Using the function

\[
f(x)=
\begin{cases}
\mathtt{true}, & \text{if $x = 0$}.\\\\
\mathtt{false}, & \text{if $ x > 0$}.
\end{cases}
\]

we can define \\( \mathcal{X}(x, y) \\) for our any \\( x, y \in \mathcal{X} \\) in our "regions of the world" preorder:

\begin{align}
d(\texttt{Boston}, \texttt{Boston}) = 0,&\quad \mathcal{X}(\texttt{Boston}, \texttt{Boston}) = \mathtt{true} \\\\
d(\texttt{Boston}, \texttt{US}) = 0,&\quad \mathcal{X}(\texttt{Boston}, \texttt{US}) = \mathtt{true} \\\\
d(\texttt{Boston}, \texttt{Spain}) \neq 0,&\quad \mathcal{X}(\texttt{Boston}, \texttt{Spain}) = \mathtt{false} \\\\
d(\texttt{US}, \texttt{Boston}) \neq 0,&\quad \mathcal{X}(\texttt{US}, \texttt{Boston}) = \mathtt{false} \\\\
d(\texttt{US}, \texttt{US}) = 0,&\quad \mathcal{X}(\texttt{US}, \texttt{US}) = \mathtt{true} \\\\
d(\texttt{US}, \texttt{Spain}) \neq 0,&\quad \mathcal{X}(\texttt{US}, \texttt{Spain}) = \mathtt{false} \\\\
d(\texttt{Spain}, \texttt{Boston}) \neq 0,&\quad \mathcal{X}(\texttt{US}, \texttt{Boston}) = \mathtt{false} \\\\
d(\texttt{Spain}, \texttt{US}) \neq 0,&\quad \mathcal{X}(\texttt{US}, \texttt{US}) = \mathtt{false} \\\\
d(\texttt{Spain}, \texttt{Spain}) = 0,&\quad \mathcal{X}(\texttt{US}, \texttt{Spain}) = \mathtt{true} \\\\
\end{align}



From this we can create the Hasse diagram for the preorder:

![](https://image.ibb.co/nHfh0J/Screenshot_20180619_012121.png)