A total order ${\displaystyle \le}$ is a relation from a set ${\displaystyle S}$ to itself that satisfies the following properties for all ${\displaystyle a, b, c \in S}$ :

1. Antisymmetry —  If ${\displaystyle a \le b}$ and ${\displaystyle b\le a}$ , then ${\displaystyle a = b}$ ;
2. Transitivity — If ${\displaystyle a \le b}$ and ${\displaystyle b\le c}$, then ${\displaystyle a \leq c}$ ;
3. Totality — Either ${\displaystyle a \le b}$ or ${\displaystyle b\le a}$ .

The totality property implies the reflexive property: ${\displaystyle a \le a}$

Since ${\displaystyle \le}$ is antisymmetric, transitive, and reflexive, it is also a partial order.

If ${\displaystyle \le}$ (less than or equal to) is a total order on a set ${\displaystyle S}$ , then we can define the following relations:

1. Greater than or equal to: define ${\displaystyle a \geq b}$ by ${\displaystyle b\le a}$ for all ${\displaystyle a, b \in S}$ ;
2. Less than: define ${\displaystyle a < b}$ by ${\displaystyle a \le b}$ , but ${\displaystyle a \ne b}$ for all ${\displaystyle a, b \in S}$ ;
3. Greater than: define ${\displaystyle a > b}$ by ${\displaystyle a \geq b}$ , but ${\displaystyle a \ne b}$ for all ${\displaystyle a, b \in S}$ .

The following results can be derived from the previous definitions:

1. The relation ${\displaystyle \geq}$ is also a total order;
2. For any ${\displaystyle a, b \in S}$ , exactly one of the following is true:
• ${\displaystyle a < b}$
• ${\displaystyle a = b}$
• ${\displaystyle a > b}$