Axiomatic characterizations of dissimilarity orderings and distances between sets

Abstract

We axiomatically characterize the orderings of pairs of sets induced by several distances: Hamming, Jaccard, Sørensen-Dice and Overlap. We also axiomatically characterize these distances. Our axioms are properties describing how a distance changes when we perform elementary modifications of the sets, like adding one element to one of the sets, to both sets, swapping both sets, permuting some elements, etc.