It looks like you're new here. If you want to get involved, click one of these buttons!
Recall the “regions of the world” Lawvere metric space from Exercise 2.33 and the text above it. We just learned that we can convert it to a preorder. Draw the Hasse diagram for the preorder corresponding to the regions: US, Spain, and Boston.
Comments
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:
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: 