# 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 $$a \leq b$$ 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 $$x \in S$$, $$a \in S$$.

Parents:

• 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 $$b \notin B$$

• 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.