Chapter 2.2, Problem 71E

The Jaccard similarity $J\left(A,B\right)$ of the finite sets A and B is $J\left(A,B\right)=\left|A\cap B\right|/\left|A\cup B\right|$ , with $J\left(\varnothing ,\varnothing \right)=1$ . The Jaccard distance $d_J\left(A,B\right)$ between A and B equals $d_J\left(A,B\right)=1-J\left(A,B\right)$ .

Find $J\left(A,B\right)$ and $d_J\left(A,B\right)$ for these pairs of sets.

   $A=\left\{1,3,5\right\},B=\left\{2,4,6\right\}$

   $A=\left\{1,2,3,4\right\},B\left\{3,4,5,6\right\}$

   $A=\left\{1,2,3,4,5,6\right\},B=\left\{1,2,3,4,5,6\right\}$

   $A=\left\{A\right\},B=\left\{1,2,3,4,5,6\right\}$