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Prove that these sets are countable :

A. Set of relations over natural which is composed by exactly one ordered pair.

B. Set of relations over natural composed by finite number of ordered pairs.


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I assume that the answer should rely on a bijection function from naturals to the sets, but don't know how to express it as answer. – adamco May 4 '12 at 15:20
up vote 2 down vote accepted

The set A is essentially the collection of all singletons from the set $\mathbb{N}\times\mathbb{N}$. (Can you see why?) So the question is: can you prove that $\mathbb{N}\times\mathbb{N}$ is countable?

The second set is the collection of all finite subsets of $\mathbb{N}\times\mathbb{N}$. Perhaps you can prove that that the set of all subsets of a given finite size is countable, and then take a union?

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For the second set, can I prove that the set is countable relying on that |N^k| (k in N) =|N|? – adamco May 5 '12 at 7:10
@galeck: I would say that if you can prove that $\mathbb{N}^k$ is countable for any finite $k$, then you can rely on that fact. If you cannot prove it, then I would be wary of relying on it... – Arturo Magidin May 5 '12 at 20:24

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