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countably infinite union of countably infinite sets is countable
proof that union of a sequence of countable sets is countable.

I'm a newbie who try to understand Set Theory. Is there anybody who can explain the solution for the following problem?

Assume that C is a countable set of countable pairwise disjoint sets, how can we prove that $\cup C$ is countable?

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@Asaf karagila Yes, I realized that later, so I deleted my comment accordingly. –  user22705 May 14 '12 at 10:50
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marked as duplicate by Zhen Lin, Benjamin Lim, Martin Sleziak, t.b., Zev Chonoles May 14 '12 at 17:03

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1 Answer

This requires a bit of the axiom of choice.

Since $C$ is countable we can write its members as $C_i$. For every $i$ fix $f_i\colon C_i\to\mathbb N\times\{i\}$ an injective function.

Now, since these are pairwise disjoint sets the union of he functions is a function, so let $f=\bigcup f_i$ be a function from $\bigcup C$ to $\mathbb N\times\mathbb N$.

Prove that this is an injective function, and use Cantor's pairing function to show that $\mathbb N\times\mathbb N$ is a countable set.

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