How is this read correctly in maths , and meaning (about: sets)? 
$J \subseteq \left\{1,2,..,n\right\}$ is an index set ....   Fix $J
\subseteq \left\{1,2,..,n\right\}$ and let $S=\left\{x_j : j \in
J\right\}$

The problem I'm having here is "Fix" , what does that mean ? I think "fix" refers to some constant value, or in this case SOME set we are chosing?
My other question is how is the set $S$ read correctly? I would read it like that: The set $S$ contains elements $x_j$ where each index $j$ is from index set $J$"
I hope someone can help :)
 A: The "fix" means "consider some particular $J$" or "for each $J$ define $S$ to be ..." . Your description of $S$ in terms of $J$ is correct.
No need to overthink this.
A: I would not use the word "fix", since logically speaking what is going on here is that we want to consider any arbitrary given $J ⊆ \{1..n\}$. Much better would be something like "Take any $J ⊆ \{1..n\}$, and let $S = \cdots$. Then . . .". But why do people use this word? It is because in their mind they are thinking that they want to temporarily focus on one specific $J$ and reason about that set together with another set $S$ associated with that particular $J$. So this becomes "let's fix $J$ for now, and define $S$ in terms of this $J$, and ...". Of course, in the end we are not talking about only one constant fixed $J$. This notion can be made clear in Fitch-style natural deduction:
[We want to prove $∀n{∈}ℕ\ ∀J{⊆}\{1..n\}\ ( \ Q(n,J) \ )$.]
Given $n{∈}ℕ$ and $J{⊆}\{1..n\}$:
  [It suffices to prove $Q(n,J)$ in this subcontext (i.e. with $n{∈}ℕ$ and $J{⊆}\{1..n\}$ given to you).]
  [Note that in this subcontext you have no control over $n,J$; they are fixed here.]
  $\vdots$
  $Q(n,J)$.
$∀n{∈}ℕ\ ∀J{⊆}\{1..n\}\ ( \ Q(n,J) \ )$.   [In the conclusion $n,J$ are of course not fixed.]
