> **Puzzle 171.** Is \\(\Phi(E,c) = \text{true}\\) ?

**No** (if I am reading the diagram correctly)

> **Puzzle 172.** Is \\(\Phi(E,e) = \text{true}\\)?

**Yes**

We have \\(E \leq N\\) so \\(\Phi(N,e) \leq \Phi(E,e)\\). Since the diagram indicates \\(\Phi(N,e)=\text{true}\\) then we know \\(\Phi(E,e) = \text{true}\\).

> **Puzzle 175.** Suppose \\(f : X \to Y\\) and \\(g : Y \to X\\) are monotone functions, and use them to build feasibility relations \\(\Phi\\) and \\(\Psi\\) as in the previous two puzzles. When is

>

> \[ \Phi = \Psi ? \]

When \\(f\\) and \\(g\\) are adjoints, that is to say \\(f \dashv g\\)

**No** (if I am reading the diagram correctly)

> **Puzzle 172.** Is \\(\Phi(E,e) = \text{true}\\)?

**Yes**

We have \\(E \leq N\\) so \\(\Phi(N,e) \leq \Phi(E,e)\\). Since the diagram indicates \\(\Phi(N,e)=\text{true}\\) then we know \\(\Phi(E,e) = \text{true}\\).

> **Puzzle 175.** Suppose \\(f : X \to Y\\) and \\(g : Y \to X\\) are monotone functions, and use them to build feasibility relations \\(\Phi\\) and \\(\Psi\\) as in the previous two puzzles. When is

>

> \[ \Phi = \Psi ? \]

When \\(f\\) and \\(g\\) are adjoints, that is to say \\(f \dashv g\\)