Notation for a binary relation depending on another set I am new to binary relations and graph theory, and I have actually not written math equations for quite some time...
Is the following notation correct ?
$R_E$ is a binary relation between edges from $E$ and edges from $E'$.
It is based an on a binary relation $R_V$ between vertices.
$R_E$ includes edges that link vertices that are related through $R_V$.
$$\begin{equation}
R_E =  \left\{ \{ e, e' \} \left| 
\begin{array}{l}
    e \in E, e' \in E',\\
    \phi(e) = \{ v_1, v_2 \}, \\
    \phi'(e') = \{ v_1', v_2' \}, \\
    \{ \{ v_1, v_1' \}, \{ v_2, v_2' \} \} \subset R_V \\
    \lor \{ \{ v_1, v_2' \}, \{ v_2, v_1' \} \} \subset R_V
\end{array} 
\right.
\right\}
\end{equation}$$
Is it mathematically correct, and is it graphically correct (with big braces and a big line) ?
Thank you
 A: Usually binary relations are ordered pairs: $(e, e')$ rather than unordered pairs $\{e, e'\}$; this allows for the possibility of the relation being asymmetric, and in the case of a relation between $E$ and $E'$ it is much easier to define $R_E$ as a subset of $E \times E'$.
Aside from that, your notation is perfectly correct...
...and a terrible idea, because if you're expressing a thought that complicated, it's better to write it out in words, rather than in symbols. Unless you're an actor playing a mathematician in a movie, the point is to be understood, rather than write an impressive formula, and so I would just write

$R_E$ is a binary relation between $E$ and $E'$; an edge $e \in E$ is related to an edge $e' \in E'$ if the two endpoints of $e$ are related via $R_V$ to the two endpoints of $e'$ (in some order).

A: As the answer by Misha suggest that you can write as follows.
$R_E =\left\{ ( e, e')\mid e \in E, e' \in E'\right\}$; Also given that $\phi(e) = ( v_1, v_2 ), \phi'(e') = (v_1', v_2' ),\{ ( v_1, v_1' ), (v_2, v_2' ) \} \subset R_V \text{ and }\{ ( v_1, v_2' ), (v_2, v_1' )\} \subset R_V.$
