Complemental Binary Operations of Sets and Their Application to Group Theory

Set theory is considered as the foundation of all mathematics since many mathematical concepts cannot be defined precisely without using set-theoretical concepts. In this study, we define new complemental binary operations, called union complements, intersection left complement, and union right complement and investigate their properties in detail. We contribute to the literature of sets by illuminating the relationships between these complemental binary operations and inclusive complements via researching the distribution rules. Moreover, we show that the set of all the sets together with these new complemental binary operations form some algebraic structures. Finally, with the inspiration of these novel concepts, we give an application to group theory as regards subgroups by defining new type of subgroups in order to prompt the reader to think via interesting questions. Since the concept of operations of soft set theory, one of the most popular theory for uncertainty modeling in the past twenty four years, is the crucial notion for developing the theory and since all the types of soft set operations are based on the classical set operations, generation of new complemental binary operations on sets, and thus on soft sets and derivation of their algebraic properties will provide new perspectives for solving problems related to parametric data.