Fuzzy Optimization and Decision Making

, Volume 17, Issue 1, pp 1–11 | Cite as

Totally ordered uncertain sets



It is known that some uncertain sets have membership functions, and some do not. How do we judge whether an uncertain set has a membership function? In order to answer this question, this paper presents a concept of totally ordered uncertain set, and shows that totally ordered uncertain sets always have membership functions if they are defined on a continuous uncertainty space. In addition, some criteria for judging the existence of membership functions for uncertain sets are provided. Several inspiring examples and counterexamples are also documented in this paper.


Uncertainty Theory Uncertain Set Membership Function 



This work was supported by National Natural Science Foundation of China Grant No.61573210.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Uncertainty Theory Laboratory, Department of Mathematical SciencesTsinghua UniversityBeijingChina

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