Foundational theories of hesitant fuzzy sets and families of hesitant fuzzy sets
Shizhan Lu, Zeshui Xu, Zhu Fu, Longsheng Cheng, Tongbin Yang
TL;DR
This work develops foundational theories for hesitant fuzzy sets (HF) and hesitant fuzzy information systems (HFIS), addressing the lack of a clear inclusion concept by introducing a spectrum of strength-based inclusion types grounded in discrete hesitant membership degrees. It establishes core operations for HF sets, derives numerous propositions about intersections, unions, and complementation, and analyzes how inclusion notions interact with equality and random augmentations. The paper also extends these ideas to families of HF sets, proving that inclusion relations behave consistently under intersections and unions across HF families and between HF1 and HF2. The results reveal that many classical set rules do not carry over to hesitant fuzzy contexts, and they propose a rich framework of inclusion notions that can underpin HFIS design and analysis. The work points to practical applications, such as multi-strength intelligent classifiers for health-state diagnosis, and outlines directions for future research in knowledge bases with varying informational strength.
Abstract
Hesitant fuzzy sets find extensive application in specific scenarios involving uncertainty and hesitation. In the context of set theory, the concept of inclusion relationship holds significant importance as a fundamental definition. Consequently, as a type of sets, hesitant fuzzy sets necessitate a clear and explicit definition of the inclusion relationship. Based on the discrete form of hesitant fuzzy membership degrees, this study proposes multiple types of inclusion relationships for hesitant fuzzy sets. Subsequently, this paper introduces foundational propositions related to hesitant fuzzy sets, as well as propositions concerning families of hesitant fuzzy sets.