Chain and Substitution Rules of Intuitionistic Fuzzy Calculus

The intuitionistic fuzzy set, originally given by Atanassov, can be considered as a set that consists of intuitionistic fuzzy numbers (IFNs), each of which is described by a membership degree, a nonmembership degree, and a hesitancy degree. Recently, the calculus of IFNs has been proposed, and the derivatives, differentials, indefinite integrals, and definite integrals with respect to IFNs have also been developed in previous research studies. In this paper, some novel results of intuitionistic fuzzy calculus are derived, which include mainly the chain rule of derivatives, the form invariance of differentials, and the substitution rules for indefinite integrals and definite integrals in intuitionistic fuzzy environment.