On the cardinalities of interval-valued fuzzy sets

  • Authors:

  • Glad Deschrijver;Pavol Král'

  • Affiliations:

  • Department of Mathematics and Computer Science, Ghent University, Fuzziness and Uncertainty Modelling Research Unit, Krijgslaan 281 (S9), B--9000 Gent, Belgium;Department of Quantitative Methods and Information Technology, Faculty of Economics, Matej Bel University, Tajovského 10, 975 90 Banská Bystrica, Slovakia

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2007

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Abstract

Wygralak has developed an axiomatic theory of scalar cardinalities of fuzzy sets with finite support which covers as particular cases all standard (historical) concepts of the scalar cardinality. In this paper we present a possible extension of this theory to interval-valued fuzzy set theory. We introduce the cardinality of interval-valued fuzzy sets as a mapping from the set of interval-valued fuzzy sets with finite support to the set of closed subintervals of [0,+~[. We study some properties of these cardinalities using t-norms, t-conorms and negations on the lattice L^I (the underlying lattice of interval-valued fuzzy set theory). In particular, the valuation property, the subadditivity property, the complementarity rule and the cartesian product rule will be discussed.