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| Title of paper:
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A descriptive definition of the probability on intuitionistic fuzzy sets
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| Author(s):
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| Beloslav Riečan
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| Matej Bel University, Tajovského 40, SK-97401 Banská Bystrica, Mathematical Institute of Slovak Acad. of Sciences, Štefánikova 49, SK-81473, Bratislava, Slovakia
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| riecan@mat.savba.sk, riecan@fpv.umb.sk
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| Presented at:
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3rd Conference of the European Society for Fuzzy Logic and Technology, Zittau, Germany, September 10-12, 2003
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| Published in:
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Conference proceedings, pages 210-213
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| Download:
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 PDF (104 Kb, File info)
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| Abstract:
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In [2] a general probability theory has been constructed for intuitionistic fuzzy events ([1]) defined on any probability space [math]\displaystyle{ (\Omega,\mathcal{S},P) }[/math]. To any element A belonging to the family [math]\displaystyle{ \mathcal{F} }[/math] of all intuitionistic fuzzy events a compact interval P(A) on the real line is assigned. In the paper we consider a mapping [math]\displaystyle{ \mathcal{P}:\mathcal{F} \to \mathcal{J} }[/math], where [math]\displaystyle{ \mathcal{J} }[/math] is the family of all compact intervals. Some properties of [math]\displaystyle{ \mathcal{P} }[/math] are postulated axiomatically. Then a representation theorem is proved stating that to any mapping [math]\displaystyle{ \mathcal{P} }[/math] satisfying the properties there exists a probability measure [math]\displaystyle{ P:\mathcal{S} \to [0;1] }[/math] such that [math]\displaystyle{ \mathcal{P}(A) }[/math] can be expressed by the help of [math]\displaystyle{ \mathcal{P} }[/math] similarly as it has been done in [2].
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| Keywords:
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Intuitionistic fuzzy sets, Intuitionistic fuzzy distance, Similarity measure.
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| References:
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- Atanassov K. (1999), Intuitionistic Fuzzy Sets: Theory and Applications. Springer-Verlag.
- Cross V. and Sudkamp T. (2002) Similarity and Compatibility in Fuzzy Set Theory. Physica-Verlag.
- Szmidt E. (2000): Applications of Intuitionistic Fuzzy Sets in Decision Making. (D.Sc.dissertation) Techn. Univ., Sofia, 2000.
- Szmidt E. and Kacprzyk J. (2000) Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems, Vol.114, No.3, pp.505—518.
- Szmidt E., Kacprzyk J. (2001) Entropy for intuitionistic fuzzy sets. Fuzzy Sets and Systems, vol. 118, No. 3, pp. 467—477.
- Szmidt E. and Kacprzyk J. (2002) Analysis of Agreement in a Group of Experts via Distances Between Intuitionistic Fuzzy Preferences. Proc. 9th Int. Conf. IPMU 2002, Annecy, France, July 1—5, pp. 1859—1865.
- Szmidt E. and Kacprzyk J.(2002b) An Intuitionistic Fuzzy Set Based Approach to Intelligent Data Analysis (an application to medical diagnosis). In A. Abraham, L. Jain, J. Kacprzyk (Eds.): Recent Advances in Intelligent Paradigms and Applications. Springer-Verlag, pp. 57-70.
- Szmidt E. and Kacprzyk J. (2002c) Evaluation of Agreement in a Group of Experts via Distances Between Intuitionistic Fuzzy Sets. Proc. IS’2002 - Int. IEEE Symposium: Intelligent Systems, Varna, Bulgaria, IEEECatalog Number 02EX499, pp. 166-170.
- Tversky A. (1977) Features of similarity. Psychol. Rev. Vol. 84, pp. 327—352.
- L.A. Zadeh (1965) Fuzzy sets. Information and Control, 8, 338—353.
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