Distance between Sets

Abstract

DISTANCE functions expressing the degree of dissimilarity of sets have found use in physical anthropology¹, psychology², numerical taxonomy³, ecology³ and elsewhere. During an ecological study by one of us, it was noticed that the similarity coefficient of Jaccard⁶, used in ecology, gives rise to a metric function satisfying the triangle inequality. For two non-empty finite sets X, Y, the Jaccard coefficient is the number of elements in the intersection X∩Y of X and Y. This coefficient (we use absolute value signs to indicate number of elements) has a heuristic interpretation. It measures the probability that an element of at least one of two sets is an element of both, and thus is a reasonable measure of similarity or “overlap” between the two. The one-complement may then be considered a measure of the dissimilarity of the two sets.