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In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c.
In notation, this is:
- <math>\forall a, b, c \in X,\ a R b \and b R c \; \Rightarrow a R c<math>
For example, "is greater than" and "is equal to" are transitive relations: if a = b and b = c, then a = c.
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.
Examples of transitive relations include:
A transitive relation that is also reflexive is a preorder. A preorder that is antisymmetric is a partial order. A preorder that is symmetric, is an equivalence relation.
See also transitive closure, Intransitivity
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