Difference between $[a,b]\in \mathbb R$ and $[a,b]\subset \mathbb R$?

Question

What is the difference between the following?

Are they both mathematically correct?

\begin{align} [a,b]\in \mathbb R \tag 1 \\ [a,b]\subset \mathbb R \tag 2 \end{align}

And also, which one should I use If I want to say "the interval between $a$ and $b$ is real"? Feel free to correct me if this phrasing is inaccurate.

Answer 1

Since $[a,b]$ is a set then only second is validate:

$$ [a,b] := \{x\in \mathbb{R}; a\leq x\leq b\} \implies [a,b]\subset \mathbb{R}$$

If the question was, is $[a,b]\in \mathcal{P}(\mathbb{R})$ (that is the power set of $\mathbb{R}$) then the answer would be yes.

Answer 2

Assuming that $a$ and be are real numbers and that $a\leqslant b$, then the assertion $[a,b]\in\mathbb R$ is false, since it means that the interval $[a,b]$ is an element of $\mathbb R$. If you want to assert that $[a,b]$ is a subset of $\mathbb R$, you write $[a,b]\subset\mathbb R$.