**Transitive Relations and Examples**

A relation *R* on a set *X* is transitive if, for all *x*,*y*,*z* in *X*, whenever *x R y* and *y R z* then *x R z*. Examples of transitive relations include the equality relation on any set, the "less than or equal" relation on any linearly ordered set, and the relation "*x* was born before *y*" on the set of all people. Symbolically, this can be denoted as: if *x* < *y* and *y* < *z* then *x* < *z*.

One example of a non-transitive relation is "city *x* can be reached via a direct flight from city *y*" on the set of all cities. Simply because there is a direct flight from one city to a second city, and a direct flight from the second city to the third, does not imply there is a direct flight from the first city to the third. The transitive closure of this relation is a different relation, namely "there is a sequence of direct flights that begins at city *x* and ends at city *y*". Every relation can be extended in a similar way to a transitive relation.

Read more about this topic: Transitive Closure

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