>**Puzzle 209.** Suppose you are trying to fry some eggs and also toast some slices of bread. Describe each process separately as a feasibility relation from \\(\mathbb{N}\\) to \\(\mathbb{N}\\) and then tensor these relations. What is the result?

The exact requirements of frying and toasting is vague... but I think this is what we were supposed to do?

Define \\(\Phi(x,x’)\\) to be feasibility for frying where you get one fried egg (x) from one egg (x’). Similarly define \\(\Psi(y,y’)\\) to be feasibility for toasting where you get one toast (y) from one slice of bread (y’).Then :

\[

\Phi(x,x') =

\begin{cases}

\texttt{true} & \mbox{if } x \leq x' \\\\

\texttt{false} & \mbox{otherwise.}

\end{cases}

\]

\[

\Psi(y,y') =

\begin{cases}

\texttt{true} & \mbox{if } y \leq y' \\\\

\texttt{false} & \mbox{otherwise.}

\end{cases}

\]

\[

(\Phi \otimes \Psi)((x,y),(x',y')) =

\begin{cases}

\texttt{true} & \mbox{if } (x \leq x') \mbox{ and } (y \leq y') \\\\

\texttt{false} & \mbox{otherwise.}

\end{cases}

\]

>**Puzzle 211.** What general mathematical result is Puzzle 209 an example of?

This is just the monoidal preorder law.

The exact requirements of frying and toasting is vague... but I think this is what we were supposed to do?

Define \\(\Phi(x,x’)\\) to be feasibility for frying where you get one fried egg (x) from one egg (x’). Similarly define \\(\Psi(y,y’)\\) to be feasibility for toasting where you get one toast (y) from one slice of bread (y’).Then :

\[

\Phi(x,x') =

\begin{cases}

\texttt{true} & \mbox{if } x \leq x' \\\\

\texttt{false} & \mbox{otherwise.}

\end{cases}

\]

\[

\Psi(y,y') =

\begin{cases}

\texttt{true} & \mbox{if } y \leq y' \\\\

\texttt{false} & \mbox{otherwise.}

\end{cases}

\]

\[

(\Phi \otimes \Psi)((x,y),(x',y')) =

\begin{cases}

\texttt{true} & \mbox{if } (x \leq x') \mbox{ and } (y \leq y') \\\\

\texttt{false} & \mbox{otherwise.}

\end{cases}

\]

>**Puzzle 211.** What general mathematical result is Puzzle 209 an example of?

This is just the monoidal preorder law.