Transitive relation

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Homogeneous relation R over a set X is transitive if for all elements a, b, c in X,
whenever R relates a to b and b to c, then R also relates a to c.

Each partial order as well as each equivalence relation needs to be transitive.

As a nonmathematical 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, too, is an ancestor of Carrie.

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

"Is greater than", "is at least as great as", and "is equal to" (equality) are
transitive relations on various sets,
for instance, the set of real numbers or the set of natural numbers:

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an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

The relation "is equal to" is the canonical example of an equivalence relation,
where for any objects a, b, and c:

a = a (reflexive property),
if a = b then b = a (symmetric property), and
if a = b and b = c, then a = c (transitive property).

As a consequence of the reflexive, symmetric, and transitive properties,
any equivalence relation provides a partition of the underlying set into
disjoint equivalence classes.

Two elements of the given set are equivalent to each other, if and only if
they belong to the same equivalence class.