Fundamental properties of intuitionistic fuzzy calculus

The intuitionistic fuzzy set (A-IFS) introduced by Atanassov (1986) is a generalization of fuzzy set (Zadeh, 1965). The basic elements of an A-IFS are intuitionistic fuzzy numbers (IFNs) (Xu and Yager, 2006), each of which is described by a membership degree, a non-membership degree and a hesitancy degree. The IFN is an effective tool in expressing fuzzy information of things. Based on IFNs and their basic operations, the paper first defines the indefinite integral and antiderivative of intuitionistic fuzzy functions (IFFs), and then gives the concept of definite integral of IFFs. Finally, the paper deduces the Newton–Leibniz formula (the fundamental theorem of calculus) in Atanassov’s intuitionistic fuzzy environment, and studies some basic properties of intuitionistic fuzzy calculus. Finally, this paper presents an aggregation method, which is based on the definite integral of IFFs, to deal with intuitionistic fuzzy information, and analyzes some basic properties of the method.