What do we mean when we say "Let $x$ be an element of the set $\mathbb{R}$"? 
What do we mean when we say "Let $x$ be an element of the set $\mathbb{R}$"?

Does $x$ represents only a single  element of set $\mathbb{R}$? or does $x$ represent all the element of set $\mathbb{R}$ simultaneously at the same time?
People say that if $x\in\mathbb{R}$ then $x$ is any real number; that means $x$ represents all real numbers. But if $x$ is any real number then let's say $x=1$; so $x$ is one, then how it can represent all real numbers?
Please help me I am very confused.
 A: The set $\mathbb{R}$ is the familiar real number line, including all "decimal expansions." When we say a number is an element of $\mathbb{R}$, we mean that it's a part of the number line. $1 \in \mathbb{R}$, $\sqrt{2} \in \mathbb{R}$, negative fractions that look weird such as $\frac{-1}{\pi}$ are in the set $\mathbb{R}$, just cause they have a decimal expansion.
When we say $x \in \mathbb{R}$, we just mean that there's a number, just like the ones I mentioned above, that is a real number. We just don't know what it is yet? What is $x$? We don't know, but we can give it a name because for whatever reason it's of interest to us. We just know it lies somewhere on the number line. We don't know what $x$ is, but noting that it's in $\mathbb{R}$ for one reason or another has importance. But "variable" in the title of your question is just giving a letter that we don't know its value.
A: *

*Suppose I say : if x // y and y // z then x // z.


*Is this sentence meaningful? In order this sentence to make sense, I should first say : Let x, y  and z be straight lines in a given plane D.


*So, saying " let x belong to R" is a way to state the domain in which a sentence will have meaning, and, therefore, will have a truth value . ( Maybe, allowing x to be some non-real number would yield a meaningless sentence.)


*Second reason: it might happen that a sentence has  meaning in some large domain D, but is false for some values of this domain. In case your goal is to establish a universally true sentence, you restrict the possible values of x to a subset D* of D.


*Third reason: sometimes , you use " Let x belong to some domain D" as an hypothesis in a conditional proof.

Let x belong to R  $\space \space $ (Hypothesis for conditional proof).

x² = 4


$\sqrt{x²} = \sqrt 4$


$|x| = 2$


$x = 2$ OR $x = -2$

$x² =4 \rightarrow (x= 2$ OR $x = -2)$

x belongs to R $\rightarrow [x² =4 \rightarrow (x= 2$ OR $x = -2) ] $
For all $x_{\in R} \space ,  \space x² =4 \rightarrow (x= 2$ OR $x = -2)$
Note : the conclusion ( for all x belonging to R ... ) is allowed on the ground that, in the hypothesis, x was arbitrary. If the concluson holds for any number in R , it also holds for all numbers in R.
A: There are already good answers, but I really struggled with this when first encountering logic and proofs, so I hope my own perspective is useful to someone.
Often we want to to prove something about all elements of a set S. That is, we want to prove that every element in $S$ satisfies some property, call it $P$. For example, say $S = \{1,2,...\}$. Now say $P(x)$ stands for the statement "$x$ is greater than $0$". Clearly $P(1)$ is true, and so is $P(2)$ is true, and indeed it should be obvious that no matter what number we choose from $S$, $P(\text{ 'that number' )}$ will be true.
We want a way to express the truth of the statement in the above example. We know that no matter what object I choose from $S$, the statement $P$ will be true for that object. We do this by reasoning about an arbitrary object from that set. We say $x \in S$, and specify nothing more. Now the key bit here is that $x$ is not simultaneously "all objects in $S$". $x$ represents a specific number from the set $S$, we just don't say which one.
Going back to our example, say I tell you $x \in S$. Now $x$ represents some specific number in $S$, but we don't know which. Suppose $x$ actually represented the number $20$. Well of course $P(20)$ is true, because $20>0$. Similarly, if $x$ actually represented the number $1727361$, then $P(1727361)$ would be true too. The pattern here is clear: no matter what number $x$ actually represents, $P(x)$ is true. It is in this sense that $x$ can represent any/all elements of $S$.
Ultimately, our example can be written as $\forall x \big(x \in S \rightarrow P(x)\big)$. This statement is really saying, "for every object $x$, if this object is a specific a number in the set $S$, then the statement $P(x)$ is true." Note the key point here is that I have said $x$ is specific, yet I have not told you what it is. The meaning here is that in the statement, we treat $x$ as a specific number. After all, we say that $x \in S$ and $P(x)$, which only make sense if $x$ is just one number. But $x$ could represent any specific number in $S$, because every number I use makes $P(x)$ true.

The above is an intuitive and informal approach; but fundamentally this question can be answered formally. This question revolves around notions of variables, quantifiers, and logic. Answers to my own questions and others' here on stack exchange do a very good job of explaining this more formal approach. For example, see this. The answer and Noah Schweber's comment are incredibly useful.

Response to comments by OP
When we say $x>-2$, yes, it is correct to say that $x$ can be any real number. $x$ could also be a function, a group, or any kind of mathematical object. $x$ is simply a symbol. There is no reason to say that $x$ must be a real number larger than $-2$. If I were to then tell you $x$ was the number $-3$, that's perfectly fine, it just means the statement is false for that 'value' of $x$.
Now, if you were to say that $x$ is an arbitrary real number such that $x>-2$, then yes, now $x$ represents a specific (yet unspecified) real number larger than $-2$. In essence we are saying $x$ is an arbitrary element of the set $\{y \in \mathbb{R}: y > -2\}$. Now it makes sense to say that $x$ is an arbitrary element greater than $-2$, because I have told you that is the case.
To summarise, the statement $\forall x \in \{y \in \mathbb{R}: y > -2 \} \big(x > -2 \big)$ is true, but the statement $\forall x \in \mathbb{R} \big(x > -2 \big)$ is not. And furthermore, $x > -2$ by itself is not comparable to the other two. This is a well formed formula, and whether it is true or not depends on what $x$ is. Of course, we may restrict $x$ to only take on those values which make $x>-2$ true. In this case, then $x$ represents some specific yet unspecified value from $\{y \in \mathbb{R}: y > -2 \}$.
A: I know an answer has already been accepted, but I'll try to answer the heart of the question anyway. All of the following statements mean the same thing:
Let $ x \in \mathbb{R}$. Then $x^2 \in$ {$u \in \mathbb{R}: u \geq 0$}.
Let $ x \in \mathbb{R}$. Then $x^2 \in \mathbb{R}$ and $x^2 \geq 0$.
$ x \in \mathbb{R} \implies x^2 \in$ {$u \in \mathbb{R}: u \geq 0$}.
$ x \in \mathbb{R} \implies \left(x^2 \in \mathbb{R} \text{ and } x^2 \geq 0 \right)$.
If $ x \in \mathbb{R}$, then $x^2 \in$ {$u \in \mathbb{R}: u \geq 0$}
If $ x \in \mathbb{R}$, then $\left(x^2 \in \mathbb{R} \text{ and } x^2 \geq 0 \right)$
If $x$ is any real number then its square is a non-negative real number.
$$$$
The statements are true. However, note that their truth value is independent to whether or not the statements mean the same thing, In other words, false statements can also be equivalent.
Personally, I avoid using the statement with "any" in it, because it can be confusing and I'd rather just get on with the maths than spend time being confused. If someone else uses the statement with "any" in it, then I translate it into one of the other statements.
