Hesitant fuzzy sets are widely used in certain instances of uncertainty and hesitation. In sets, the inclusion relationship is an important and foundational definition. Thus, as a kind of set, hesitant fuzzy sets require an explicit definition of inclusion relationship. Based on the hesitant fuzzy membership degree of discrete form, several kinds of inclusion relationships for hesitant fuzzy sets are proposed in this work. Then, some foundational propositions of hesitant fuzzy sets are presented, along with propositions of families of hesitant fuzzy sets. Some foundational propositions of hesitant fuzzy information systems are proposed with respect to parameter reductions and an example and an algorithm are given to illustrate the processes of parameter reduction. Finally, a multi-strength intelligent classifier is proposed to make health state diagnoses for complex systems.

翻译：犹豫模糊集广泛应用于存在不确定性和犹豫的特定场景。在集合论中，包含关系是一个重要且基础的定义。因此，作为一种集合形式，犹豫模糊集需要明确包含关系的定义。本文基于离散形式的犹豫模糊隶属度，提出了几种犹豫模糊集的包含关系。随后，给出了犹豫模糊集的一些基础命题，以及犹豫模糊集族的相关命题。针对参数约简问题，提出了犹豫模糊信息系统的基础命题，并通过实例和算法说明了参数约简的过程。最后，提出了一种多强度智能分类器，用于对复杂系统进行健康状态诊断。