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A total order is a relation from a set to itself that satisfies the following properties for all  :

  1. Antisymmetry —  If and , then  ;
  2. Transitivity — If and , then  ;
  3. Totality — Either or .

The totality property implies the reflexive property:

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

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

  1. Greater than or equal to: define by for all  ;
  2. Less than: define by , but for all  ;
  3. Greater than: define by , but for all .

The following results can be derived from the previous definitions:

  1. The relation is also a total order;
  2. For any , exactly one of the following is true: