What is the difference between an ordering and ordered set?

Question

Given a set $X$ and an ordering $R$ on $X$, what is the difference between $R$ and $(X, R)$? Is $(X, R)$ just a way to say that the $X$ is special cause it can be ordered?

If so, why did we choose to represent this with an ordered pair? Is there a reason or just convenient notation?

Answer 1

As a concrete object, $R$ is a binary relation on $X$, so we can view it as a subset of $X\times X$.

So $X$ is a set, $R$ is a subset of $X\times X$, and $(X,R)$ is a pair consisting of the two objects $X$ and $R$.

In some cases you might be able to reverse engineer $X$ from $R$. For example if I'm told $R$ is a partial order then $(a,a)\in R$ for all $a\in X$. So I can find all elements of $X$ just by examining $R$.

One might say:

$X$ is a set.

$R$ is an ordering on the set $X$.

$(X,R)$ is an ordered set.

It's similar to how one might say that a group is a pair $(G,\ast)$ where $G$ is a set and $\ast$ is a binary relation on that set. (If you're familiar with groups.)

Answer 2

Formal definition is following:

Ordering is a correspondence $\mathcal{G}=(R, X, X)$ such that relation $(x,y) \in R$ is an order relation on $X$. Sometimes, by abuse of language, graph $R$ is called an ordering on $X$. We say that set $X$ is ordered by ordering $\mathcal{G}$.