Consider that we have a set whose elements are the vowels of the English alphabet - "a", "e", "i", "o", "u". Here, "a" denotes the letter a.
In a book that I'm reading right now, this set is denoted as follows:
$$\{x | x \text{ is a vowel in the English Alphabet}\}$$
My question is what exactly $"x"$ is denoting here?
Does it denote, a single arbitrary letter of the alphabet, a single arbitrary vowel, all letters, all vowels or something else?
The $x$ used in the formula above is a variable that is ranging over all letters of the English Alphabet.
The set-builder notation selects all values of the variables that fit with the "specification": in this case all those letters of the English alphabet that are vowels.
So if the English alphabet is the set $\{ a,b,c,…,z \}$, then the above set will be: $\{ a,e,i,o,u \}$. We read it as: "the set of all those letters of the English Alphabet that are vowels".
The set-builder notation can be used in two ways: either 1) $\{ a,b,c,…,z \}$. In this case we list all elements: of course, "it works" if the number of elements is finite (and not huge).
Or 2) to "carve out" the set from another already existing set using a formula that specify the criteria to be used for selecting the elements: $\{ x \mid \text { formula with } x \}$
Thus, if we start with the set $\text {EL}$ of all the letters of English alphabet, we can use the notation to extract from it the set of vowels: $\{ x \mid x \in \text { EL and} x \text { is a vowel} \}$.