Contingency table for binary data

Symmetrical binary variables: $d(x,y)=\frac{r+s}{q+r+s+t}$ Asymmetrical binary variables: $d(x,y)=\frac{r+s}{q+r+s}$

Jaccard coefficient for asymmetrical binary similarity:

$\mathrm{sim}(i,j)=\mathrm{jaccard}(i,j)=1-d(i,j)=\frac{q}{q+r+s}$

Example