Exercise 8.2.3

Property (i) and (ii) are trivial. For property (iii), there are two cases - either $x=y$ or $x\ne y$. If $x=y$ then certainly $\rho (x,z)+\rho (z,y)\ge 0$ by property (i). If $x\ne y$ then at least one of $z\ne y$ and $z\ne x$ is true, so