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Possible Duplicate:
Countable Sets and the Cartesian Product of them
Inductive Proof of a countable set Cartesian product

Let $A$ and $B$ be countable sets.

(a) Show that $A \times B$ is countable. Hint: Show that there is a bijection from $A\times B$ onto a subset of $\Bbb Z \times\Bbb Z$:

(b) Use induction on $n$ to show that $A_1 \times A_2 \times \ldots \times A_n$ is countable if $A_1, A_2,\ldots, A_n$ are countable.

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    Have you tried searching the site?2012-12-14
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    Was the hint not helpful enough?2012-12-14
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    Note, though that the proofs in the answers to the question that @Amr mentions are very different from the one suggested in your hint. They are direct proofs; yours makes use of the result that $\Bbb Z\times\Bbb Z$ is countable, which presumably you’ve already proved.2012-12-14
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    Note that the question found by @Asaf covers only (b).2012-12-14
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    @Brian: Thanks, luckily Amr's duplicate covers (a).2012-12-14
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    @Asaf: But not in a way compatible with the hint in the problem, which is why I won’t vote to close.2012-12-14

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