双模糊概率集与3-值r-模糊集之间的关系
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摘要
本文建立了双模糊概率集和3-值r-模糊集之间的联系。首先,研究了3-值r-模糊集的定义和运算,说明了3-值r-模糊集与r-直觉模糊集是一一对应的。然后,给出了3-值r-模糊集合套的概念,证明了双模糊概率集是3-值r-模糊集合套的等价类。
In this paper, the connection between bifuzzy probabilistic sets and three-valued r-fuzzy sets is established. Firstly, we introduce the notion and its operations of three-valued r-fuzzy sets, and show that there is a one-to-one relationship between three-valued r-fuzzy sets and r-intuitionistic fuzzy sets.Secondly, the concept of three-valued r-fuzzy nested sets is given, and it is proved that a bifuzzy probabilistic set is an equivalent class of a three-valued r-fuzzy set.
In this paper, the connection between bifuzzy probabilistic sets and three-valued r-fuzzy sets is established. Firstly, we introduce the notion and its operations of three-valued r-fuzzy sets, and show that there is a one-to-one relationship between three-valued r-fuzzy sets and r-intuitionistic fuzzy sets.Secondly, the concept of three-valued r-fuzzy nested sets is given, and it is proved that a bifuzzy probabilistic set is an equivalent class of a three-valued r-fuzzy set.
引文
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[12] Li H X.Probability representations of fuzzy systems[J].Science China Information Sciences,2006,49(3):339～363.
[13] Li H X.The united theory of uncertainty systems[J].Chinese Journal of Engineering Mathematics,2007,24(1):1～21.
[14] Dong X M.Bifuzzy probabilistic sets and r-intuitionistic fuzzy sets[C]//2009 Sixth International Conference on Fuzzy Systems and Knowledge Discovery,2009:123～127.
[15] 袁学海,李洪兴,孙凯彪.直觉模糊集和区间值模糊集的截集、分解定理和表现定理[J].中国科学:信息科学,2009,39(9):933～945.
[16] Ciungu,Lavinia,Riecan B.General form of probabilities on IF-Sets[C]//International Workshop on Fuzzy Logic and Applications.Springer-Verlag,2009:101～107.
[17] Ciungu L C,Riecan B.Representation theorem for probabilities on IFS-events[J].Information Sciences,2010,180(5):793～798.
[18] Wang Q P,Wang G J.The probability theories for IVFSs and IVIFSs[M].Pergamon Press,Inc,2011,62(12):4535～4538.
[19] Yu B,Yuan X H.Three-valued random fuzzy sets[C]//Fuzzy Systems & Operations Research and Management.Springer International Publishing,2016:183～198.
[20] 罗承忠.模糊集引论(上)[M].北京:北京师范大学出版社,1989.
[21] 罗承忠.可数集合套[J].模糊系统与数学,1991,5(1):1～6.
[2] Atanassov K T,Gargov.Elements of intuitionistic fuzzy logic,Part I[J].Fuzzy Sets and Systems,1998,95:39～52.
[3] Mitchell H B.Ranging intuitionistic fuzzy number[J].International Journal of Uncertainty,Fuzziness and Knowledge-Based,Systems,2004,12:377～386.
[4] Yuan X H,Li H X,Lee E S.On the definition of intuitionistic fuzzy subgroups[J].Computer and Mathematics with Applications,2010:3117～3129.
[5] Coker D.An introduction to intuitionistic fuzzy topological,spaces[J].Fuzzy Sets and Systems,1997,88:81～89.
[6] Hung W L,Lee J S,Fuh C D.Fuzzy clustering based on intuitionistic fuzzy relations[J].International Journal of Uncertainty,Fuzziness and Knowledge-Based Systems,2004,12:513～529.
[7] Erstenkorn T,Manko J.Bifuzzy probabilistic sets[J].Fuzzy Sets,and Systems,1995,71:207～214.
[8] 汪培庄.模糊集与随机集落影[M].北京:北京师范出版杜,1985.
[9] K Hirota.Concepts of fuzzy probabilistic sets[J].Fuzzy Sets and Systems,1981,5:31～46.
[10] Li Q,Wang P,Lee E S.r-fuzzy sets[J].Computers and Mathematics,1996,31:49～61.
[11] Yuan X H,Li H X,Zhang C.Fuzzy probabilistic and r-fuzzy sets[C]//Proceedings of the 5th International Conference on Fuzzy Systems and Knowledge Discovery.Dalian:Dalian University of Technology Press,2008:271～276.
[12] Li H X.Probability representations of fuzzy systems[J].Science China Information Sciences,2006,49(3):339～363.
[13] Li H X.The united theory of uncertainty systems[J].Chinese Journal of Engineering Mathematics,2007,24(1):1～21.
[14] Dong X M.Bifuzzy probabilistic sets and r-intuitionistic fuzzy sets[C]//2009 Sixth International Conference on Fuzzy Systems and Knowledge Discovery,2009:123～127.
[15] 袁学海,李洪兴,孙凯彪.直觉模糊集和区间值模糊集的截集、分解定理和表现定理[J].中国科学:信息科学,2009,39(9):933～945.
[16] Ciungu,Lavinia,Riecan B.General form of probabilities on IF-Sets[C]//International Workshop on Fuzzy Logic and Applications.Springer-Verlag,2009:101～107.
[17] Ciungu L C,Riecan B.Representation theorem for probabilities on IFS-events[J].Information Sciences,2010,180(5):793～798.
[18] Wang Q P,Wang G J.The probability theories for IVFSs and IVIFSs[M].Pergamon Press,Inc,2011,62(12):4535～4538.
[19] Yu B,Yuan X H.Three-valued random fuzzy sets[C]//Fuzzy Systems & Operations Research and Management.Springer International Publishing,2016:183～198.
[20] 罗承忠.模糊集引论(上)[M].北京:北京师范大学出版社,1989.
[21] 罗承忠.可数集合套[J].模糊系统与数学,1991,5(1):1～6.