\begin{equation} \Phi(i,j) = \begin{cases} 1, & \textrm{if } \sum_{f_k \in \mathcal{F}} x_k(i,j) > 0, \\ 0, & \textrm{otherwise.} \end{cases} \end{equation} \begin{equation} \alpha = \sum_{(i,j) \in \mathcal{L}} \Phi(i,j) \end{equation}
I need $\alpha$. Is it possible to write a compact expression $\alpha$ without explicitly defining $\Phi(i,j)$
Edit:
$\mathcal{L}$ is the set of links in a graph. $x_k(i,j)$ denotes if a link is active for a particular flow $f_k \in \mathcal{F}$ where $\mathcal{F}$ is the set of all flows. $x_k(i,j)$ can either be 0 (False) or 1 (True) denoting link $(i,j)$ is inactive or active. Thus, $\alpha$ can be at most $|\mathcal{L}|$.
One way of doing it would be the following $\alpha = \sum\limits_{(i,j) \in \mathcal{L}}(1-\prod \limits_{f_k \in \mathcal{F}}(1 - x_k(i,j))$. Basically you use $\prod$ as big $\bigwedge$ and use subtraction from one as a not to create a $\bigvee$.