Daniel Michael Cicala wrote:

> Here are two more examples of preorders. Take your favorite set X.

> 1) For all x in X, setting \\(x \leq x\\) gives you a preorder.

> 2) For all x,y in X, setting \\(x \leq y \\) gives you a preorder.

Nice! This leads to some other puzzles:

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

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

Clearly Puzzle 9 relies on getting the "right" answer to Puzzle 8, meaning the one that I'm thinking of. There could be other correct answers to Puzzle 8 that I haven't thought of.

> Here are two more examples of preorders. Take your favorite set X.

> 1) For all x in X, setting \\(x \leq x\\) gives you a preorder.

> 2) For all x,y in X, setting \\(x \leq y \\) gives you a preorder.

Nice! This leads to some other puzzles:

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

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

Clearly Puzzle 9 relies on getting the "right" answer to Puzzle 8, meaning the one that I'm thinking of. There could be other correct answers to Puzzle 8 that I haven't thought of.