(untitled) (European Journal of Pure and Applied Mathematics, 2025)
In order to solve problems and provide practical solutions, researchers attempt to accurately 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 conventional 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 one another as well as to earlier paradigms. Next, we describe 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.