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 $a\le b$ and $b\le c$, then $a\le c$) 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 $x\in S$, $a\in S$.