Abstract
The conditions of the equivalence relation limit the application fields of the methodology of Pawlak's rough sets. So, to expand the application areas of this theory, it is generalized to any binary relation. Neighborhoods induced from the relations represent a core bridge between rough sets and application since it represents easy tools for dealing with daily-life problems. Accordingly, the first core objective of the current research is to propose a novel neighborhood (so-called an initial-neighborhood) generated from any binary relation. Based on this neighborhood, we suggest a new generalization to Pawlak rough sets and some of their extensions. The proposed approaches satisfy all properties of classical rough sets without adding any extra restrictions and hence we can apply them in any real-life problem. The second aim is to generalize the notion of nano-topology into any binary relation to extend the applications of this concept. Properties of the suggested methods are introduced with many counter-examples. Comparisons between the suggested techniques and the others studies published in the literature are examined. We proved that the proposed techniques are extra precise than the earlier approaches. Finally, the medical application of COVID-19 is provided to illustrate the significance of our approaches in deciding the impact factors for COVID-19 infection. The proposed application is based on a reflexive relation, so Pawlak rough sets and some of its generalizations couldn't be applied to solve this problem. Accordingly, we have successes in solving this problem using the suggested techniques. Hence, we write an algorithm to be a useful tool that may help the doctor in diagnosing the infection of COVID-19.