Belief function of Pythagorean fuzzy rough approximation space and its applications

Rough set theory and evidence theory are two approaches to handle decision making and reduction problems of imprecise and uncertain knowledge. It is motivated that Pythagorean fuzzy set excels at describing the situation where the sum of non-membership degree and membership degree is greater than 1, and may have wider applications than intuitionistic fuzzy set. So in this paper we study probability measure of Pythagorean fuzzy sets and belief structure of Pythagorean fuzzy information systems based on rough set theory, and discuss the reduction of Pythagorean fuzzy information systems. First, we review the properties of Pythagorean fuzzy sets and the upper and lower Pythagorean fuzzy rough approximation operators on the level sets. Then using these properties, probability measure of Pythagorean fuzzy sets are constructed. And the belief and plausibility functions are studied by using the Pythagorean fuzzy rough upper and lower approximation operators. Finally, we apply the belief function to construct an attribute reduction algorithm, and an example is employed to illustrate the feasibility and validity of the algorithm.