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.

fuzzy set, divergence measure, scalar cardinality, fuzzy cardinality

03B52, 03E75

- S. Ashraf and T. Rashid: Fuzzy similarity measures. LAP LAMBERT Academic Publishing, 2010. CrossRef
- J. Casasnovas and J. Torrens: 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
- B. De Baets, H. De Meyer and H. Naessens: A class of rational cardinality-based similarity measures. J. Comput. Appl. Math. 132 (2001), 51-69. DOI:10.1016/s0377-0427(00)00596-3
- B. De Baets, S. Janssens and H. De Meyer: 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
- G. Deschrijver and P. Král': On the cardinalities of interval-valued fuzzy sets. Fuzzy Sets and Systems 158 (2007), 1728-1750. DOI:10.1016/j.fss.2007.01.005
- E. P. Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer Academic Publishers, London 2000. DOI:10.1007/978-94-015-9540-7
- V. Kobza, V. Janiš and S. Montes: Generalizated local divergence measures between fuzzy subsets. J. Intelligent and Fuzzy Systems (2017), accepted, in press. CrossRef
- I. Montes: Comparison of Alternatives under Uncertainty and Imprecision. PhD Thesis, University of Oviedo 2013. CrossRef
- S. Montes, I. Couso, P. Gil and C. Bertoluzza: Divergence measure between fuzzy sets. Int. J. Approx. Reasoning 30 (2002), 91-105. DOI:10.1016/s0888-613x(02)00063-4
- S. Montes and P. Gil: Some classes of divergence measures between fuzzy subsets and between fuzzy partitions. Mathware and Soft Computing 5 (1998), 253-265. CrossRef
- D. Ralescu: Cardinality, quantifiers and the aggregation of fuzzy criteria. Fuzzy Sets and Systems 69 (1995), 355-365. DOI:10.1016/0165-0114(94)00177-9
- G. Shang, Z. Zhang and C. Cao: Multiplication operation on fuzzy numbers. J. Software 4 (2009), 331-338. DOI:10.4304/jsw.4.4.331-338
- M. Wygralak: Cardinalities of Fuzzy Sets. Springer, Berlin, Heidelberg, New York 2003. DOI:10.1007/978-3-540-36382-8
- M. Wygralak: 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
- M. Wygralak: Questions of cardinality of finite fuzzy sets. Fuzzy Sets and Systems 102 (1999), 185-210. DOI:10.1016/s0165-0114(97)00097-3
- L. Zadeh: Fuzzy sets and systems. System Theory, Brooklyn, Polytechnic Press (1965), 29-39. CrossRef
- L. Zadeh: Fuzzy logic and its application to approximate reasoning. Inform. Process. 74 (1974), 591-594. CrossRef