Fuzzy metric and convergences based on the symmetric difference

Abstract

In this paper, by applying the Lebesgue's measure on the symmetric difference of sets, we introduce the notions of the symmetric difference metric d △ and p-mean symmetric difference metric d △p in order to measure the difference between fuzzy numbers. The concept of the platform type of fuzzy number ( p-fuzzy number) is introduced, and we show that d △ and d △p are zero if and only if two fuzzy numbers are equal or have the same p-fuzzy numbers. Further, the relations between d △ and the uniform Hausdorff metric D H (cf. Diamond and Kloeden, 1989), d △p and D p (cf. Nola and Ventre, 1987) are discussed. For the non- p-fuzzy numbers, the equivalences of the metric convergences on d △ and D H , d △p and D p are provided.