Today's lecture will be very short, consisting solely of some puzzles about _prices_.

We often compare resources by comparing their prices. So, we have some set of things \\(X\\) and a function \\(f: X \to \mathbb{R}\\) that assigns to each thing a price. Given two things in the set \\(X\\) we can then say which costs more... and this puts a preorder on the set \\(X\\). Here's the math behind this:

**Puzzle 75.** Suppose \\( (Y, \le_Y) \\) is a preorder, \\(X\\) is a set and \\(f : X \to Y\\) is any function. Define a relation \\(\le_X\\) on \\(X\\) by

\[ x \le_X x' \textrm{ if and only if } f(x) \le_Y f(x') .\]

Show that \\( (X, \le_X ) \\) is a preorder.

Sometimes this trick gives a poset, sometimes not:

**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?

We often have a way of combining things: for example, at a store, if you can buy milk and you can buy eggs, you can buy milk _and_ eggs. Sometimes this makes our set of things into a monoidal preorder:

**Puzzle 77.** Now suppose that \\( (Y, \le_Y, \otimes_Y, 1_Y) \\) is a monoidal preorder, and \\( (X,\otimes_X,1_X ) \\) is a monoid. Define \\(\le_X\\) as above. Under what conditions on \\(f\\) can we conclude that \\( (X,\le_X\otimes_X,1_X) \\) is a monoidal preorder?

We will come back to these issues in a bit more depth when we discuss Section 2.2.5 of the book.

**[To read other lectures go here.](**