About: Total order   Sponge Permalink

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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 .

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  • Total order
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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 .
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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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