## FANDOM

1,141 Pages

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 $a\le c$ ;
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 $a\ge b$ 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 $a\ge b$ , but $a\ne b$ for all $a,b\in S$ .

The following results can be derived from the previous definitions:

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