Some Studies on Roughness of Bipolar Fuzzy and Bipolar Soft Substructures

Abstract

Theories of rough sets [41, 42], fuzzy sets [48] and soft sets [37] are the most eminent and dynamic mathematical tools for modeling various types of data with uncertainty. Uncertainty and vagueness is often faced in the data assembled and studied for various purposes. The classical mathematical tools are not always convenient to describe such aspects of the real world problems. These theories are presented to handle the uncertainty in data in order to construct a productive mathematical model. The fuzzy set theory reflects the uncertain knowledge in a very fruitful way by grading the elements of the universe on the basis of their characteristics; where grades are assigned from the interval [0, 1]. The rough set theory acquires a completely different mathematical approach towards the uncertainty and ambiguity of the data. In this theory, the level of definiteness of the information associated to a set of objects is interpreted by two definable sets, called the lower and upper rough approximations, subject to the available information