I'm trying to understand some graph theory and set notation. Does the statement below read like this for calculating the flow in a graph?
Given some fixed input value j, sum those elements in f whose index (s,j) satisfies (s,j)∈ E
So loop over all the s's (j is fixed) and add the terms corresponding to those js's which satisfy (s,j)∈E.
Which effectively means sum all the flows on the edges to get the flow in a network.
It does not read like this. The comma symbol, in relation to set membership, usually satisfies this implication: $x, y \in A \implies x \in A $ and $y \in A$. We know $j$ cannot be an edge, since it's also used in the node pair $(i,j)$ representing an edge in $E$. It's strange to have an element $j$ be both a node and an edge, but if the comma is interpreted like this, that's what we get.
To fix a variable, usually you would say $f_j = \sum_{(s,j)\in E} f_{s,j}$. Since $j$ is affixed as a subscript of $f$, it is made clear that $j$ is fixed for the right side of the equation.