Complete Binary Relation

Definition

Complete Binary Relation

A binary relation $R\subseteq X\times X$ is complete if every pair of elements is comparable, i.e.,

$$\forall x,y\in X:xRy\vee yRx.$$

Examples

$(\mathbb{R},\le )$

The relation $\le$ on $\mathbb{R}$ is complete because for any $x,y\in \mathbb{R}$,

$$x\le y\text{or}y\le x.$$