Is value in the matrix notation

Question

Given an $m\times n$ matrix $A=\begin{bmatrix} a_{1,1}&...&a_{1,n} \\ \vdots&\ddots&\vdots \\ a_{m,1}&...&a_{m,n} \\ \end{bmatrix}$, say that I wanted to describe a set generator $S(A)$ which only consists of every individual entry in A, using set builder notation. I want to write something like this...

$$S(A) = \left\{s\in\mathbb{R}\ :\ s\in A\right\}$$

But I know that '$s\in A$' is incorrect bc '$\in$' should only ever be used wrt sets; is there a common convention (perhaps another symbol) that I could use?

Answer 1

What is a matrix, exactly?

A good formal definition of a matrix is a function

$$A : [m] \times [n] \rightarrow \mathbb{R}$$

where $[m] = \{1,2,\cdots,m\}$ and $[n] = \{1,2,\cdots,n\}$. The codomain could be any set, but let's use $\mathbb{R}$ for sake of example.

Thus, when you see an entry $a_{i,j}$, you can essentially think of it as the output $A(i,j) \in \mathbb{R}$. In this case, you can easily see that the notation you're looking for is simply the range $ran(A)$of $A$:

$$ ran(A) = \{A(i,j) | (i,j) \in [m]\times [n]\} = \{a_{i,j} | (i,j) \in [m] \times [n]\}$$