\(\Rightarrow\left|X\cap Y\right|=\left|X\cup Y\right|\)
This is only possible when X equals Y . Therefore X=Y and property of isolation holds for the function.
iii. Symmetry
\(d\left(x,y\right)=d\left(y,x\right)\)
\(LHS=JD\left(X,Y\right)=1-\frac{\left|X\cap Y\right|}{\left|X\cup Y\right|}\)
\(RHS=JD\left(Y,X\right)=1-\frac{\left|Y\cap X\right|}{\left|Y\cup X\right|}\)
Since \(X\cap Y=Y\cap X\ and\ X\cup Y=\ Y\cup X\) we can rewrite RHS as below
\(RHS=1-\frac{\left|X\cap Y\right|}{\left|X\cup Y\right|}\)
Since LHS=RHS property of symmetry holds.