Rough Pythagorean fuzzy approximations with neighborhood systems and information granulation

A rough set approximates a subset of a universal set on the basis of some binary relation and is significant for the reduction of attributes of an information system. On the other hand, a Pythagorean fuzzy set provides information about the extent of truthness and falsity of a statement. Both these theories deal with different forms of uncertainty and can be united to get their combined benefits. This paper contributes a new blend of rough sets and Pythagorean fuzzy sets namely, rough Pythagorean fuzzy sets. This model can encapsulate two distinct types of uncertainties that appear in imprecise available data through the approximation of Pythagorean fuzzy sets in crisp approximation space. We define rough Pythagorean fuzzy sets on the basis of equivalence relation and generalize it for arbitrary binary relations. The manuscript also provides a general framework to study rough Pythagorean fuzzy approximations of different k-step neighborhood systems. The identities and properties of upper and lower rough Pythagorean fuzzy approximation operators are discussed for the neighborhood systems induced from different types of binary relations. Further, we develop algorithms that compute reduct family, core and rough Pythagorean fuzzy approximations of single-valued and set-valued information systems using indiscernibility relation and similarity relation, respectively. These algorithms are subjected to simple yet interesting applications.