Abstract
The intuitionistic fuzzy calculus (IFC), based on the basic operational laws of intuitionistic fuzzy numbers (IFNs), has been put forward. However, the interval‐valued IFC (IVIFC), based on the basic operational laws of interval‐valued IFNs (IVIFNs), is only in the original stage. To further develop the theory of the IVIFC and make it be rigorous, the primary task is to systematically investigate the characteristics of the limits and differentials, which is a foundation of the IVIFC. Moreover, there is quite a lot of literature on IVIFNs; however, the scholars did not reveal the relationships between IFNs and the IVIFNs. To do that, we first investigate the limit of interval‐valued intuitionistic fuzzy sequences, and then, we focus on investigating the limit, the continuity, and the differential of IVIFFs in detail and reveal their relationships. After that, due to the fact that the IFC and the IVIFC are based on the basic operational laws of IFNs and IVIFNs, respectively, we reveal the relationships between the IFNs and the IVIFNs via some homomorphic mappings. Finally, a case study about continuous data of IVIFNs is provided to illustrate the advantages of continuous data.
Highlights
Since Zadeh [1] introduced fuzzy sets (FSs) in 1965, a lot of theories addressing vagueness and uncertainties of the data have been proposed [2,3,4,5,6]. e intuitionistic fuzzy set (IFS) [2] is one of the most widely used extensions of the FSs
We have described the characterization of the limit, the continuity, and the differential with respect to the interval-valued intuitionistic fuzzy functions (IVIFFs) and revealed their relationships
Considering that both the intuitionistic fuzzy calculus (IFC) and the interval-valued IFC (IVIFC) are based on the basic operational laws of the intuitionistic fuzzy numbers (IFNs) and interval-valued IFNs (IVIFNs), respectively
Summary
Since Zadeh [1] introduced fuzzy sets (FSs) in 1965, a lot of theories addressing vagueness and uncertainties of the data have been proposed [2,3,4,5,6]. e intuitionistic fuzzy set (IFS) [2] is one of the most widely used extensions of the FSs. The scholars have investigated a series of fundamental properties with respect to the intuitionistic fuzzy calculus (IFC) [16,17,18,19,20,21,22]; the key characteristic of the IFC is that it is based on the basic operational laws of intuitionistic fuzzy numbers (IFNs). With these operational laws, Lei and Xu [16] introduced the limits, the Complexity derivatives, and the differentials in intuitionistic fuzzy circumstance and gave their basic properties.
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