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fuzzy set; divergence measure; scalar cardinality; fuzzy cardinality
In this paper we extend the concept of measuring difference between two fuzzy subsets defined on a finite universe. The first main section is devoted to the local divergence measures. We propose a divergence measure based on the scalar cardinalities of fuzzy sets with respect to the basic axioms. In the next step we introduce the divergence based on the generating function and the appropriate distances. The other approach to the divergence measure is motivated by class of the rational similarity measures between fuzzy subsets expressed using some set operations (namely intersection, complement, difference and symmetric difference) and their scalar cardinalities. Finally, this concept is extended into the fuzzy cardinality in the last part. Some open problems omitted in this paper are discussed in the concluding remarks section.
[1] Ashraf, S., Rashid, T.: Fuzzy similarity measures. LAP LAMBERT Academic Publishing, 2010.
[2] Casasnovas, J., Torrens, J.: An axiomatic approach to fuzzy cardinalities of finite fuzzy sets. Fuzzy Sets and Systems 133 (2003), 193-209. DOI 10.1016/s0165-0114(02)00345-7 | MR 1949022
[3] Baets, B. De, Meyer, H. De, Naessens, H.: A class of rational cardinality-based similarity measures. J. Comput. Appl. Math. 132 (2001), 51-69. DOI 10.1016/s0377-0427(00)00596-3 | MR 1834802
[4] Baets, B. De, Janssens, S., Meyer, H. De: On the transitivity of a parametric family of cardinality-based similarity measures. In. J. Approx. Reasoning 50 (2009), 104-116. DOI 10.1016/j.ijar.2008.03.006 | MR 2519040 | Zbl 1191.68706
[5] Deschrijver, G., Král', P.: On the cardinalities of interval-valued fuzzy sets. Fuzzy Sets and Systems 158 (2007), 1728-1750. DOI 10.1016/j.fss.2007.01.005 | MR 2341334
[6] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, London 2000. DOI 10.1007/978-94-015-9540-7 | MR 1790096 | Zbl 1087.20041
[7] Kobza, V., Janiš, V., Montes, S.: Generalizated local divergence measures between fuzzy subsets. J. Intelligent and Fuzzy Systems (2017), accepted, in press. MR 3684678
[8] Montes, I.: Comparison of Alternatives under Uncertainty and Imprecision. PhD Thesis, University of Oviedo 2013.
[9] Montes, S., Couso, I., Gil, P., Bertoluzza, C.: Divergence measure between fuzzy sets. Int. J. Approx. Reasoning 30 (2002), 91-105. DOI 10.1016/s0888-613x(02)00063-4 | MR 1906630
[10] Montes, S., Gil, P.: Some classes of divergence measures between fuzzy subsets and between fuzzy partitions. Mathware and Soft Computing 5 (1998), 253-265. MR 1704068
[11] Ralescu, D.: Cardinality, quantifiers and the aggregation of fuzzy criteria. Fuzzy Sets and Systems 69 (1995), 355-365. DOI 10.1016/0165-0114(94)00177-9 | MR 1319236
[12] Shang, G., Zhang, Z., Cao, C.: Multiplication operation on fuzzy numbers. J. Software 4 (2009), 331-338. DOI 10.4304/jsw.4.4.331-338
[13] Wygralak, M.: Cardinalities of Fuzzy Sets. Springer, Berlin, Heidelberg, New York 2003. DOI 10.1007/978-3-540-36382-8
[14] Wygralak, M.: Fuzzy sets with triangular norms and their cardinality theory. Fuzzy Sets and Systems 124 (2001), 1-24. DOI 10.1016/s0165-0114(00)00108-1 | MR 1859773
[15] Wygralak, M.: Questions of cardinality of finite fuzzy sets. Fuzzy Sets and Systems 102 (1999), 185-210. DOI 10.1016/s0165-0114(97)00097-3 | MR 1674931
[16] Zadeh, L.: Fuzzy sets and systems. System Theory, Brooklyn, Polytechnic Press (1965), 29-39. MR 0256772
[17] Zadeh, L.: Fuzzy logic and its application to approximate reasoning. Inform. Process. 74 (1974), 591-594. MR 0408358
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