Sets and set-builder notation I have few questions about sets and set-builder notations.

*

*How to name the sets correctly ?
I know that typically we use capital letters $A, B, C, ...$ , but I wonder if it is allowed to name sets like $AT, A1, A_1, A^1$ or some of them are reserved for representing some operation on set.


*How to show that the set-builder for set B depends on set-builder from set A ?
For example:
$A = \{n, n \in Z\}$
$B = \{\frac{2n + 10}{7}, n \in Z\}$
Where set-builder from set B is connected with set-builder from set A by the function:
$f(n) = \frac{2n + 10}{7}$,
or it is actually shown by naming the variables from both set-builders the same "n" ?


*Is it right that if I would like to check if set A and set B have common values I have to compare their set-builders ?
For example:
$A = \{n, n \in Z\}$
$B = \{2n, n \in Z\}$
$n = 2m$
There exists solution for all $n,m \in Z$, so actually set B is subset of set A.
 A: (1) The notation $A, B, C$ are fine. I would only use $AT$ if it can be seen as a mnemonic to describe the set, like in
$$
  AT = \bigl\{ \mathrm{arctan}(x) \mid -1 \leqslant x \leqslant 1 \bigl\}
$$
I would avoid $A1$. The next one, $A_1$, is common if you have a family of sets, like in
$$
\text{For $0 \leqslant r < 5$, let $A_r = \{ n \in \Bbb N \mid n \equiv r \bmod 5\}$. Then $A_1 = \{ 1, 6, 11, 16, \dotsm \}$}
$$
The notation $A^1$ should generally be avoided, since $A^n$ usually denotes the Cartesian product of $n$ copies of $A$, that is $\underbrace{A \times {}\dotsm{} \times A}_{\text{$n$ times}}$.
(2) The correct notation should be
$$
A = \{ n \mid n \in \Bbb Z \} \qquad B = \Bigl\{ \frac{2n + 10}{7} \mid n \in \Bbb Z \Bigr\}
$$
although the set $A$ is simply $\Bbb Z$. If you want to insist that you use different set builders on the same domain, you could first introduce the functions you intend to use, and then define the sets, like in
$$
\text{Let $f_1(n) = n$, $f_2(n) = 2n$ and $f_3(n) = \frac{2n + 10}{7}$. For $1 \leqslant i \leqslant 3$, set $A_i = \bigl\{ f_i(n) \mid n \in \Bbb Z \bigr\}$ }.
$$
(3) To compare the sets $A_1$, $A_2$ and $A_3$, it would indeed be sufficient to compare the images of the functions $f_1$, $f_2$ and $f_3$.
