Countable sets question

Asked Jan 11 '21 at 09:29

Active Jan 11 '21 at 09:29

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Let $C$ be a certain set of pairwise co-prime balanced rectangles in a plane. need to prove that C is a Countable set. The meaning of balanced rectangles is a set of the next form: $$B=<x,y>\in \mathbb{R}^2:a\le x\le b, c\le y\le d$$ for $a,b,c,d$ such that: $a<b \ and \ c<d$.

I need to use the density of $\mathbb{Q} \ in \ (\mathbb{R},\le)$.

I don't know how to even start proving. Therefore, I will be happy to get some help.

asked Jan 11 '21 at 09:29

user12121

If you don't require that $a, b, c ,d \in \mathbb Q$, this can't be true as the sets $[0, a] \times [0, a]$ are in $B$ for any real number $a>0$, so we have found an uncountably infinite subset of $B$. – ViktorStein Jan 11 '21 at 09:49
@Ramanujan and if it is required? – user12121 Jan 11 '21 at 09:59
1

what does "pairwise co-prime" mean for rectangles? – shortmanikos Jan 11 '21 at 10:03
@shortmanikos meaning that the intersection between each couple is empty. – user12121 Jan 11 '21 at 10:07
I've voted to close this question as a duplicate; formally, of this, but this is more directly relevant (has no upvoted answer), and links to some other questions that should be helpful. – Carl Schildkraut Jan 11 '21 at 10:16

Countable sets question

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