Let $\langle V\_1, E\_1 \rangle, \langle V\_2, E\_2 \rangle$ and $\langle V\_3, E\_3\rangle$ be any three undirected simple graphs with $m$, $n$ and $p$ number of edges, respectively such that $E\_2$ and $E\_3$ have no edge in common. Then what would be the number of edges in
$$
\bigl[(E\_1\cup E\_2)\cup\lbrace (a,b): \text{$a\in V\_1$ and $b\in V\_2$} \rbrace\bigr]\cap \bigl[(E\_1\cup E\_3)\cup\lbrace (c,d): \text{$c\in V\_1$ and $d\in V\_3$}\rbrace\bigr],
$$
where the operations $\cup$ and $\cap$ are usual set union and join, respectively? Also, given that $V\_2$ and $V\_3$ have no vertex in common. Or, how to simplify and formulate 
$$
\bigl[(E\_1\cup E\_2)\cup\lbrace (a,b): \text{$a\in V\_1$ and $b\in V\_2$} \rbrace\bigr]\cap \bigl[(E\_1\cup E\_3)\cup\lbrace (c,d): \text{$c\in V\_1$ and $d\in V\_3$} \rbrace\bigr]
$$
to put into a more compact form?



Find the number of edges
The number of edges is $m$ plus all the possible edges between $V\_1 \cap V\_2$ and $V\_1 \cap V\_3$. For the ease of notation, I will define $V\_{1,2}$ as the set of all possible edges between $V\_1$ and $V\_2$, with a similar definition for $V\_{1,3}$. 


The intersection of unions is equal to the union of the intersections. Therefore, the set of edges described contains $E\_1$, since there is an $E\_1$ in both terms of the intersection. Any other term intersected with $E\_1$ will be a subset of $E\_1$ and so is already taken care of. Therefore, we need only consider the intersections of the remaining terms. By assumption, $E\_2 \cap E\_3$ is empty. The intersection $V\_{1,2} \cap V\_{1,3}$, will only contain edges between $V\_1 \cap V\_2$ and $V\_1 \cap V\_3$. This means the only intersections left to worry about are the intersection between $E\_2$ and $V\_{1,3}$, and the same with 2 and 3 switched. However the edges in $E\_2$ have both endpoints in $V\_2$, which is disjoint from $V\_3$, and so the intersection is empty. Therefore, the only edges in the intersection are in $E\_1$, which has size $m$, or between $V\_1 \cap V\_2$ and $V\_1 \cap V\_3$. 


