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$~$