FANDOM


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 $