Ordered set

Definition

Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following two properties:

1. If $x\in S$ and $y\in S$, then one and only one of the statements: $$\begin{array}{r}x<y,x=y,y<x\end{array}$$ is true.
2. If $x,y,z\in S$, and if both $x<y$ and $y<z$, then $x<z$.

An ordered set is a set $S$ in which an order is defined.