Matthew Doty wrote:

> > > **Puzzle 8.** What simple law do Daniel's preorders obey, that does not hold for the real numbers with its usual notion of \\(\leq\\)?

> > Both obey the *symmetry* rule: if \\(x \leq y\\) then \\(y \leq x\\)

> > **Puzzle 9.** What do you call preorders that obey this law?

> They are [*equivalence relations*](https://en.wikipedia.org/wiki/Equivalence_relation).

Right! This is what I was looking for. But Dan Schmidt was on the right track, because any partition determines an equivalence relation, and vice versa. They are two ways of thinking about the same thing.

This sort of preorder is sort of the extreme opposite case from a total order, since total orders obey the **antisymmetry** rule: if \\(x \leq y\\) and \\(y \leq x\\) then \\(x = y\\).

What are all the total orders that are also equivalence relations?

> > > **Puzzle 8.** What simple law do Daniel's preorders obey, that does not hold for the real numbers with its usual notion of \\(\leq\\)?

> > Both obey the *symmetry* rule: if \\(x \leq y\\) then \\(y \leq x\\)

> > **Puzzle 9.** What do you call preorders that obey this law?

> They are [*equivalence relations*](https://en.wikipedia.org/wiki/Equivalence_relation).

Right! This is what I was looking for. But Dan Schmidt was on the right track, because any partition determines an equivalence relation, and vice versa. They are two ways of thinking about the same thing.

This sort of preorder is sort of the extreme opposite case from a total order, since total orders obey the **antisymmetry** rule: if \\(x \leq y\\) and \\(y \leq x\\) then \\(x = y\\).

What are all the total orders that are also equivalence relations?