Suppose $R$ is a relation on $X$. What does it mean if $R$ is both a partial order and an equivalence?

Question

Suppose $R$ is a relation on $X$. What does it mean if $R$ is both a partial order and an equivalence?

I couldn't think of anything else other than $\emptyset$, is this correct?

Answer 1

$\emptyset$ isn't an equivalence relation (unless $X$ is empty), since it's not reflexive: any equivalence relation must contain at least the "diagonal set" $\{(x, x): x\in X\}$, this is exactly what reflexivity demands.

In fact, if we think about the diagonal set, it should quickly become clear that it's an example of an equivalence relation which is also a partial order! So we have one example.

Now, we want to figure out if there are any others. Here's the crucial point:

Suppose $R$ is a partial order. Given $x$, how many $y$ can there be such that both $xRy$ and $yRx$ hold?

Once you answer this, you should - together with the fact that any equivalence relation contains the diagonal set - be able to figure out what all the appropriate relations are ...

Answer 2

The difference between a partial order and an equivalence relation is that one is antisymmetric and the other is symmetric.

This means that if $x\mathrel{R}y$, then also $y\mathrel{R}x$ by symmetry; and therefore $x=y$ by antisymmetry.

Therefore $R$ can only be one thing.

(Also note that $R$ cannot be $\varnothing$, as long as $X$ is not empty, since both things require reflexivity, which ensures that $x\mathrel{R}x$ for all $x\in X$.)