Transitive relation

A binary relation \(R\) is transitive if whenever \(aRb\) and \(bRc\), \(aRc\).

The most common examples or transitive relations are partial orders (if \(a \leq b\) and \(b \leq c\), then \(a \leq c\)) and equivalence relations (if \(a \sim b\) and \(b \sim c\), then \(a \sim c\)).

A transitive relation that is also reflexive is called a preorder.

A transitive set \(S\) is a set on which the element-of relation \(\in\) is transitive; whenever \(a \in x\) and \(x \in S\), \(a \in S\).