ii. Isolation
 \(d\left(x,y\right)=0\ \Leftrightarrow\ x=y\)
\(\triangle\left(x,y\right)=\left|x\backslash\ y\ \right|+\left|y\backslash x\right|=0\) 
Since sum of two positive terms can only be zero if both of them are zero we can say that \(\left|x\backslash y\right|=0\) and\(\left|y\backslash x\right|=0\)
This means  that there is no uncommon element  between x and y which implies that x=y.
iii. Symmetry
\(d\left(x,y\right)=d\left(y,x\right)\)
\(LHS=\triangle\left(x,y\right)=\left|x\backslash\ y\ \right|+\left|y\backslash x\right|\)
\(RHS=\triangle\left(y,x\right)=\left|y\backslash\ x\ \right|+\left|x\backslash y\right|\)
Clearly LHS=RHS therefore  \(d\left(x,y\right)=d\left(y,x\right)\)