Transitive relation
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) are transitive relations on various sets.As are the set of real numbers or the set of natural numbers:

More examples of transitive relations:

Examples of non-transitive relations:

The empty relation on any set X {\displaystyle X} is transitive because there are no elements a, b, c ∈ X {\displaystyle a,b,c\in X} such that a R b {\displaystyle aRb} and b R c {\displaystyle bRc}, and hence the transitivity condition is vacuously true. A relation R containing only one ordered pair is also transitive: if the ordered pair is of the form (x, x) {\displaystyle (x,x)} for some x ∈ X {\displaystyle x\in X} the only such elements a, b, c ∈ X {\displaystyle a,b,c\in X} are a = b = c = x {\displaystyle a=b=c=x}, and indeed in this case a R c {\displaystyle aRc}, while if the ordered pair is not of the form (x, x) {\displaystyle (x,x)} then there are no such elements a, b, c ∈ X {\displaystyle a,b,c\in X} and hence R {\displaystyle R} is vacuously transitive.
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