Simple examples of preorders that are not partial orders? What are some simple examples of preorders — that is, binary relations that are reflexive and transitive — that are not partial orders (and hence not total orders, either)?
I'm looking for a couple of examples that do not involve graph theory or other less basic ideas in math. And preferably examples simpler than the relation $\preccurlyeq$ on the power set of a given set that declares $A \preccurlyeq B$ iff there exists an injection from $A$ to $B$.

Any real function (let's say a "measure") $\mu:X\to\mathbb{R}$ defines a preorder on $X$ by $x\preceq y$ iff $\mu(x)\leq\mu(y)$. For this not to be a partial order, your measure must not be injective.
This can be almost anything. Couple examples:


*

*diameter of compact subsets of $\mathbb{R}^n$ is a preorder on them.

*volume of compact measurable subsets of $\mathbb{R}^n$ is a preorder on them.

*number of distinct divisors is a preorder on $\mathbb{N}$ (well, any arithmetic function defines a preorder);

*...


In general, preorder is a very generic notion, so thinking out preorders is very easy.
