A note on the triangle inequality for the Jaccard distance

Two simple proofs of the triangle inequality for the Jaccard distance in terms of nonnegative, monotone, submodular functions are given and discussed. The Jaccard index [8] is a classical similarity measure on sets with a lot of practical applications in information retrieval, data mining, machine learning, and many more (cf., e.g., [7]). A very simple, elementary proof of the triangle inequality was given in [5] using an appropriate partitioning of sets. Here, we give two more simple, direct proofs of the triangle inequality. One proof comes without any set difference or disjointness of sets.