What is the cardinality of the set S of all closed intervals on the real number line with rational (positive) lengths?
I believe the number of intervals with a specific fixed length but varying start points is |R|, and the number of intervals with a fixed start point but varying lengths is |Q| = |N|. I think the answer to the question would then be |S| = |R x N| = |R|? I'm not quite sure how to prove the last equality though. Would I have to define a bijective function that maps elements of (R, N) to R? How would I approach doing so?
Hint:
You can get a bijection from $\mathbb{R}\times\mathbb{N}$ to $[0,\infty)$ by bijectively mapping (for each $n\in\mathbb{N}$), $\mathbb{R}\times\{n\}$ to $[n,n+1)$.