In set notation could somebody explain the meaning of $\mid$ in the equation below please? How does it read?
I read it as $s$ and $j$ are an element of $E$ but what does the $j \mid$ mean?
Similarly in this equation what does the comma mean?
Set notation what does the bar symbol mean
elementary-set-theorynotation
Best Answer
Suppose the $m_{j,i}$'s are elements of the set $M$, and that each pair $(j,i) \subset I \subset \Bbb{N}^2$ is an index of an element in $M$, where $I$ is an indexing set. Then
$$Inputs(i) = \sum_{j|(j,i) \in E} m_{j,i}$$ means "Given some fixed input value $i$, sum those elements in $M$ whose index $(j,i)$ satisfies $(j,i) \in E$.
Here $E$ is some subset of $\Bbb{N}^2$. So we loop over all the $j$'s ($i$ is fixed) and add the terms corresponding to those $j$'s which satisfy $(j,i) \in E$.
In the second picture, the comma can be read as a vertical bar. They have the same meaning in this case.
EDIT - in response to Shaun:
$I$ is some set used for indexing the elements in $M$, which must exist since otherwise the subscript of ordered pairs doesn't make sense. For example, take a $2\times2$-matrix. Then if you want to take the matrix element of the first row, second column, one writes $a_{1,2}$. But what is really going on, is that you have the index set $I = \{ (1,1),(1,2),(2,1),(2,2)\}$, where there is a one-to-one correspondence between elements in $I$ and the set of matrix elements.
For example, returning to our situation, taking the same indexing set $I$ as above and letting $E = \{(1,1),(2,1),(2,2)\}$, then $Inputs(2) = m_{2,2}$.
Another example of indexing sets: $$1/2+1/4+1/8 +... = \sum_{n \in \Bbb{N} } \frac{1}{2^n}$$
Here $\Bbb{N}$ is the indexing set.