Cardinality rough neighborhoods with applications

Rough set theory serves as an effective method for managing complicated real-world data. Through rough approximation operators, it discerns both confirmed and possible data attainable through subsets. Earlier studies have presented several rough approximation models, drawing inspiration from neighborhood systems aimed at enhancing accuracy degree and satisfying the axioms of traditional approximation spaces (TAS) that were initiated by Pawlak. This article proposes an easy method to deal with information systems in most cases, wherein it introduces a new forming of generalized approximation spaces, namely, cardinality rough neighborhoods. It is defined depending on the cardinal number of the ${\mathcal{N}}_{\sigma }$-neighborhoods of elements that are established under an arbitrary relation. Their main features are investigated and the connections between them, as well as their relationships with the preceding kinds of neighborhood systems, are uncovered with the aid of some examples. Then, novel rough set paradigms induced by cardinality rough neighborhoods are displayed that satisfy most properties of Pawlak's paradigm. Next, a topological method to study these paradigms is provided, wherein this method produces approximation operators similar to the given paradigms in six cases that are proved. Additionally, both paradigms in a practical example concerning books and the authors who authored them or participated in their authorship are applied. To illuminate the need for the current concepts, we elaborate on their advantages from different views. Finally, a summary of the obtained results and relationships and suggestions for some forthcoming work are offered.