Total relation: Difference between revisions
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:<math>\forall a, b \in X,\ a R b \or b R a.</math> |
:<math>\forall a, b \in X,\ a R b \or b R a.</math> |
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For example, "is less than or equal to" is a total relation over the set of real numbers, because for two numbers either the first is less than equal than the second, or the second is less than or equal to the first. |
For example, "is less than or equal to" is a total relation over the set of real numbers, because for two numbers either the first is less than equal than the second, or the second is less than or equal to the first. On the other hand, "is less than" is not a total relation, since one can pick two equal numbers, and then neither the first is less than the second, nor is the second less than the first (but note that "is less than" is a weak order which gives rise to a total order, namely "is less than or equal to". The relationship between strict orders and weak orders is discussed at [[partially ordered set]]). The relation "is a subset of" is also not total. |
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Total relations are sometimes said to have ''comparability''. |
Total relations are sometimes said to have ''comparability''. |
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Revision as of 23:52, 22 November 2005
In mathematics, a binary relation R over a set X is total if it holds for all a and b in X that a is related to b or b is related to a (or both).
In mathematical notation, this is
For example, "is less than or equal to" is a total relation over the set of real numbers, because for two numbers either the first is less than equal than the second, or the second is less than or equal to the first. On the other hand, "is less than" is not a total relation, since one can pick two equal numbers, and then neither the first is less than the second, nor is the second less than the first (but note that "is less than" is a weak order which gives rise to a total order, namely "is less than or equal to". The relationship between strict orders and weak orders is discussed at partially ordered set). The relation "is a subset of" is also not total.
Total relations are sometimes said to have comparability.
A common total relation is the total order.