Determine if it is countable or uncountable The set $E$ of all circle in $R$$^2$ with centers at rational coordinate points and positive rational radius.
I have no idea about this type of question.
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Hint: if you can find a surjective mapping from a countable set to the set $E$ you are looking at, you will prove the set $E$ is countable.
Try finding a surjection from $\mathbb Q\times\mathbb Q\times\mathbb Q_+$ to $E$.
5xum
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Don't you need $\mathbb Q\times\mathbb Q\times\mathbb Q$ to $E$? As you have a rational radius as well. – Cruncher Feb 24 '14 at 15:53
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1Correct. Even more correct, it's $\mathbb Q\times \mathbb Q\times \mathbb Q_+$ – 5xum Feb 24 '14 at 19:15
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Still not 100% understand... How about the $R$$^2$?? Is it mean real number? So it is uncountable?? – user109403 Feb 25 '14 at 02:04
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No, because you can't have a circle with the center at $(\sqrt 2, 5)$. – 5xum Feb 25 '14 at 06:26
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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 $r\in \mathbb Q^{+}$
drhab
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