Total orders

Definition. A partial order $\le$ on $A$ is total if for all $x,y\in A$, $x\le y$ or $y\le x$.

Example total. $(\mathbb{R},\le)$.

Example not total. $(\mathcal{P}(\{1,2\}),\subseteq)$: $\{1\}$ and $\{2\}$ are incomparable.

Conclusion. Total orders are partial orders with comparability; the subset relation is typically not total.