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 \(x \in S\), \(a \in S\).