Total order

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

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

The totality property implies the reflexive property: $a \le a$

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

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

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

The following results can be derived from the previous definitions:

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