Keith - that's interesting, and you're on the right track. You're also the only person so far who had the guts to try these puzzles, and I don't think I should keep cranking out lectures until people solve these puzzles.

But please clarify what you mean by saying

> \$$\texttt{false}\$$ maps to everything less than \$500

You're making it sound like a feasibility relation is a 'multi-valued function' where \$$\texttt{false}\$$ can map to lots of different things. That's a really cool way of talking, which we may be able to make sense of if we work a little - but that's not what we defined a feasibility relation to be.

We defined a feasibility relation \$$\Phi \colon \textbf{Bool} \to [0,\infty) \$$
to be a monotone function

$\Phi \colon \textbf{Bool}^{\text{op}} \times [0,\infty) \to \textbf{Bool} .$

So, I'd be happiest if you said what \$$\Phi(x,y)\$$ equals for each \$$x \in \textbf{Bool}^{\text{op}} \$$ and \$$y \in [0,\infty) \\$$. Then we could stare at that and see if it defines a monotone function.