For example, "is greater than," "is at least as great as," and "is equal to" (equality) are transitive relations:
For some time, economists and philosophers believed that preference was a transitive relation however there are now mathematical theories which demonstrate that preferences and other significant economic results can be modelled without resorting to this assumption.
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 always 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 two elements.
Unlike other relation properties, no general formula that counts the number of transitive relations on a finite set is known. However, there is a formula for finding the number of relations which are simultaneously reflexive, symmetric, and transitive – in other words, equivalence relations – , those which are symmetric and transitive, those which are symmetric, transitive, and antisymmetric, and those which are total, transitive, and antisymmetric. Pfeiffer has made some progress in this direction, expressing relations with combinations of these properties in terms of each other, but still calculating any one is difficult. See also.
Inference Based on Transitive Relation in Tree Shrews (Tupaia Belangeri) and Rats (Rattus Norvegicus) on a Spatial Discrimination Task
Mar 22, 2008; To use transitive inference (TI) is to deduce a conclusion from premises that have a nonequivalent relationship; for example, a...