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$~$.
Comments
Martin Epstein
Is this what is meant by transitive and nontransitive set?
Transitive:
$~$A = \{ \{ 1,2 \}, \{ 3,4 \}, 1, 2, 3, 4 \}$~$
$~$x = \{1,2\}$~$
$~$a = 2$~$
$~$a \in x$~$, $~$x \in A$~$ and $~$a \in A$~$
Nontransitive:
$~$B = \{ \{ 1,2 \}, \{ 3,4 \} \}$~$
$~$y = \{1,2\}$~$
$~$b = 2$~$
$~$b \in y$~$, $~$y \in B$~$ but $~$b \notin B$~$
Kevin Clancy
Yes, that's correct. I wonder if it is even a good idea to talk about transitive sets in the transitive relation page, as most people who are interested in transitive relations are not likely to care about transitive sets. When this page is expanded beyond stub status, I hope that it will focus mostly on transitivity, rather than related concepts such transitive sets, posets, and preorders.