axiomatic sets 12 (relations)

In axiomatic set theory, a relation is also a set.

Ordinally we will look at what is a relation between the element $x$ and $y$ .

For instance,  in real numbers an ordered relation is satisfied.
Namely, in an arbitrary pair $\lt x,y\gt$  of two elements of real numbers
 only one of the three relations
\[ x=y,\quad x\lt y,\quad x\gt y \]
 must occure.

However, in axiomatic set theory we are interested in
the set $R$ which all $\left\{ \lt x,y\gt \right\}$ belongs to.

Please note that there is a set before a relation.

If $R$ is a binary relation,
\[ \forall z[z\in R\rightarrow \exists x\exists y[z=\lt x,y\gt ] ] . \]
 ,where $\lt x,y\gt$  is an ordered pair, which has been explained in the preceding post.

We will state it $xRy$ . $ xRy$ if and only if $\lt x,y\gt\in R$ .

If $R$ is a ternary relation,
\[ \forall z [z\in R\rightarrow \exists x\exists y\exists w[z=\lt\lt x,y\gt,w\gt ] ] . \]


In addition, well-known properties of a relation are as follow;

$R$ is reflexive in $A$ if and only if $\forall z[z\in A\rightarrow zRz ]$ .
$R$ is symmetric in $A$ if and only if $\forall x\forall y[x,y\in A , xRy \rightarrow yRx] $
$R$ is transitive in $A$  if and only if $\forall x\forall y\forall z[x,y,z\in A , xRy , yRz\rightarrow xRz] $

You must know the relation which satisfies above three conditions is an equivalence relation "=" on the set.

(This is not an axiom.)