Correction: Decision-making in diagnosing heart failure problems using basic rough sets

A correction on

Decision-making in diagnosing heart failure problems using basic rough sets

By D. I. Taher, R. Abu-Gdairi, M. K. El-Bably and M. A. El-Gayar, 2024, 9(8): 21816‒21847.

Doi: 10.3934/math.20241061

In the original article ^[1], we introduced novel rough approximation operators inspired by topological structures to enhance the applicability of rough set theory through a general binary relation. The methods presented are rooted in the concept of "basic-minimal neighborhoods", which builds upon the idea of a "basic-neighborhood", first introduced by Abu-Gdairi et al. ^[2] as a counterpart to "initial neighborhoods" ^[3]. Interestingly, the same concept of "basic-neighborhoods" was also proposed in 2021 under the term "containment neighborhoods" in ^[4], where applications of rough sets were explored using the $k$-neighborhood space ($k$-$\boldsymbol{N}\boldsymbol{S}$) concept developed by Abd El-Monsef et al. ^[5]. However, this work did not thoroughly examine the related topological properties or methods for constructing associated topologies. El-Gayar et al. ^[6] later extended the concept of "basic-neighborhoods" by introducing four distinct types of neighborhoods and examining their topological structures. They also explored methods for constructing related topologies and applied these concepts to decision-making in economic contexts. Furthermore, the term "initial-neighborhoods" was subsequently referred to as "subset neighborhoods" in 2022 in ^[7]. This correction includes a citation for "basic-neighborhoods" in Definition 3.1, further strengthening the conceptual foundation of our study.

Definition 3.1. ^[4] Assume that $\mathcal{R}$ is a binary relation on $\mathcal{\mho }$. Then, we define the following neighborhoods of $\mathcal{p}\in \mathcal{\mho }$:

$\left(i\right)$ basic $\left\langle{r}\right\rangle$-neighborhood: ${\mathbb{N}}_{\left\langle{r}\right\rangle}^{\mathfrak{b}}\left(\mathcal{p}\right) = \left\{\mathcal{q}\in \mathcal{\mho }\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{ }:{\mathbb{N}}_{\left\langle{r}\right\rangle}\left(\mathcal{q}\right)\subseteq {\mathbb{N}}_{\left\langle{r}\right\rangle}\left(\mathcal{p}\right)\right\}$.

$\left(ii\right)$ basic $\left\langle{l}\right\rangle$-neighborhood: ${\mathbb{N}}_{\left\langle{l}\right\rangle}^{\mathfrak{b}}\left(\mathcal{p}\right) = \left\{\mathcal{q}\in \mathcal{\mho }\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{ }:{\mathbb{N}}_{\left\langle{l}\right\rangle}\left(\mathcal{q}\right)\subseteq {\mathbb{N}}_{\left\langle{l}\right\rangle}\left(\mathcal{p}\right)\right\}$.

$\left(iii\right)$ basic $\left\langle{i}\right\rangle$-neighborhood: ${\mathbb{N}}_{\left\langle{i}\right\rangle}^{\mathfrak{b}}\left(\mathcal{p}\right) = {\mathbb{N}}_{\left\langle{r}\right\rangle}^{\mathfrak{b}}\left(\mathcal{p}\right)\cap {\mathbb{N}}_{\left\langle{l}\right\rangle}^{\mathfrak{b}}\left(\mathcal{p}\right)$.

$\left(iv\right)$ basic $\left\langle{u}\right\rangle$-neighborhood: ${\mathbb{N}}_{\left\langle{u}\right\rangle}^{\mathfrak{b}}\left(\mathcal{p}\right) = {\mathbb{N}}_{\left\langle{r}\right\rangle}^{\mathfrak{b}}\left(\mathcal{p}\right)\cup {\mathbb{N}}_{\left\langle{l}\right\rangle}^{\mathfrak{b}}\left(\mathcal{p}\right)$.

The change has no material impact on the conclusion of this article. The original manuscript will be updated ^[1]. We apologize for any inconvenience caused to our readers by this change.

Conflict of interest

The authors declare that they have no conflicts of interest.