axiomatic sets 36 (a totally ordered relation)

We have already gotten the equivalent relation.
For a set $\Omega$ and elements $a,b,c\in \Omega$ ,

(1)if $a\sim b$ ,then $b\sim a$ .
(2)for all $a$ , $a\sim a$ .
(3)if $a\sim b$ and $b\sim c$ , then $a\sim c$ .

We get the ordering relation by changing the condition.
The difference is only (1).

(1)if $a\preceq b$ and $b\preceq a$ , then $a\sim b$
(2)for all $a$ , $a\preceq a$ .
(3)if $a\preceq b$ and $b\preceq c$ , then $a\preceq c$ .

Namely, $a\preceq b$ does not always mean $b\preceq a$ .

A totally ordered relation of the set is for any $a,b\in\Omega$ , $a\preceq b$ or $b\preceq a$ is true.

If it is not true, the set is incomparable and called the partially ordered set.

In axiomatic set theory, we often use a pair $(\Omega,\preceq)$ of a set $\Omega$ and a relation $\preceq$ .
(you must note that a relation is also a set. )