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

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.

• 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.