When you write $x\in R$, this reads $x$ is an element of $R$. This specifies a domain for $x$ which is simply a variable.
When you define a set $X = \{x\}$ , the set of all $x$, or $X = \{x\in R\}$ it seems as though to define the set you are specifying a domain for $x$, the values $x$ can take is the set. My question is how would you refer to $x$? I know you can refer to it as the general element but I can't really understand this.
Best Answer
$X = \{x\}$ says $X$ is a set and it has one element in it.
If you want to define a set $X$ and then say for all $x\in X$ that is fine. You don't need anything else other than "for set $X$" or similar. Heck, you often don't even need to say that much.
And if you need to be more specific, you may want to say that $X$ is a subset of the natural numbers. i.e. $X\subset \mathbb N$
And maybe you want to define a set like the set of point on the unit circle. $X=\{(x,y)|x^2 + y^2 = 1 \}$