Total order

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

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

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

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

If $\leq$ (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 $b\leq a$ for all $a,b\in S$ ;
2. Less than: define $a<b$ by $a\leq b$ , but $a\neq b$ for all $a,b\in S$ ;
3. Greater than: define $a>b$ by $a\geq b$ , but $a\neq b$ 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:
   • $a<b$
   • $a=b$
   • $a>b$