Options

Exercise 64 - Chapter 2

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.

Previous Next

Comments

  • 1.
    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)
Sign In or Register to comment.