Transitive relation

In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c. Transitivity (or transitiveness) is a key property of both partial order relations and equivalence relations.

In terms of set theory, the transitive relation can be defined as:

For example, "is greater than", "is at least as great as," and "is equal to" (equality) are transitive relations:

On the other hand, "is the mother of" is not a transitive relation, because if Alice is the mother of Brenda, and Brenda is the mother of Claire, then Alice is not the mother of Claire. What is more, it is antitransitive: Alice can never be the mother of Claire.

Then again, in biology we often need to consider motherhood over an arbitrary number of generations: the relation "is a matrilinear ancestor of". This is a transitive relation. More precisely, it is the transitive closure of the relation "is the mother of".

More examples of transitive relations:

The converse of a transitive relation is always transitive: e.g. knowing that "is a subset of" is transitive and "is a superset of" is its converse, we can conclude that the latter is transitive as well.

The intersection of two transitive relations is always transitive: knowing that "was born before" and "has the same first name as" are transitive, we can conclude that "was born before and also has the same first name as" is also transitive.

The union of two transitive relations is not always transitive. For instance "was born before or has the same first name as" is not generally a transitive relation.

The complement of a transitive relation is not always transitive. For instance, while "equal to" is transitive, "not equal to" is only transitive on sets with at most one element.

A transitive relation is asymmetric if and only if it is irreflexive.

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