Regarding this puzzle:

> **Puzzle 76.** Now suppose \\( (Y, \le_Y) \\) is a poset. Under what conditions on \\(f\\) can we conclude that \\( (X, \le_X ) \\) defined as above is a poset?

Sophie wrote:

> **Puzzle 76**. I agree with Marius that \\(f\\) must be injective. In fact I think this is a necessary and sufficient condition!

That's right! But you don't need me to tell you this, since you proved it, so you _know_ it's right.

(Of course sometimes we screw up when proving things, but writing down a proof and carefully checking the logic can reduce the chance of error quite dramatically.)