Rough groups

Mathematical theories not only enable the understanding of abstract structures but also make it possible to apply these structures in different fields. In this study, the foundations of rough sets and rough group theory have been examined, and the mathematical properties of these structures have been addressed from both a theoretical and applied perspective. The first chapter summarizes the literature on rough sets and related theories, presenting the fundamental principles of set theory and its connection to daily life. The second chapter provides definitions, theorems and foundational reminders to establish a basis for the concepts discussed in subsequent chapters. The third chapter begins with interpretations of information tables and basic sets, followed by definitions and examples related to the lower and upper approximations of rough sets using equivalence relations. It also introduces concepts such as the positive, negative, and boundary regions of sets, along with proofs of relevant theorems and includes calculations and interpretations of approximate membership functions in rough sets. The fourth chapter examines the concepts of rough groups, rough subgroups, rough normal subgroups, and rough cosets within a theoretical framework, supported by examples.