Issue:The Inclusion–Exclusion principle for general IF-states

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Title of paper: The Inclusion–Exclusion principle for general IF-states Author(s): Daniela Kluvancová Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01, Banská Bystrica, Slovakia daniela.kluvancova@umb.sk Presented at: 11th International Workshop on Intuitionistic Fuzzy Sets, Banská Bystrica, Slovakia, 30 Oct. 2015 Published in: "Notes on Intuitionistic Fuzzy Sets", Volume 21, 2015, Number 5, pages 24–32 Download:  PDF (180  Kb, File info) Abstract: Any real state on intuitionistic fuzzy sets (IF-sets) can be represented by integrals. L. Ciungu in [3] proved that for any real state on IF-sets and for a pair of binary operations which satisfy some special conditions holds an Inclusion–Exclusion principle. In [10], J. Považan proved that also any state on IF-sets with values from the arbitrary Riesz space we can represented by integrals. But could we consider Inclusion–Exclusion principle for any IF-state? In this paper we will prove this property for general case in very similar way as for real. Keywords: IF-set, IE-pair, Inclusion–Exclusion principle, Riesz space, Representation theorem. AMS Classification: 03E72. References: Atanassov, K. (1986) Intuitionistic fuzzy sets. Fuzzy sets and systems, 20, 87–96. Boccuto, A., B. Riečan & M. Vrábelová. (2009) Kurzweil–Henstock Integral in Riesz spaces. Bentham Science Publishers Ltd. Ciungu, L.C., J. Klemenová & B. Riečan. (2012) A New Point of View to the Inclusion–Exclusion principle. Proc. of 6th IEEE International Conference of Intelligent Systems IS, Varna, Bulgaria, 142–144. Ciungu, L.C. & B. Riečan. (2009) General form of probability on IF-sets. Fuzzy logic and applications, Proc.8th Int.Workshop WILF, Lecture Notes in Artificial Intelligence, 101–107. Ciungu, L.C. & B. Riečan. (2014) The Inclusion–Exclusion principle for IF-states. Iranian Journal of Fuzzy systems, 11(2), 17–25. Grzegorzewski, P. & E. Mrówka. (2002) Probability on intuitionistic fuzzy events. Soft Methods in Probability, Statistics and Data Analysis, 105–115. Grzegorzewski, P. (2011) The Inclusion–Exclusion principle on IF-sets. Information Sciences, 181, 536–546. Kuková, M. & M. Navara. (2013) Principles of inclusion and exclusion for fuzzy sets. Fuzzy sets and systems, 232, 98–109. Petrovičová, J. & B. Riečan. (2010) On the Łukasiewicz probability theory on IF-sets. Tatra Mountains Mathematical Publications, 46, 125–146. Považan, J. (2014) Representation theorem of general states on IF-sets. Intelligent systems, 322, 141–147. Riečan, B. (2003) A descriptive definition of probability on intuitionistic fuzzy sets. In: Proc. EUSFLAT, 236–266. Riečan, B. (2014) A general point of view to Inclusion–Exclusion property. In: Modern Approaches in Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets and Related Topics, 1, 171–180. Riečan, B. & T. Neubrunn. (1998) Integral, Measure and Ordering. Ister Science, Bratislava. Riečan, B. (2006) On a problem of Radko Mesiar: General form of IF-probabilities. Fuzzy Sets and Systems, 157, 1485–1490. Riečan, B. (2004) Representation of probabilities on IFS events. Advances in Soft Computing, Soft Methodology and Random Information Systems, 243–246. Citations: The list of publications, citing this article may be empty or incomplete. If you can provide relevant data, please, write on the talk page.