Hesitant fuzzy sets are widely used in the instances of uncertainty and hesitation. The inclusion relationship is an important and foundational definition for sets. Hesitant fuzzy set, as a kind of set, needs explicit definition of inclusion relationship. Base on the hesitant fuzzy membership degree of discrete form, several kinds of inclusion relationships for hesitant fuzzy sets are proposed. And then some foundational propositions of hesitant fuzzy sets and the families of hesitant fuzzy sets are presented. Finally, some foundational propositions of hesitant fuzzy information systems with respect to parameter reductions are put forward, and an example and an algorithm are given to illustrate the processes of parameter reductions.

翻译：犹豫模糊集广泛应用于不确定性和犹豫性场景。包含关系是集合的重要基础定义。作为集合的一种，犹豫模糊集需要明确的包含关系定义。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。进而给出了犹豫模糊集及犹豫模糊集族的一些基础命题。最后，提出了关于参数约简的犹豫模糊信息系统的一些基础命题，并通过实例和算法说明了参数约简的过程。