A mathematical relationship between set-theoretic and metric compatibility measures

Because of the diversity of fuzzy set applications, a wide variety of compatibility measures have been employed to assess the degree of agreement between two fuzzy sets. Set theoretic compatibility measures are the ones predominantly found in the literature and are based on determining the commonality between two fuzzy sets. Recently, however, metric compatibility measures are being used to assess agreement as a function of the distance between the fuzzy sets. The Jaccard index from the set theoretic class and the dissemblance index from the metric class are described and a mathematical relationship between these two is established.