Equivalence and Order Relations I have the following problem: Provide an example of a set $S$ and a relation $\sim$ on $S$ such that $\sim$ is both an equivalence relation and an order relation.  Conjecture for which sets and this is possible, and prove your conjecture. 
Below is the example I came up with but I don't know how to prove my conjecture and I don't even know if I am correct, I would really appreciate some help!
The only set that satisfies the reflexive property of an equivalence relation and the nonreflexive property of an order relation is the $\emptyset$.  For any relation on that set, it is vacuously true that is is both an equivalence relation and an order relation. 

Assuming your order relations are required to be strict (that is, irreflexive), then you're almost done.
You have already shown that $\varnothing$ has an appropriate relation. What remains is to show that no nonempty set can have a relation that is both an equivalence and an order.
Thus assume that $A$ is a set, $x\in A$ (such an $x$ will exist if $A$ is nonempty), and $R\subseteq A^2$ is a relation on $A$, and that $R$ is a equivalence relation, and that $R$ is a (partial?) order. Use these assumptions to reach a contradiction.
