⊑ ℵ, G ⊑ J then, the following claims are true for all ϱ. (1) GM̃Cϱ (D) ⊑ JM̃Cϱ (D). (2) GM̃Cϱ (D) ⊑ GM̃Cϱ (D). Proof. This is evident. Remark 3. Let G = {{b}, {b, e}, {b, n}, {b,m}, {b, e, n}, {b, e,m}, {b, n,m},ℵ}, and J = 2ℵ\ϕ be grills on a ϱ-NS (ℵ,Γ, ξϱ). (1) If D = {b}, then GM̃Cr (D) = ℵ ⊈ ϕ = JM̃Cr (D). (2) If D = {b, n}, then GM̃Cr (D) = {b, n,m} ⊈ {b,m} = GM̃Cr (D). 3.2. The second class of Rs paradigms Some drawbacks and unacceptable characteristics of the first category of Rs paradigms include: The property that GM̃Cϱ (D) ⊆ D ⊆ GM̃Cϱ ({D}) does not apply to all subsets, consequently, particularly when GM̃Cϱ (ϕ) ̸= ϕ and GM̃Cϱ (ℵ) ̸= ℵ, results in irrational de- scriptions of those basic set models or mistrust of the information gleaned from them. Since the original accuracy measure formula yields values greater than one or undefined cases for certain subsets, we are unable to apply it, i.e. in Example 3.11, we have |GM̃Cr ({b})| |GM̃Cr ({b})| = 4 2 > 1, and |GM̃Cr (ϕ)| |GM̃Cr (ϕ)| = 2 0 . These situations have little practical significance and are worthless. This section aims to improve the initial Rs paradigm by enhancing Lw-Ap and Up-Ap, while maintaining its advantages. Definition 18. Let G be a grill on a ϱ-NS (ℵ,Γ, ξϱ). Based on grills and cardinality nbds, the GCϱLw −Ap (GMCϱ ()), and GCϱUp-Ap GMCϱ () of ∆ ⊑ ℵ are respectively computed by: GMCϱ (∆) = {δ ∈ ∆ : Cϱ(δ) ⊓ ∆c /∈ G} GMCϱ (∆) = GM̃Cϱ (∆) ⊔ ∆ A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 16 of 31 Definition 19. The GCϱ-boundary, GCϱ-positive, and GCϱ-negative regions of a subset ∆ within a ϱ-NS (ℵ,Γ, ξϱ) with grill G on ℵ are respectively computed by: GBCϱ (∆) = GMCϱ (∆) \ GMCϱ (∆) GPCϱ (∆) = GMCϱ (∆) GNCϱ (∆) = ℵ\GMCϱ (∆) The following measurements can be used to numerically characterize Rss in relation to Cϱ-nbds and grills. Definition 20. The GCϱ-accuracy, and GCϱ-roughness criteria of ∆ ̸= of a ϱ-NS (ℵ,Γ, ξϱ) with grill G on ℵ are provided, respectively, by: GACϱ (∆) = |GMCϱ (∆)| |G MCϱ (∆)| , |GMCϱ (∆)| ̸= 0. GRCϱ (∆) = 1 − GACϱ (∆). The statement of Pawlak’s properties using the GCϱ-Lw and GCϱ-Up-Ap will be looked at in the following theorem. Theorem 4. For a ϱ-NS (ℵ,Γ, ξϱ) and D,∆ ⊑ ℵ, let G = 2ℵ\ϕ be a grill. Next, we have the properties listed below. (1) GMCϱ (∆) ⊑ ∆ ⊑ GMCϱ (∆). (2) GMCϱ (ϕ) = ϕ, and GMCϱ (ϕ) = ϕ. (3) GMCϱ (ℵ) = ℵ, and GMCϱ (ℵ) = ℵ. (4) If D ⊑ ∆, then GMCϱ (D) ⊑ GMCϱ (∆), and GMCϱ (D) ⊑ GMCϱ (∆). (5) GMCϱ (GMCϱ (∆)) = GMCϱ (∆) for ϱ ∈ {r, l, i, ⟨l⟩, ⟨i⟩}. (6) GMCϱ (GMCϱ (∆)) ⊑ GMCϱ (∆) for ϱ ∈ {u, ⟨u⟩, ⟨r⟩}. (7) Let π ∈ ℵ. Then GMCϱ (Cϱ(π)) = Cϱ(π) for ϱ ∈ {r, l, i, ⟨l⟩, ⟨i⟩}. (8) Let π ∈ ℵ. Then GMCϱ (Cϱ(π)) ⊑ Cϱ(π) for ϱ ∈ {u, ⟨u⟩, ⟨r⟩}. A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 17 of 31 (9) MCϱ(MCϱ(∆)) = MCϱ(∆) for ϱ ∈ {r, l, i, ⟨l⟩, ⟨i⟩}. (10) MCϱ(MCϱ(∆)) ⊒ MCϱ(∆) for ϱ ∈ {u, ⟨u⟩, ⟨r⟩}. (11) GMCϱ (D) ⊓ GMCϱ (∆) = GMCϱ (D ⊓ ∆) for ϱ. (12) GMCϱ (D) ⊔ GMCϱ (∆) = GMCϱ (D ⊔ ∆) for ϱ. Proof. Validating (1), (2), (3), (4), (11), and (12) is simple in accordance with Defini- tion 3.14. (5) Let ϱ be a member of {r, l, i, ⟨l⟩, ⟨i⟩}. Using the current theorem’s characteristics (1) and (4), GMCϱ (GMCϱ (∆)) ⊑ GMCϱ (∆). The GMCϱ (GMCϱ (∆)) = GMCϱ (∆) is the result of utilizing (6) of Theorem 3.3 to show the other direction. (6) Let ϱ ∈ {u, ⟨u⟩, ⟨r⟩}. Using the current theorem’s characteristics (1) and (4), GMCϱ (GMCϱ (∆)) ⊑ GMCϱ (∆). (7) Let ϱ ∈ {r, l, i, ⟨l⟩, ⟨i⟩}. Applying the current theorem’s property (1), we have GMCϱ (Cϱ(π)) ⊑ Cϱ(π), ∀π ∈ ℵ. Theorem 2’s (7) is used to demonstrate the opposite direction. Thus, GMCϱ (Cϱ(π)) = Cϱ(π). (8) Assume that ϱ ∈ {u, ⟨u⟩, ⟨r⟩}. By property (1) of the current theorem, we, have GMCϱ (Cϱ(π)) ⊑ Cϱ(π)∀π ∈ ℵ. (9) Identical to evidence of property (5). (10) Identical to evidence of property (5). Proposition 12. For a ϱ-NS (ℵ,Γ, ξϱ) and ∆ ⊑ ℵ, let G be a grill, then (1) GMCu (∆) ⊑ GMCr (∆) ⊓ GMCl (∆) ⊑ GMCr (∆) ⊔ GMCl (∆) ⊑ GMCi (∆). (2) GMCi (∆) ⊑ GMCr (∆) ⊓ GMCl (∆) ⊑ GMCr (∆) ⊔ GMCl (∆) ⊑ GMCu (∆). (3) GMC⟨u⟩ (∆) ⊑ GMC⟨r⟩ (∆) ⊓ GMC⟨l⟩ (∆) ⊑ GMC⟨r⟩ (∆) ⊔ GMC⟨l⟩ (∆) ⊑ GMC⟨i⟩ (∆). (4) GMC⟨i⟩ (∆) ⊑ GMC⟨r⟩ (∆) ⊓ GMC⟨l⟩ (∆) ⊑ GMC⟨r⟩ (∆) ⊔ GMC⟨l⟩ (∆) ⊑ GMC⟨u⟩ (∆). Proof. Derived from Definition 3.14 and Proposition 3.5. Corollary 6. For a ϱ-NS (ℵ,Γ, ξϱ) and ∆ ⊑ ℵ, let G be a grill, then A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 18 of 31 (1) GACu (∆) ≤ GACr (∆) ≤ GACi (∆). (2) GACu (∆) ≤ GACl (∆) ≤ GACi (∆). (3) GAC⟨u⟩ (∆) ≤ GAC⟨r⟩ (∆) ≤ GAC⟨i⟩ (∆). (4) GAC⟨u⟩ (∆) ≤ GAC⟨l⟩ (∆) ≤ GAC⟨i⟩ (∆). Proposition 13. If ∆ is a nonempty subset of ℵ, then for any ϱ, 0 ≤ GACϱ (∆) ≤ 1. Proof. Derives from the reality that GMCϱ (∆) ⊑ ∆ ⊑ GMCϱ (∆). Definition 21. We refer to a subset ∆ GCϱ-exact if GACϱ (∆) = 1. Otherwise it’s known GCϱ-rough. The opposite side of Corollary 3.19 generally fails, as the following example shows. Example 4. Extended in Example 3.11. Tables 4 and 5 calculate the accuracy criteria GACϱ (∆) for each ϱ. Table 4: GACϱ (∆) for ϱ ∈ {r, l, i, u}. ∆ GACr (∆) GACl (∆) GACi (∆) GACu (∆) {b} 1 1 1 1 {n} 1 1 1 1 {m} 1 1 1 0 {e} 1 1 1 1 {b, n} 1 1 1 1 {b,m} 1 1 1 1 {b, e} 1 1 1 1 {n,m} 1 2 1 1 1 2 {n, e} 1 1 1 1 {m, e} 1 2 1 1 1 {b, n,m} 1 1 1 1 {b, n, e} 1 1 1 1 {b,m, e} 1 1 1 1 {n,m, e} 2 3 1 1 1 ℵ 1 1 1 1 The following theorem describes how the current models outperform the models pro- vided in [24] in terms of App operators. They clearly boost the accuracy measures of subsets and make a true shrink (or removal) for the Brs. Theorem 5. Let G be a grill for a ϱ-NS (ℵ,Γ, ξϱ) and D ⊑ ℵ. Consequently, the following assertion is valid for all ϱ. A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 19 of 31 Table 5: GACϱ (∆) for ϱ ∈ {⟨r⟩, ⟨l⟩, ⟨i⟩, ⟨u⟩}. ∆ GAC⟨r⟩ (∆) GAC⟨l⟩ (∆) GAC⟨i⟩ (∆) GAC⟨u⟩ (∆) {b} 1 1 1 1 {n} 1 1 1 1 {m} 1 1 1 1 {e} 1 1 1 1 {b, n} 1 1 1 1 {b,m} 1 1 1 1 {b, e} 1 1 1 1 {n,m} 1 1 1 1 {n, e} 1 1 1 1 {m, e} 1 1 1 1 {b, n,m} 1 1 1 1 {b, n, e} 1 1 1 1 {b,m, e} 1 1 1 1 {n,m, e} 1 1 1 1 ℵ 1 1 1 1 (1) MCϱ(D) ⊑ GMCϱ (D). (2) GMCϱ (D) ⊑ MCϱ(D). Proof. By using Theorem 3.8, then MCϱ(D) ⊑ GM̃Cϱ (D). Since MCϱ(D) ⊑ D, then MCϱ(D) ⊑ GMCϱ (D). It is possible to prove the second statement by following the same logic. Corollary 7. For a ϱ-NS (ℵ,Γ, ξϱ) and D ⊑ ℵ, let G be a grill, then ACϱ(D) ≤ GACϱ (∆)∀ϱ. To demonstrate that the inverse of the aforementioned theorem and corollary fails, use the following example. Example 5. Extending from Example 3.11. If ∆ = {b, n}, then GMCϱ (∆),GMCϱ (∆), and GACϱ (∆) are computed for ϱ ∈ {r, u} as follows: (1) GMCϱ (∆) = {b, n}, GMCϱ (∆) = {b, n}, and GACϱ (∆) = 1. (2) MCϱ(∆) = {n}, MCϱ(∆) = {b, n,m}, and GACϱ (∆) = 1 3 . Proposition 14. For any ϱ, suppose that G is a grill on a ϱ-NS (ℵ,Γ, ξϱ). If ∆ ⊑ ℵ, then A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 20 of 31 (1) GMCϱ (∆) ⊑ GM∦ϱ (∆). (2) GM∦ϱ (∆) ⊑ GMCϱ (∆). (3) GACϱ (∆) ≤ GA∦ϱ (∆). Proof. (1) Let δ ∈ GMCϱ (∆), then Cϱ(δ) ⊓ ∆c /∈ G (by Definition 4.1), ∦ϱ(δ) ⊓ ∆c /∈ G. This is show that δ ∈ GM∦ϱ (∆). (2) Let δ ∈ GM∦ϱ (∆), i.e., ∦ϱ ⊓ ∆ ∈ G, Hence, Cϱ ⊓ ∆ ∈ G, δ ∈ GM̃Cϱ (∆). So, δ ∈ GMCϱ (∆),(by Definition 3.14). (3) The evidence is obvious. Remark 4. Let G1, and G2 be grills on a ϱ-NS (ℵ,Γ, ξϱ), ∆ ⊑ ℵ. If G1 ⊑ G2 then {G1}ACϱ (∆) ≥ {G2}ACϱ , for each ϱ. 4. Various topologies created using grills and cardinality nbds In this section, for any given relation, we use grills and cardinal nbds to build a variety of topologies that are finer than those previously produced by cardinal nbds as detailed in [46]. Theorem 6. For a ϱ-NS (ℵ,Γ, ξϱ), let G be a grill. The family GΨCϱ = {∆ ⊑ ℵ : ∀δ ∈ ∆, Cϱ(δ) ⊓ ∆c /∈ G} a topology on ℵ. Proof. First, for any ϱ, it is obvious that ℵ, ϕ ∈ GΨCϱ . Second, define ∆1,∆2 as elements of GΨCϱ , and δ ∈ ∆1 ⊓ ∆2. Then Cϱ(δ) ⊓ ∆c 1 /∈ G, and Cϱ(δ) ⊓ ∆c 2 /∈ G. Hence, Cϱ(δ) ⊓ (∆1 ⊓ ∆2) c /∈ G. This implies that ∆1 ⊓ ∆2 ∈ GΨCϱ . Lastly, for each i ∈ I, assume that ∆i belonged to GΨCϱ . Let δ ∈ ⊔i∈I∆i, then there is i0 ∈ I s.t. δ ∈ ∆i0 and Cϱ(δ)⊓∆c i0 /∈ G. Since ∆i0 ⊑ ⊔i∈I∆i. Then, Cϱ(δ)⊓ (⊔i∈I∆i) c /∈ G, and ⊔i∈I∆i ∈ GΨCϱ . The GΨCϱ -open sets are the members of GΨCϱ , and the GΨCϱ -closed sets are its comple- ment. Proposition 15. For any ϱ, suppose that G is a grill on a ϱ-NS (ℵ,Γ, ξϱ). Then (1) For each ϱ, ΨCϱ ⊑ GΨCϱ . (2) For each ϱ, GΨCϱ ⊑ GΨ∦ϱ . Proof. (1) Cϱ(e) ⊓Dc /∈ G ∀e ∈ D is implied by the fact that Cϱ(e) ⊑ D ∀e ∈ D. (2) According to Proposition 2.24, for any ϱ, we have ∥ϱ(e) ⊑ Cϱ(e). Therefore, using the grill property, we determine that Cϱ(e) ⊓Dc /∈ G. A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 21 of 31 Example 6. From Example 3.11, we have: ΨCr = {{n}, {e}, {n, e}, {b,m}, {b,m, n}, {b,m, e},ℵ, ϕ}. GΨCr = {{n}, {b}, {e}, {b, n}, {b,m}, {b, e}, {n, e}, {b,m, n}, {b, n, e}, {b,m, e},ℵ, ϕ}. ΨCl = {{n}, {b}, {m, e}, {b, n}, {b,m, e}, {n,m, e},ℵ, ϕ}. GΨCl = 2ℵ. ΨCi = 2ℵ. GΨCi = 2ℵ. ΨCu = {{n}, {b,m, e},ℵ, ϕ}. GΨCu = {{n}, {b}, {e}, {b, n}, {b,m}, {b, e}, {n, e}, {b,m, n}, {b, n, e}, {b,m, e},ℵ, ϕ}. ΨC⟨r⟩ = {{b}, {m}, {n, e}, {b,m}, {b, n, e}, {n,m, e},ℵ, ϕ}. GΨC⟨r⟩ = 2ℵ. ΨC⟨l⟩ = {{b}, {e}, {n,m}, {b, e}, {n,m, e}, {n,m, b},ℵ, ϕ}. GΨC⟨l⟩ = 2ℵ. ΨC⟨i⟩ = 2ℵ. GΨC⟨i⟩ = 2ℵ. ΨC⟨u⟩ = {{b}, {n,m, e}, ϕ,ℵ}. GΨC⟨u⟩ = 2ℵ. Lemma 2. Let G,J be grills on a ϱ-NS (ℵ,Γ, ξϱ) such that G ⊑ J for any ϱ. JΨCϱ ⊑ GΨCϱ follows. Proof. Straight to prove. The following example shows that Lemma 4.4’s inverse implication is not always true. Example 7. Keeping with Example 3.11. Let G = {{b}, {b, e}, {b, n}, {b,m}, {b, e, n}, {b, e,m}, {b, n,m},ℵ}, and J = 2ℵ\ϕ, and ϱ = r. A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 22 of 31 Then, GΨCr = {{n}, {b}, {e}, {n, b}, {b,m}, {b, e}, {n, e}, {b, n,m}, {b, n, e}, {b,m, e},ℵ, ϕ} ⊈ {{n}, {e}, {b,m}, {n, e}, {b, n,m}, {b,m, e},ℵ, ϕ} = JΨCr . Theorem 7. Topologies satisfy the following relations: (1) GΨCu ⊑ GΨCr ⊓ GΨCl ⊑ GΨCr ⊔ GΨCl ⊑ GΨCi . (2) GΨC⟨u⟩ ⊑ GΨC⟨r⟩ ⊓ GΨC⟨l⟩ ⊑ GΨC⟨r⟩ ⊔ GΨC⟨l⟩ ⊑ GΨC⟨i⟩ . Proof. Proposition 2.14’s first item justifies these relationships. Example 4.3 displays that GΨCr ̸= GΨCl , GΨCr ̸= GΨCi , GΨCu ̸= GΨCl , GΨCu ̸= GΨCi , GΨCr = GΨCu , GΨCi = GΨCl , and GΨC⟨u⟩ = GΨC⟨r⟩ = GΨC⟨l⟩ = GΨC⟨i⟩ . Next, crude Aps will be created using topologies based on grills and cardinal nbds. Additionally, the features of these crude Aps will be examined. Definition 22. Let GΨCϱ represent a topology induced by grills and cardinality nbds. Then, for each ϱ, Lw and Up-Ap of a st ∆ ⊑ ℵ are respectively given by: Gξϱ(∆) = GintCϱ (∆), Gξϱ(∆) = GclCϱ (∆), where GintCϱ (∆), GclCϱ (∆) respectively represent the interior and closure of a set ∆ with respect the topology GΨCϱ . Furthermore, ∆’s accuracy criterion is set as follows: Gλξϱ (∆) = |Gξϱ (∆)| |Gξϱ (∆)| , |Gξϱ(∆)| ̸= 0. 0 ≤ Gλξϱ ≤ 1 is clearly. ∆ is called a GCϱ-exact set if Gλξϱ (∆) = 1. On the other hand, ∆ is referred to as a GCϱ-Rs. Concerning to Definition 4.7, the following results can be proven using the topological characteristics of interior and closure operators. It is noteworthy that certain properties absent in the GM̃Cϱ -,GM̃Cϱ -Ap are still valid for the Gξϱ-, Gξϱ-Ap that as item (1) of Theorem 3.3. Theorem 8. For each ϱ, let GΨCϱ represent a topology induced by grills and cardinality nbds, and let D,∆ ⊑ ℵ. Next are the following properties: (1) Gξϱ(∆) ⊑ ∆. (2) Gξϱ(ϕ) = ϕ. (3) Gξϱ(ℵ) = ℵ. (4) If D ⊑ ∆, then Gξϱ(D) ⊑ Gξϱ(∆). (5) Gξϱ(D ⊓ ∆) = Gξϱ(D) ⊓ Gξϱ(∆). (6) Gξϱ(∆c) = (Gξϱ(∆))c. (7) Gξϱ(Gξϱ(∆)) = Gξϱ(∆) ∀ϱ. A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 23 of 31 Proof. These connections are true because they correspond to interior topology and Lw-Ap operators. Corollary 8. For each ϱ, let GΨCϱ represent a topology induced by grills and cardinality nbds. Then, Gξϱ(D) ⊔ Gξϱ(∆) ⊑ Gξϱ(D ⊔ ∆) for any D,∆ ⊑ ℵ. Theorem 9. For each ϱ, let GΨCϱ represent a topology induced by grills and cardinality nbds, and let D,∆ ⊑ ℵ. Next are the following properties: (1) ∆ ⊑ Gξϱ(∆). (2) Gξϱ(ϕ) = ϕ. (3) Gξϱ(ℵ) = ℵ. (4) If D ⊑ ℵ, then Gξϱ(D) ⊑ Gξϱ(∆). (5) Gξϱ(D ⊔ ∆) = Gξϱ(∆) ⊔ Gξϱ(∆). (6) Gξϱ(∆c) = (Gξϱ(∆))c. (7) Gξϱ(Gξϱ(∆)) = Gξϱ(∆) ∀ϱ. Proof. These connections are true because they correspond to closure topology and Up-Ap operators. Corollary 9. For each ϱ, let GΨCϱ represent a topology induced by grills and cardinality nbds. Then, Gξϱ(D ⊓ ∆) ⊑ Gξϱ(D) ⊓ Gξϱ(∆) for any D,∆ ⊑ ℵ. Proposition 16. For ϕ ̸= ∆ ⊑ ℵ, 0 ≤ Gλξϱ (∆) ≤ 1, and Gλξϱ (ℵ) = 1 for each ϱ. Proposition 17. The following inclusion relations are valid for any subset D of a topo- logical space (ℵ,GΨCϱ ): (1) Gξu(D) ⊑ Gξr(D) ⊓ Gξl(D) ⊑ Gξr(D) ⊔ Gξl(D) ⊑ Gξi(D). (2) Gξl(D) ⊑ Gξr(D) ⊓ Gξi(D) ⊑ Gξr(D) ⊔ Gξi(D) ⊑ Gξu(D). (3) Gξ⟨u⟩(D) = Gξ⟨r⟩(D) = Gξ⟨l⟩(D) = Gξ⟨i⟩(D). (4) Gξ⟨u⟩(D) = Gξ⟨r⟩(D) = Gξ⟨l⟩(D) = Gξ⟨i⟩(D). Corollary 10. The following inequalities are valid for any nonempty subset D of a topo- logical space (ℵ,GΨCϱ ): A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 24 of 31 (1) Gλξu (D) ≤ Gλξr (D) ≤ Gλξi (D). (2) Gλξu (D) ≤ Gλξl (D) ≤ Gλξi (D). (3) Gλξ⟨u⟩ (D) = Gλξ⟨r⟩ (D) = Gλξ⟨i⟩ (D) = Gλξ⟨l⟩ (D). The approaches covered in the preceding section will now be compared with the Ap and accuracy standards described in this section, which are based on topological spaces. Proposition 18. The relations for each ϱ and D ⊑ ℵ are as follows: (1) Gξϱ(D) ⊑ GMCϱ (D). (2) GMCϱ (D) ⊑ Gξϱ(D). Proof. (1) Let b ∈ Gξϱ(D). Then we find a subset V ∈ GΨCϱ with b ∈ V ⊑ D. We derive Cϱ(b) ⊓ V c /∈ G from the topology structuring method. Now, we get Cϱ(b) ⊓Dc /∈ G since V ⊑ D. Hence, b ∈ GM̃Cϱ (D). Since b ∈ D, then b ∈ GMCϱ (D), and Gξϱ(D) ⊑ GMCϱ (D). By the same manner, one can prove (2). The converse of Proposition 4.15 need not be true, refer to Table 2 and Example 4.3. Suppose that σ = r and D = {b, n}. Then GMCϱ (D) = ℵ, Gξϱ(D) = {b, n}. Hence,The opposite is therefore untrue. Corollary 11. For any ϱ, suppose that G is a grill on a ϱ-NS (ℵ,Γ, ξϱ). If D ⊑ ℵ, then Gλξϱ (D) ≤ GACϱ (D). 5. Diagnostic analysis of heart failure as a medical application In this section, we assess the efficiency of suggested models in managing heart failure information systems for specific patients [47]. We demonstrate how the current approach improves decision-making and how we use a topological strategy to pinpoint the most important symptoms for identifying heart failure disease. The investigation that follows leads us to the conclusion that the suggested Rs paradigms perform better than their counterparts that are based on cardinality nbds without grills. We also make reference to the restrictions connected to the approach described in Sec. 3.1. Table 6 shows data for eight patients (ℵ = {p1, p2, p3, p4, p5, p6, p7, p8}) and their associated symptoms (con- ditional attributes): breathlessness (Br), orthopnea (Or), paroxysmal nocturnal dyspnea (Pnd), impaired exercise tolerance (Iet), and ankle swelling (As). Heart failure is regarded as the deciding attribute. We add a value of s+ or s− to each conditional property (symp- tom) to indicate whether the patient has the symptom or not. In a similar manner, the decision attribute is labeled with ++ or −− to indicate a heart failure report that is positive or negative. Suppose the system’s expert proposed the next relation Γ on the set of patients ℵ to describe the links between them based on their symptoms: A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 25 of 31 Table 6: Information system for heart failure Patients Br Or Pnd Iet As Decision. p1 s+ s+ s+ s+ s− ++ p2 s− s− s− s+ s+ −− p3 s+ s+ s+ s+ s+ ++ p4 s− s− s− s+ s− −− p5 s+ s− s− s+ s+ −− p6 s− s− s− s+ s− −− p7 s+ s+ s+ s+ s+ ++ p8 s+ s+ s− s+ s+ ++ prΓpt ⇔ there are more than two frequent positive symptoms that distinguish pr from pt. Then, Γ = {(p1, p1), (p3, p3), (p5, p5), (p7, p7), (p8, p8), (p1, p3), (p3, p1), (p1, p7), (p7, p1), (p1, p8), (p8, p1), (p3, p5), (p5, p3), (p3, p7), (p7, p3), (p3, p8), (p8, p3), (p5, p7), (p7, p5), (p5, p8), (p8, p5), (p7, p8), (p8, p7)}. Note that Γ is a symmetric relation that is neither transitive nor reflexive. We start by building the Cϱ-nbd systems in order to process the data that is described by the specified relation. As stated in Proposition 2.14, we deduce that all Cϱ-nbds are similar because of the symmetry of the suggested relation. The Cϱ-nbd for every patient is shown in Table 7. Let G = {{p2, p4, p5, p6}, {p1, p2, p4, p5, p6}, {p2, p3, p4, p5, p6}, {p2, p4, p5, p6, p7}, {p2, p4, p5, p6, p8}, {p1, p2, p3, p4, p5, p6}, {p1, p2, p4, p5, p6, p7}, {p1, p2, p4, p5, p6, p8}, {p2, p3, p4, p5, p6, p7}, {p2, p3, p4, p5, p6, p8}, {p2, p4, p5, p6, p7, p8}, {p1, p2, p3, p4, p5, p6, p7}, {p1, p2, p3, p4, p5, p6, p8}, {p1, p2, p4, p5, p6, p7, p8}, {p2, p3, p4, p5, p6, p7, p8}, ϕ,ℵ}. Using Rs models and [21], we determine approximations Lw,Up, and accuracy for D = {p1, p3, p5}, a set of patients with a positive report of heart failure: (1) MCϱ(D) = {p1, p5}. (2) MCϱ(D) = {p1, p3, p5, p7, p8}. (3) BCϱ(D) = MCϱ(D)\MCϱ(D) = {p3, p7, p8}. (4) ΣCϱ(D) = 2 5 . In Sec 3.1, we showed our preliminary set model. (1) GM̃Cϱ (D) = ℵ, and A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 26 of 31 (2) GM̃Cϱ (D) = ϕ. In Sec. 3.2, we showed our rough set model. GMCϱ (D) = D, (2) GMCϱ (D) = D, (3) GBCϱ (D) = GMCϱ (∆) \ GMCϱ (D) = ϕ, and (4) GACϱ (D) = |GMCϱ (D)| |G MCϱ (D)| = 1. Models of topology developed in Sect.4. In order to use these models, we first set up a topology as shown in Table 7: Table 7: Cϱ for each patient. p1 p2 p3 p4 p5 p6 p7 p8 Mr() {p1, p3, p7, p8} ϕ {p1, p3, p5, p7, p8} ϕ {p3, p5, p7, p8} ϕ {p1, p3, p5, p7, p8} {p1, p3, p5, p7, p8} M⟨r⟩() {p1, p3, p7, p8} ϕ {p3, p7, p8} ϕ {p3, p5, p7, p8} ϕ {p3, p7, p8} {p3, p7, p8} Cϱ() {p1, p5} {p2, p4, p6} {p3, p7, p8} {p2, p4, p6} {p1, p5} {p2, p4, p6} {p3, p7, p8} {p3, p7, p8} GΨCϱ = {ϕ,ℵ, {p1, p5}, {p2, p4, p6}, {p3, p7, p8}, {p1, p2, p4, p5, p6}, {p1, p3, p5, p7, p8}, {p2, p3, p4, p6, p7, p8}}. Then we calculate, in the following items, the Ap(Lw and Up), and accuracy for D: Gξϱ(D) = GintCϱ (D) = {p1, p5}, Gξϱ(D) = GclCϱ (D) = {p1, p5}, BCϱ(D) = GclCϱ (D)\GintCϱ (D) = ϕ, Gλξϱ (∆) = |Gξϱ (∆)| |Gξϱ (∆)| = 1. It is clear from the aforementioned results that these calculations support the conclu- sions made in Theorem 3.23 and Corollary 3.24. In summary, it can be observed that, in contrast to the Rs models discussed by [46], the models suggested in Sec. 3.2 improve the accuracy measures of subsets by strengthening the Lw, and Up-Ap. Furthermore, the Rs paradigms shown in Sec. 3.2 produce the same Ap spaces as those shown in Sec. 4 due to the topological technique. The computations from the four Rs models discussed above will be compared. We discovered that the greatest estimates (Lw and Up) are produced by the model shown in Sec. 3.2. It is important to note that Model 3.1 has certain shortcomings, including the inability to preserve the essential elements of Ap procedures and excessive accuracy of measurements. In the remaining portion of this part, we determine the primary symptoms for determin- ing whether a patient has heart failure disease by using the topological spaces that were previously created using grills and cardinality nbds. The initial topology is: A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 27 of 31 GΨCϱ = {ϕ,ℵ, {p1, p5}, {p2, p4, p6}, {p3, p7, p8}, {p1, p2, p4, p5, p6}, {p1, p3, p5, p7, p8}, {p2, p3, p4, p6, p7, p8}}. Next, we will contrast the topologies produced from the same database after eliminating each symptom separately with the initial topology created from the patient information system shown in Table 6. This procedure will be carried out once again for every symptom. (1) If the symptom ”breathlessness” is removed from the input attributes, then ΓBr = {(p1, p1), (p3, p3), (p7, p7), (p8, p8), (p1, p3), (p3, p1), (p1, p7), (p7, p1), (p3, p7), (p7, p3), (p3, p8), (p8, p3), (p7, p8), (p8, p7)}. It is clear that GΨCϱ -Br ̸= GΨCϱ . (2) If the symptom ”orthopnea” is removed from the input attributes, then ΓOr = Γ. Consequently, GΨCϱ -Br = GΨCϱ . (3) If the symptom ”paroxysmal nocturnal dyspnea” is removed from the input attributes, then ΓPnd = Γ. Consequently, GΨCϱ -Pnd = GΨCϱ . (4) If the symptom ”impaired exercise tolerance” is removed from the input attributes, then ΓIet ̸= Γ. Consequently, GΨCϱ -Pnd ̸= GΨCϱ . (5) If the symptom ”ankle swelling” is removed from the input attributes, then ΓAs ̸= Γ. Consequently, GΨCϱ -Pnd ̸= GΨCϱ . The computations indicate that the main symptoms are Br, Iet, and As. Stated dif- ferently, these symptoms are recognized as the primary markers for identifying if a patient has heart failure disease. On the other hand, eliminating the Or and Br symptom does not change the topology’s structure; as a result, it can be skipped during tests. 6. Conclusion and future work Rsst, proposed by Pawlak in 1982, is a powerful tool for handling inaccurate and confusing information. The capacity of Rst to represent data using granular structure without having any a priori knowledge outside of the data set itself is one of its main advantages. As is well known, neighborhood systems modeled after arbitrary relations have been used to update the detail that equivalency classes reflect. This helps to eliminate a rigid constraint of an equivalence relation. Recent models have failed to maintain the original paradigm’s major elements and have flawed formulas for measuring proven and potential knowledge. In this work, we provide new kinds of generalized Ap spaces that are inspired by the concepts of cardinality nbds and grills. The first kind has proven to be effective in extracting as many details as possible from dataset subsets. Nevertheless, it has flaws in that it violates certain of the original model’s characteristics. In order to get over this problem, we have introduced the second kind of Ap space, which maintains the features of the original model while appreciably expanding the knowledge that has been gathered. To improve the Ap operators, we inferred the associated positive properties of these models and verified their validity. The suggested crude set model is then represented A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 28 of 31 by a topological frame that we have constructed. We have presented the crude topological model’s characteristics and clarified how it differs from its counterpart that is defined without a grill structure. We may conclude that the Rs models used here perform better than the current models based on the analysis presented in the medical case of heart failure condition. Here is our strategy for the future: † To improve accuracy, the existing models are being expanded using various relations. ‡ In connection with the concepts that were given, the concepts of soft and pythagorean fuzzy soft settings will be discussed. †‡ Investigate innovative rough set models produced by Cϱ-neighborhoods and a grill structure created by two grills, G∗ and G∗∗, as outlined below. G . = {X ⊔ Y : X ∈ G∗, Y ∈ G∗∗}. ‡‡ In connection with the concepts that were given, the concepts of soft and Picture fuzzy soft settings will be discussed. Acknowledgements The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/2025/01/34111). Conflict of interest Regarding the publishing of this work, the authors affirm that they have no conflicts of interest. References [1] Zdzis law Pawlak. Rough sets. International journal of computer & information sciences, 11(5):341–356, 1982. [2] Zdzislaw Pawlak. Rough sets and decision analysis. INFOR: Information Systems and Operational Research, 38(3):132–144, 2000. [3] EA Abo-Tabl. A comparison of two kinds of definitions of rough approximations based on a similarity relation. Information Sciences, 181(12):2587–2596, 2011. [4] B Almarri and AA Azzam. Energy saving via a minimal structure. Mathematical Problems in Engineering, 2022(1):5450344, 2022. [5] Jianhua Dai, Shuaichao Gao, and Guojie Zheng. Generalized rough set models de- termined by multiple neighborhoods generated from a similarity relation. Soft Com- puting, 22(7):2081–2094, 2018. [6] Tareq M Al-shami, Wen Qing Fu, and EA Abo-Tabl. New rough approximations based on e-neighborhoods. Complexity, 2021(1):6666853, 2021. [7] AA Azzam. A topological tool to develop novel rough set. Journal of Mathematics and Computer Science, 33(2):204–216, 2024. A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 29 of 31 [8] Jianhua Dai and Qing Xu. Approximations and uncertainty measures in incomplete information systems. Information Sciences, 198:62–80, 2012. [9] Keyun Qin, Jilin Yang, and Zheng Pei. Generalized rough sets based on reflexive and transitive relations. Information sciences, 178(21):4138–4141, 2008. [10] Roman Slowinski and Daniel Vanderpooten. A generalized definition of rough approx- imations based on similarity. IEEE Transactions on knowledge and Data Engineering, 12(2):331–336, 2002. [11] Hua-Peng Zhang, Yao Ouyang, and Zhudeng Wang. Note on “generalized rough sets based on reflexive and transitive relations”. Information Sciences, 179(4):471–473, 2009. [12] AA Allam, MY Bakeir, and EA Abo-Tabl. New approach for basic rough set concepts. In International Workshop on Rough Sets, Fuzzy Sets, Data Mining, and Granular- Soft Computing, pages 64–73. Springer, 2005. [13] AA Allam, MY Bakeir, and EA Abo-Tabl. New approach for closure spaces by relations. Acta Mathematica Academiae Paedagogicae Nyiregyháziensis, 22(3):285– 304, 2006. [14] Huishan Wu and Guilong Liu. The relationships between topologies and generalized rough sets. International Journal of Approximate Reasoning, 119:313–324, 2020. [15] YY Yao. Two views of the theory of rough sets in finite universes. International journal of approximate reasoning, 15(4):291–317, 1996. [16] Mohammed Atef, Ahmed Mostafa Khalil, Sheng-Gang Li, AA Azzam, and Abd El Fattah El Atik. Comparison of six types of rough approximations based on j- neighborhood space and j-adhesion neighborhood space. Journal of Intelligent & Fuzzy Systems, 39(3):4515–4531, 2020. [17] R Mareay. Generalized rough sets based on neighborhood systems and topological spaces. Journal of the Egyptian Mathematical Society, 24(4):603–608, 2016. [18] Abdel Fatah A Azzam. Rough neighborhood ideal and its applications. International Journal of Fuzzy Logic and Intelligent Systems, 24(1):43–49, 2024. [19] Mohamed Abd Elaziz, Hassan M Abu-Donia, Rodyna A Hosny, Saeed L Hazae, and Rehab Ali Ibrahim. Improved evolutionary-based feature selection technique using extension of knowledge based on the rough approximations. Information Sciences, 594:76–94, 2022. [20] Seiki Akama, Tetsuya Murai, and Yasuo Kudo. Reasoning with rough sets. Logical Approaches to Granularity-Based Framework, 2018. [21] Mona Hosny and Tareq M Al-shami. Rough set models in a more general manner with applications. Aims Math, 7(10):18971–19017, 2022. [22] Rodyna A Hosny, Mohamed Abd Elaziz, and Rehab Ali Ibrahim. Enhanced feature selection based on integration containment neighborhoods rough set approximations and binary honey badger optimization. Computational Intelligence and Neuroscience, 2022(1):3991870, 2022. [23] Rodyna A Hosny, Tareq M Al-shami, AA Azzam, and Ashraf S Nawar. Knowledge based on rough approximations and ideals. Mathematical Problems in Engineering, 2022(1):3766286, 2022. A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 30 of 31 [24] Mona Hosny. Generalization of rough sets using maximal right neighborhood systems and ideals with medical applications. AIMS Math, 7(7):13104–13138, 2022. [25] Marzena Kryszkiewicz. Rough set approach to incomplete information systems. In- formation sciences, 112(1-4):39–49, 1998. [26] Roshdey Mareay. Soft rough sets based on covering and their applications. Journal of Mathematics in Industry, 14(1):4, 2024. [27] Antoni Wiweger. On topological rough sets. Bulletin of the Polish Academy of Sciences. Mathematics, 37(1-6):89–93, 1989. [28] EA Abo-Tabl. Rough sets and topological spaces based on similarity. International Journal of Machine Learning and Cybernetics, 4(5):451–458, 2013. [29] Tareq M Al-shami. Improvement of the approximations and accuracy measure of a rough set using somewhere dense sets. Soft Computing, 25(23):14449–14460, 2021. [30] Tareq M Al-shami. Topological approach to generate new rough set models. Complex & Intelligent Systems, 8(5):4101–4113, 2022. [31] EF Lashin, AM Kozae, AA Abo Khadra, and Tamer Medhat. Rough set theory for topological spaces. International Journal of Approximate Reasoning, 40(1-2):35–43, 2005. [32] AS Salama. Topological solution of missing attribute values problem in incomplete information tables. Information Sciences, 180(5):631–639, 2010. [33] MM El-Sharkasy. Minimal structure approximation space and some of its application. Journal of Intelligent & Fuzzy Systems, 40(1):973–982, 2021. [34] Tareq M Al-Shami and Ibtesam Alshammari. Rough sets models inspired by supra- topology structures. Artificial Intelligence Review, 56(7):6855–6883, 2023. [35] Tareq M Al-Shami and Abdelwaheb Mhemdi. Approximation operators and accuracy measures of rough sets from an infra-topology view. Soft Computing, 27(3):1317–1330, 2023. [36] Kamalpreet Kaur, Asha Gupta, Tareq M Al-shami, and M Hosny. A new multi- ideal nano-topological model via neighborhoods for diagnosis and cure of dengue. Computational and Applied Mathematics, 43(7):400, 2024. [37] AS Salama. Bitopological approximation space with application to data reduction in multi-valued information systems. Filomat, 34(1):99–110, 2020. [38] Pankaj Kumar Singh and Surabhi Tiwari. Topological structures in rough set theory: a survey. Hacettepe Journal of Mathematics and Statistics, 49(4):1270–1294, 2020. [39] Yan-Lan Zhang, Jinjin Li, and Changqing Li. Topological structure of relation-based generalized rough sets. Fundamenta Informaticae, 147(4):477–491, 2016. [40] G Choquet. Sur les notions de filter et, 1947. [41] AA Azzam. Comparison of two types of rough approximation via grill. Italian J. Pure Appl. Math, 47:258–270, 2022. [42] AA Azzam. Grill nano topological spaces with grill nano generalized closed sets. Journal of the Egyptian Mathematical Society, 25(2):164–166, 2017. [43] AS Salama and Mohamed Mohamed Ezzat Abd El-Monsef. New topological ap- proach of rough set generalizations. International Journal of Computer Mathematics, 88(7):1347–1357, 2011. A. A. Azzam, M. Aldawood, B. Alreshidi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6997 31 of 31 [44] Yi Yu Yao. Relational interpretations of neighborhood operators and rough set ap- proximation operators. Information sciences, 111(1-4):239–259, 1998. [45] Tareq M Al-shami and Davide Ciucci. Subset neighborhood rough sets. Knowledge- Based Systems, 237:107868, 2022. [46] Tareq M Al-shami, Rodyna A Hosny, Abdelwaheb Mhemdi, and M Hosny. Cardinality rough neighborhoods with applications. AIMS Math, 9(11):31366–31392, 2024. [47] AA Azzam, Ahmed Mostafa Khalil, and Sheng-Gang Li. Medical applications via minimal topological structure. Journal of Intelligent & Fuzzy Systems, 39(3):4723– 4730, 2020.