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.