Wikipedia defines the [Jaccard distance][1] between sets *A* and *B* as $$J_\delta(A,B)=1-\frac{|A\cap B|}{|A\cup B|}.$$ There's also a [book][2] claiming that this is a metric. However, I couldn't find any explanation of why $J_\delta$ obeys the triangle inequality. The naive approach of writing the inequality with seven variables (e.g., $x_{001}$ thru $x_{111}$, where $x_{101}$ is the number of elements in $(A\cap C) \backslash B$) and trying to reduce it seems hopeless for pen and paper. In fact it also seems hopeless for Mathematica, which is trying to find a counterexample for 20 minutes and is still running. (It's supposed to say if there isn't any.)

Is there a simple argument showing that this is a distance? Somehow, it feels like the problem shouldn't be difficult and I'm missing something.

  [1]: https://en.wikipedia.org/wiki/Jaccard_index
  [2]: https://books.google.com/books?id=DGjbibiS-S0C&pg=PA38&dq=jaccard+similarity&as_brr=3&ei=iNGbS96kNYfIywTVjZ2cCg&cd=1#v=onepage&q=jaccard%20similarity&f=false