Inclusion Relationship of Uncertain Sets

Introduction In order to indicate human’s belief degree on the chance that an event happens, an uncertain measure was defined by Liu [1, 2] as a set function satisfying normality, duality, subadditivity and product axioms. Uncertain set, initialized by Liu [3], is a set-valued function on an uncertainty space to model the unsharp concepts such as “old people” and “heavy vehicles”. In order to describe an uncertain set, a concept of membership function was proposed by Liu [4]. Uncertain sets are the theoretical foundations of uncertain inference control and uncertain logic. In 2010, Liu [3] proposed some inference rules for uncertain systems, which were extended to the rules with multiple antecedents by Gao et al. [5]. Following that, Gao [6] balanced an inverted pendulum by using the uncertain controller. In addition, Peng and Chen [7] showed that the uncertain inference system is a universal approximator. In 2011, Liu [8] applied uncertain set to uncertain logic for dealing with human language by using the truth value formula of uncertain propositions. The concept of independence for uncertain sets was defined by Liu [9]. Then, the set operational law and arithmetic operational law of independent uncertain sets were given by Liu [4]. The concept of expected value for uncertain set was defined by Liu [3], and formulas to calculate the expected value using the membership function and inverse membership function were given by Liu [10] and Liu [4], respectively. Besides, the variance of uncertain set was proposed by Liu [8] and was investigated by Yang and Gao [11]. In addition, the entropy of uncertain set was presented by Liu [4] and was further studied by Yao and Ke [12], Yao [13], Wang and Ha [14], and Peng and Li [15]. For more details about uncertain sets, please refer to Liu [16] and Yang and Gao [17]. In this paper, we will study the inclusion relationship of uncertain sets. The rest of this paper is organized as follows. The next section is intended to introduce some basic concepts about uncertain sets. Then the inclusion relationship is introduced in