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.