Listing elements from set-builder notation, and vice versa I have trouble translating from a set-builder notation to a "dotted set"
$$\{\ldots,v_1,v_2,v_3,\ldots\}$$
and vice-versa.
Set-builder to dotted set:
$$\begin{align*}
A &= \{5a+ 2b : a,b \in \mathbb{Z}\}\\  
B &= \{6a + 2b : a,b \in \mathbb{Z}\}
\end{align*}$$
Here, my problem is that I don't know how to list every element of the set. I get confused because there are 2 variables.
Dotted set to set-builder:
$$\begin{align*}
C &= \{0,4,16,36,64,100, \ldots\} \\
D &= \{0,1,4,9,16,25, \ldots\}  \\
E &= \{3,6,11,18,27,38, \ldots\}
\end{align*}$$
Here, my problem is that I know that this set follows a certain pattern, but I don't know how to extract this pattern, and express it in set builder notation.
 A: For the two set-builder $\to$ dotted set questions you mentioned, you should use Bezout's identity (Wikipedia link), which essentially says that
$$\{ma+nb:a,b\in\mathbb{Z}\}=\{dc:c\in\mathbb{Z}\}$$
where $d=\gcd(m,n)$. Thus, concrete knowledge about this particular situation is necessary to simplify the representation of the set.
For the reverse, it is impossible in principle because an infinite number of patterns will match a given finite set of data, but you are expected to make a "reasonable" guess as to what the question-writer was thinking of. For example, the probable intention was
$$D=\{n^2:n\in\mathbb{Z},n\geq 0\}$$
Any polynomial-type patterns are findable via the calculus of finite differences.
A: For your Set-builder to dotted set problem, you just need (1) a reliable pattern to list all the integers (i.e. 0, 1, -1, 2, -2...), then you a reliable way to mix two of those reliable lists, which we can copy from the standard trick to list all the rational numbers (i.e. diagonalise)
( 0, 0) ( 1, 0) (-1, 0) ( 2, 0) (-2, 0)...
( 0, 1) ( 1, 1) (-1, 1) ( 2, 1)...
( 0,-1) ( 1,-1) (-1,-1)...
( 0, 2) ( 1, 2)...
from which we can take the top left (0,0), then the diagonal from top second (1,0) to the (0, 1) and so on:
(0,0), (1,0), (0, 1), (-1, 0), ( 1, 1), ( 0,-1), ( 2, 0), (-1, 1), ( 1,-1), ( 0, 2)...
now apply $A$'s rule
5(0) + 2(0)
5(1) + 2(0)
5(0) + 2(1)
5(-1) + 2(0)
5(1) + 2(1)
...

For your Dotted set to set-builder problem, I can't really help. I usually figure them out my looking at them, or if that doesn't work, thinking about the difference between each item in the list. Here's the answer to those problems.
$$\begin{align*}
C &= \{ (2i)^2 : i \in Z \text{ and } i \ge 0  \} \\
D &= \{ i^2 : i \in Z \text{ and } i \ge 0 \} \\
E &= \{ 2 + i^2 : i \in Z \text{ and } i \ge 1 \}
\end{align*}$$
