How to write $A=\{...,A_{-2},A_{-1},A_{0},A_{1},A_{2},...\}$ using the union notation?

Question

Let, $A_m$ where $m\in \mathbb Z$ be the set $\{(m,n):n\in\mathbb Z\}$.For eg. $A_1={...,(1,-2),(1,-1),(1,0),(1,1),(1,2),...}$

Now I define the set $A$ to be the collection of all such sets ie. $A=\{...,A_{-2},A_{-1},A_{0},A_{1},A_{2},...\}$. How do I write A using the union notation?

Answer 1

$A$ is not a union of the sets $A_m$. Instead, each $A_m$ is an element of the set $A$. With that said, the most natural way to describe the set $A$ is to say that $A = \{A_m : m \in \Bbb Z\}.$