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Determine if it is countable or uncountable The set $E$ of all circle in $R$$^2$ with centers at rational coordinate points and positive rational radius.

I have no idea about this type of question.

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2 Answers

Hints:

1) If sets $A$ and $B$ are countable then their cartesian product $A\times B$ is countable.

2) Note that every element of $E$ is represented by a triple $(x,y,r)$ with $x,y\in \mathbb Q$ and $r\in \mathbb Q^{+}$

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Hint: if you can find a surjective mapping from a countable set to the set $E$ you are looking at, you will prove the set $E$ is countable.

Try finding a surjection from $\mathbb Q\times\mathbb Q\times\mathbb Q_+$ to $E$.

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Don't you need $\mathbb Q\times\mathbb Q\times\mathbb Q$ to $E$? As you have a rational radius as well. –  Cruncher Feb 24 at 15:53
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Correct. Even more correct, it's $\mathbb Q\times \mathbb Q\times \mathbb Q_+$ –  5xum Feb 24 at 19:15
    
Still not 100% understand... How about the $R$$^2$?? Is it mean real number? So it is uncountable?? –  user109403 Feb 25 at 2:04
    
No, because you can't have a circle with the center at $(\sqrt 2, 5)$. –  5xum Feb 25 at 6:26
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