Transitive relation

https://arbital.com/p/transitive_relation

by Dylan Hendrickson Jul 7 2016 updated Dec 19 2016

If a is related to b and b is related to c, then a is related to c.

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 $b \notin B$