Quantifiers and set theory

Question

Given the following statement,

$$\forall x \, (x \in S)$$

When we say "for every x in S", are we saying that every possible element is in S -- and there are no other elements, except x?

Answer 1

What you wrote amounts to

Every $x$ in our domain of discourse is an element of $S$

i.e., $S$ is a "set of everything".

Answer 2

Typically this is written more simply as $$\forall x\in S$$ On its own, the expression is meaningless. It only has meaning when coupled with other statements. For example, consider the following: $$|x|\geq x~~,\forall x\in\Bbb{R}$$ In words, this means "For any real number - $5$, $\log(\gamma)$, $\sqrt{\pi}$, etc, its absolute value is greater than or equal to the number itself." The important point is the fact we used $x$ is irrelevant - the statement has exactly the same meaning if we write $$|\text{banana}|\geq \text{banana}~~,\forall ~\text{banana}\in\Bbb{R}$$ In other words, the statement says nothing about "$x$" itself. $x$ is simply a placeholder that we use to represent "something contained in the set $S$".