As a powerful tool to model some unsharp concepts in real life, uncertain sets have been studied by more and more scholars. In order to characterize the degree of difficulty of uncertain sets, the hyperbolic entropy of an uncertain set and the hyperbolic relative entropy of uncertain sets are introduced in this paper. After that, this paper derived a key formula to calculate the hyperbolic entropy of an uncertain set via membership function, and some mathematical properties of hyperbolic entropy are also investigated in this paper. Finally, the hyperbolic entropy is applied in some research fields such as uncertain learning curve, clustering of rare books and portfolio selection of collecting rare books.
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