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