Jaccard Similarity Function in Python 3 (builtins only)

jaccard_similarity.py

"""Jaccard similarity of two sets using builtin Python 3 only."""

import doctest
  

def jaccard_similarity(x: set, y: set) -> float:
    """Get the Jaccard similarity of two sets.

    Example:
      >>> jaccard_similarity({1,2,3,4}, {2,3,5,7})
      0.3333333333333333
      >>> jaccard_similarity({1,2,3,4}, {2,4,6})
      0.4
      >>> jaccard_similarity({2,3,5,7}, {2,4,6})
      0.16666666666666666

    """

    intersection_cardinality = len(x.intersection(y))
    union_cardinality = len(x) + len(y) - intersection_cardinality
    return intersection_cardinality / union_cardinality
  

if __name__ == "__main__":
    doctest.testmod()

Line #20 is tricky. The formula to get the union_cardinality, while counter-intuitive, is precisely cardinality (x union y) - cardinality (x intersect y). This is because a union produces a set (whose members are unique) and thus by subtracting the intersection set (whose elements are also unique) you subtracked the duplicates from the union set so that one may finally correctly obtain the union_cardinality.