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Let $A$ be an countable set, and let $B_n$ be the set of all $n$-tuples $\left(a_1,\ldots,a_n\right)$

$B_n$ is the union of a countable set of countable sets.

This question maybe about the English.

Is my rephrase right?

$B_n$ is a countable set as the union of countable sets.

I think the quoted sentece has some problems in grammar or others?

Of course, that's only my judge, I'm not a native speaker and not so confident in the judge, so I asked this.

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Here are the two sentences in question:

  1. $B_n$ is the union of a countable set of countable sets.
  2. $B_n$ is a countable set as the union of countable sets.

Sentence (1) is just fine, and sentence (2) is not a correct paraphrase of sentence (1).

  • Sentence (2) says that $B_n$ is a countable set; sentence (1) does not say this.
  • Sentence (1) says that $B_n$ is the union of countably many countable sets; sentence (2) merely says that $B_n$ is the union of some arbitrary family of countable sets, not necessarily a countable family.
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  • $\begingroup$ aha, I misunderstood, thanks. $\endgroup$ – HyperGroups Jul 8 '13 at 4:15
  • $\begingroup$ @HyperGroups: You’re welcome. $\endgroup$ – Brian M. Scott Jul 8 '13 at 4:21
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Let me rephrase the question. Let $A$ be a countable set and let $B_n$ be the set of all $n$-tuples of elements of $A$.

For a fixed $n$ the set $B_n$ is countable by a diagonal argument.

I suspect the poser really wants to know about the set $B$ which is the union of all of the $B_n$. This set $B$ is also countable. To phrase this in terms of cardinal arithmetic,

$$ \begin{align} & \operatorname{Card}(B_1) + \operatorname{Card}(B_2) + \operatorname{Card}(B_3) + \cdots \\[10pt] & = \aleph_0 + \aleph_0 \cdot \aleph_0 + \aleph_0 \cdot \aleph_0 \cdot \aleph_0 + \cdots \\[10pt] & = \aleph_0 \end{align} $$

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