| rdfs:comment | - 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 .
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| abstract | - 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: 3.
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