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Let $A=:\{(x,x)|x\in[0,1]\}\subset[0,1]\times[0,1]$

I see A as $A=\{(x,y)|x\in[0,1]|x=y\}$ so it is a straight line bounded by points $(0,0)$ and $(1,1)$ if I understand this construction.

Is this set countable?

I could not find a suitable injective function from $A$ to $\mathbb{N}$ so I was thinking of taking other approach through countable unions.

Can we write $A$ as a countable union of points it contains $A=\displaystyle\bigcup_{i=1}^{\infty}\{(x_{i},y_{i})|x_{i}\in[0,1]|x_{i}=y_{i}\}$ would that work? How to show it is countable or show otherwise?

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HINT:

Note that there is an obvious bijection between $A$ and $[0,1]$. So the question is, is $[0,1]$ countable?

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  • $\begingroup$ no, it is not because it is a subset of an uncountable set $\endgroup$ – H.E Sep 23 '13 at 16:25
  • $\begingroup$ @Heidi: But $\Bbb N$ is a subset of an uncountable set. Is $\Bbb N$ uncountable? What about $\varnothing$ or singletons? $\endgroup$ – Asaf Karagila Sep 23 '13 at 16:26
  • $\begingroup$ hm $\mathbb{N}$ is countable, the empty set is also countable (strange, because it does not have any elements), I am not sure about singletons. I remember that [0,1] is uncountable $\endgroup$ – H.E Sep 23 '13 at 16:30
  • $\begingroup$ @Heidi: Being a subset of an uncountable set doesn't imply anything about cardinality (finite sets, including singletons are of course countable). However having an uncountable subset does imply that a set is uncountable. Which sets do you know are uncountable? Can you find an injection from one of them into $[0,1]$? $\endgroup$ – Asaf Karagila Sep 23 '13 at 16:35
  • $\begingroup$ Yes I know that, I mixed that up. Uncountable are $\mathbb{C}$,$\mathbb{R}$,$\mathbb{R\setminus Q}$, Cantor set and so on. $\endgroup$ – H.E Sep 23 '13 at 16:44

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