determining the cardinality

Question

Let $S$ be the collection of closed intervals in the real line whose lengths are positive rational numbers. Determine the cardinality of $S$. Justify your answer

As I understand, $S$ will be an infinite set of positive rationals. Would that mean that its cardinality will be equal to $c$?

Answer 1

HINT 1: Let $C = \{[r,r+1]: r \in \mathbb{R}\}$ and $C \subset S$.

HINT 2: Write $S = \bigcup_n C_n$, where $C_n = \{[r,r+q_n]:r \in \mathbb{R}\}$, where $(q_n)_{1}^{\infty}$ is an enumeration of rationals.

Answer 2

No, it's a set of intervals, not of rational numbers.

Consider $[x, x+1]$ where $x \in \mathbb{R}$. Are these in $S$? So what is the minimal and maximal cardinality of $S$?

Answer 3

$S$ is in obvious bijection to $\mathbb R\times\mathbb Q_{>0}$. So you need to be aware that $\mathfrak c\cdot \aleph_0=\mathfrak c$.