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# Exercise 64 - Chapter 2

edited June 2018

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.

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edited June 2018

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: Comment Source: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)