What is a metric to measure the similarity of any two sets

Question

I seek a function that will assign a real number to any two sets based on their elements alone. I need it to be sensitive to the extent of the intersection of the sets but to penalize if the sets have extraneous items. In other words I want to count both recall and precision in the same metric.

Answer 1

What you are looking for is the Jaccard index:

J(A, B) := |A ∩ B| / |A ∪ B|

Hence it counts how many elements both sets have in common and divides it by the number of unique elements.

J(A, B) is at maximum 1 if both sets are identical. The minimum is zero, if they don't share any element. You might want to assign a number for the case that both sets are empty.