# Lecture 24 - Pricing Resources

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.**