Editing Transitive relation (section)

==Examples==
As a non-mathematical example, the relation "is an ancestor of" is transitive. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy is also an ancestor of Carrie.

On the other hand, "is the birth mother of" is not a transitive relation, because if Alice is the birth mother of Brenda, and Brenda is the birth mother of Claire, then it does not follow that Alice is the birth mother of Claire. In fact, this relation is [[antitransitive]]: Alice can ''never'' be the birth mother of Claire.

Non-transitive, non-antitransitive relations include sports fixtures (playoff schedules), 'knows' and 'talks to'.

The examples "is greater than", "is at least as great as", and "is equal to" ([[equality (mathematics)|equality]]) are transitive relations on various sets.
As are the set of real numbers or the set of natural numbers:

: whenever ''x'' &gt; ''y'' and ''y'' &gt; ''z'', then also ''x'' &gt; ''z''
: whenever ''x'' &ge; ''y'' and ''y'' &ge; ''z'', then also ''x'' &ge; ''z''
: whenever ''x'' = ''y'' and ''y'' = ''z'', then also ''x'' = ''z''.

More examples of transitive relations:
* "is a [[subset]] of" (set inclusion, a relation on sets)
* "divides" ([[divisor|divisibility]], a relation on natural numbers)
* "implies" ([[material conditional|implication]], symbolized by "⇒", a relation on [[proposition]]s)

Examples of non-transitive relations:
* "is the [[successor function|successor]] of" (a relation on natural numbers)
* "is a member of the set" (symbolized as "∈")<ref>However, the class of [[von Neumann ordinal]]s is constructed in a way such that ∈ ''is'' transitive when restricted to that class.</ref>
* "is [[perpendicular]] to" (a relation on lines in [[Euclidean geometry]])

The [[empty relation]] on any set <math>X</math> is transitive<ref>{{harvnb|Smith|Eggen|St. Andre|2006|loc=p. 146}}</ref> because there are no elements <math>a,b,c \in X</math> such that <math>aRb</math> and <math>bRc</math>, and hence the transitivity condition is [[vacuous truth|vacuously true]]. A relation {{math|''R''}} containing only one [[ordered pair]] is also transitive: if the ordered pair is of the form <math>(x, x)</math> for some <math>x \in X</math> the only such elements <math>a,b,c \in X</math> are <math>a=b=c=x</math>, and indeed in this case <math>aRc</math>, while if the ordered pair is not of the form <math>(x, x)</math> then there are no such elements <math>a,b,c \in X</math> and hence <math>R</math> is vacuously transitive.