Total Orders

What is a Total Order?

A total order is a partial order where every pair is comparable.

For any $a$ and $b$: either $a \leq b$ or $b \leq a$ (or both, if $a = b$).

No incomparable elements allowed.

Visualizing a Total Order

In a total order, everything forms a single chain. No branches, no side-by-side elements.

Compare this to the Hasse diagram for partial orders, which had branches.

Example: $\leq$ on Numbers

Pick any two numbers. One is always less than or equal to the other.

Every pair is comparable. So $\leq$ is a total order.

Non-Example: $\subseteq$ on Sets

Consider $\{1, 2\}$ and $\{2, 3\}$:

• Is $\{1, 2\} \subseteq \{2, 3\}$? No
• Is $\{2, 3\} \subseteq \{1, 2\}$? No

These sets are incomparable.

$\subseteq$ is a partial order, but not a total order.

Summary

A total order is a partial order with no incomparable pairs.