Notation in set theory. What is the difference between these two notations? 


*

*$\{a_k\}_{k\in K}$

*$\{a_k, k\in K\}$


Is this correct?


*

*$\{a_1\}$, $\{a_2\}$, $\dotsc$

*$\{a_1, a_2, \dotsc\}$


Or 1. and 2. are the same? Is it a standard notation or anyone can use the notation he wants?
 A: The notation $\{a_k\}_{k\in K}$ is usually used to denote an index family, which is the same as a map $a:K\to A$ that maps every $k\in K$ to $a_k\in A$. So we have a map lurking in the background. We can of course form the set of images, which is $\{a_k : k\in K\}$, a plain set.
On the other hand, given just the set of images, we are not able to reconstruct the indexing map $I\to K$, since we lost all the indexing information.
So strictly speaking, the two things are different. On the other hand, the notation $\{a_k : k\in K\}$, while just defining a set, already implies we have some way of associating to every $k\in K$ an element $a_k\in A$, so we have an indexing map lurking as well. If we are just given $\{1,2,3\}$ though, there's no way to figure out any indexing from this.

Let me give you a quick example why we sometimes need to talk about indexed families instead of sets. Consider the two vectors $\pmatrix{1\\1}$ and $\pmatrix{2\\2}$. I'm sure you agree that these two are linearly dependent, so the set $\left\{\pmatrix{1\\1}, \pmatrix{2\\2}\right\}$ is linearly dependent. That's all fine. Now, again, take the two vectors $v_1=\pmatrix{3\\4}$ and $v_2=\pmatrix{3\\4}$. Again, those two vectors are linearly dependent, they are the same. The set $\{v_1, v_2\}=\left\{\pmatrix{3\\4}\right\}$ contains only one non-zero vector though, and is linearly independent. To talk about $v_1, v_2$ as two vectors, we say the family $\{v_i\}_{i\in\{1,2\}}$ is linearly dependent.

What you wrote in 1. is certainly not what we have here, it is a list of singleton sets.
