In the puzzles [last time](https://forum.azimuthproject.org/discussion/2281/lecture-57-chapter-4-feasibility-relations/p1) we saw something nice: a [feasibility relation](https://forum.azimuthproject.org/discussion/2281/lecture-57-chapter-4-feasibility-relations#latest) between preorders is a generalization of something we learned about a long time ago: a [monotone function](https://forum.azimuthproject.org/discussion/1828/lecture-4-chapter-1-galois-connections/p1) between preorders. But the really cool part is this: given preorders \\(X\\) and \\(Y\\), we can get a feasibility relation \\(\Phi : X \nrightarrow Y\\) either from a monotone function \\(f : X \to Y\\) or from a monotone function \\(g: Y \to X\\). So, feasibility relations put monotone functions going _forwards_ from \\(X\\) to \\(Y\\) and those going _backwards_ from \\(Y\\) to \\(X\\) into a common framework! Even better, we saw that one of our favorite themes, namely _adjoints_, is deeply connected to this idea. Let me state this as a theorem: **Theorem.** Let \\(f : X \to Y \\) and \\(g: Y \to X\\) be monotone functions between the preorders \\(X\\) and \\(Y\\). Define the feasibility relations \\(\Phi : X \nrightarrow Y\\) by \[ \Phi(x,y) \text{ if and only if } f(x) \le y \] and \[ \Psi(x,y) \text{ if and only if } x \le g(y) .\] Then \\(\Phi = \Psi\\) if and only if \\(f \\) is the left adjoint of \\(g\\). **Proof.** We have \\(\Phi = \Psi\\) iff \[ \Phi(x,y) \text{ if and only if } \Psi(x,y) \] for all \\(x \in X, y \in Y\\), but by our definitions this is true iff \[ f(x) \le y \text{ if and only if } x \le g(y) \] which is true iff \\(f\\) is the left adjoint of \\(g\\). \\( \qquad \blacksquare \\) Ah, if only all proofs were so easy! Now, to make feasibility relations into a truly satisfactory generalization of monotone functions, we should figure out how to _compose_ them. Luckily this is easy, because we already know how to compose relations from [Lecture 40](https://forum.azimuthproject.org/discussion/2223/lecture-40-chapter-3-relations/p1). So, we should try to prove this: **Theorem.** Suppose that \\(\Phi : X \nrightarrow Y, \Psi : Y \nrightarrow Z\\) are feasibility relations between preorders. Then there is a **composite** feasibility relation \[ \Psi \Phi : X \nrightarrow Z \] defined as follows: \[ (\Psi \Phi)(x,z) = \text{true} \] if and only if for some \\(y \in Y\\), \[ \Phi(x,y) = \text{true} \text{ and } \Psi(y,z) = \text{true}. \] **Puzzle 176.** Prove this! Show that \\(\Psi \Phi\\) really is a feasibility relation. I hope you see how reasonable this form of composition is. Think of it in terms of our pictures from last time: