Exercise 2.34#

For 1., in the following table we use 0 for no, 0.5 for maybe, and 1 for yes:

import pandas as pd

preorder = [0, 0.5, 1]
min = pd.DataFrame(index=preorder, columns=preorder, data=[[0,0,0], [0,0.5,0.5], [0,0.5,1]])
display(min)

	0.5	1.0
0.0	0.0	0.0
0.5	0.5	0.5
1.0	0.5	1.0

For 2 notice the min function is associative; see Associative property - Examples. The yes element can be used as a monoidal unit. It’s easy to see the symmetry/commutativity property is satisfied in the truth table.

To see the monotonicity condition is satisfied we’ll use as an inference rule that a partially-applied min function is order-preserving (i.e. monotone). That is, if we have that $a_1 \leq a_2$, it follows that $min(a_1, b) \leq min(a_2, b)$. Then, starting from these two premises: $$ \begin{align} \\ x_1 & \leq y_1 \\ x_2 & \leq y_2 \\ \end{align} $$

Applying $min(\_, y_2)$ to the first premise:

\[\begin{split} \begin{equation} \\ min(x_1, y_2) \leq min(y_1, y_2) \\ \label{eq:34a} \tag{a} \end{equation} \end{split}\]

Applying $min(x_1, \_)$ to the second premise:

\[\begin{split} \begin{equation} \\ min(x_1, x_2) \leq min(x_1, y_2) \\ \label{eq:34b} \tag{b} \end{equation} \end{split}\]

And combining $\eqref{eq:34a}$ and $\eqref{eq:34b}$ with transitivity:

\[ min(x_1, x_2) \leq min(x_1, y_2) \leq min(y_1, y_2) \]