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.](http://www.azimuthproject.org/azimuth/show/Applied+Category+Theory#Chapter_2)**