> **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.