Sørensen-Dice coefficient equivalence

Question

I'm working on a open source custom NLP engine: https://github.com/nicrusso7/Mistyca (source code coming soon!). Trying to figure out some similarity measures, I wrote this formula (SIM) used to compare the similarity of two sets:

$SIM = 1 - \frac{|A-B|+|B-A|}{|A|+|B|}$

I'm trying a to prove that this formula is equivalent to the Sørensen-Dice coefficient:

$\frac{2|A∩B|}{|A|+|B|} = \frac{|A|+|B|-|A-B|-|B-A|}{|A|+|B|}$

Can anyone help me to demonstrate that?

Thank you!

Answer 1

In other words, you want to prove $|A|+|B|-|A-B|-|B-A|=2|A\cap B|$. This follows from $|A\cap B|=|A|-|A-B|=|B|-|B-A|$.