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$~$