> **Puzzle 205.** Suppose you are trying to buy either one or two loaves of bread - or perhaps none. Suppose bread costs $2 per loaf. Describe this using a feasibility relation \\(\Phi : \lbrace 0,1,2\rbrace \nrightarrow [0,\infty) \\). Here we make \\( \{0,1,2\}\\) into a poset with its usual ordering.

We can define

\[ \Psi \colon \\{0,1,2\\} \times [0,\infty) \to \textbf{Bool} \]

as follows

\[
\Psi(x, y) =
\begin{cases}
\texttt{false} & \mbox{if } (x = 1 \mbox{ and } y \le 2) \mbox{ or } (x = 2 \mbox{ and } y \le 4) \\\\
\texttt{true} & \mbox{otherwise.}
\end{cases}
\]

We can represent the mapping \\(\Psi\\) graphically – the red color denotes \\(\texttt{false}\\) and green denotes \\(\texttt{true}\\):

![Plot of psi](https://doneata.bitbucket.io/applied-category-theory/puzzle-205.png)

I think that the visual representation shows that the mapping is monotone.