Total relation: Difference between revisions
No edit summary |
Note that this implies reflexivity. |
||
| Line 3: | Line 3: | ||
In [[mathematical notation]], this is |
In [[mathematical notation]], this is |
||
:<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> |
||
Note that this implies [[Reflexive relation|reflexivity]]. |
|||
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 or equal to 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 proper subset of" is also not total. |
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 or equal to 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 proper subset of" is also not total. |
||
Revision as of 08:46, 7 May 2007
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
Note that this implies reflexivity.
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 or equal to 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 proper subset of" is also not total.
Total relations are sometimes said to have comparability.
A total order provides a common example of a total relation.