  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.
