Meaning of “is an element of” This is a rather simple question I suppose, but i just feel stuck.
When we say for example that “x ∈ A where A is the set of reals.” Is it possible that this x equals only some of the values in the reals or is it all values?
Because when I actually read the sentence x ∈ A, I read it as x is an element of A and the word “an”, I believe is putting me off, as I keep thinking that x can only be only one of these values.
Can someone please help me understand.
Thank you in advance.
 A: The meaning of "$x\in A$" by itself is that $x$ is a (single) element of $A$. If this is all you know about $x$, then you might need to keep in mind several possible values of $x$. It's analogous to something like "$x > 10$".
For example, if you're given

Let $x\in\Bbb Z$. Prove that $x\le x^2$.

then the task is to prove $x\le x^2$, for the single unspecified number $x$, using only the information $x\in\Bbb Z$. If you can do this, you've effectively proved that every integer is less than or equal to its own square. In other words, you've proved

For all $x$, if $x\in\Bbb Z$, then $x\le x^2$.

At a grammatical level, the $x$ still refers to a single object (i.e. it is a term), but the enclosing "for all $x$" (a quantifier) compels us to consider more than one instance of the proposition "$x\le x^2$".
The logical language surrounding symbolic expressions is extremely important in determining what's being conveyed.
A: The meaning of $x \in A$ depends largely on context.
The answer by Karl gives an example of a sentence structure in which there is an implied (or tacit) universal quantifier:  "Let $x \in \mathbb Z$. Then $x \le x^2$."  As that answer explains, this really means "For any integer $x$, the property $x \le x^2$ is true."
But there are other contexts in which there is an implied (or tacit) existential quantifier.  An example of this might be:

Let $m, n \in \mathbb Z^+$ be relatively prime. Then for $x \in \mathbb Z^+$, we can write $x = ab + mn$ for $a, b \in \mathbb Z$.

In this sentence, the variables $m, n$ and $x$ are universally quantified (the claimed property is true for every relatively prime positive integers $m, n$ and for every positive integer $x$), but the variables $a, b$ are existentially quantified: for each suitable choice of $m, n, x$ the claim guarantees us that there are some integers $a, b$ that will work. (We are not claiming that every pair of integers $a, b$ will work; that would be absurd.)
This is one of the reasons why well-written mathematics requires clear use of words and sentences.  The symbols by themselves do not always carry all of the meaning in an unambiguous way.
Sometimes the distinction can be quite subtle.  For example, in

We define the set of $n^{th}$ roots of unity to be $R_n(z) = \{ w \in \mathbb C \vert w^n = z \}$, for $z \in \mathbb C$ and $n \in \mathbb N$

the variables $z$ and $n$ are universally quantified (read it as: "for any $z \in \mathbb C$ and any $n \in \mathbb N$"), but $w$ is not: only those complex numbers $w$ that satisfy the condition $w^n = z$ are included in the set $R_n(z)$.  (Certainly we're not claiming that every complex number $w$ satisfies this condition!)  The distinction largely rests on the fact that $z$ and $n$ have their meaning determined outside the definition of the set, where their scopes are essentially unrestricted, while $w$ has its meaning determined inside the definition of the set, where its scope is restricted by the condition that follows the vertical bar $\vert$.
