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. 

You can do as follows: Let $S$ be a nonempty set. Set $$\sim:=\{(x,x)\mid x\in S\}.$$ That is, $\sim$ is just the diagonal of $S\times S.$  Then $\sim$ is clearly reflexive. The transitivity and symmetry and antisymmetry of $\sim$ is surely vacuously true.  In conclusion, $\sim$ is an equivalence relation on $S,$ so is a partial order relation. 
