Fuzzy Distance and Fuzzy Norm

Abstract

We consider fuzzy sets on a metric, vector, or normed space. It is not assumed that the fuzzy sets have compact supports. In the present paper, a fuzzy distance and a fuzzy norm are proposed in order to measure the difference between two fuzzy sets, and their fundamental properties are investigated. Their definitions are based on Zadeh's extension principle. Although they are different from the classical ones based on the Hausdorff metric, they are suitable for data containing uncertainty or vagueness. The obtained results can be expected to be useful for analyzing such data when the data are represented as fuzzy sets.