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### If $A$ and $B$ are countable sets, show that $A \times B$ is countable [duplicate]

My question is If $A$ and $B$ are countable sets, show that $A \times B$ is countable. I know the definitions to be a countable set are: A set $A$ is countable if $A$ is finite or countably infinite....
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### Bijection between $\mathbb N^2$ and $\mathbb N$ [duplicate]

A question regarding a bijection between $\mathbb N^2$ and $\mathbb N$. I know about cantor pairing function but I wanted to ask about a bijection I have seen around the site which is $n=2^{u-1}(2v-1)$...
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### Is the set of all pairs of natural numbers countable? [duplicate]

Say that $\Bbb N \times \Bbb N$ is the set of all pairs $(n_1, n_2)$ of natural numbers. Is it countable? My hypothesis is yes it is countable because sets are countable. But I am unable to come up ...
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### Prove countable set function: natural numbers and pairs of natural numbers

Can someone please explain me the proof, where there is a 1-1 Correspondence between the set of natural numbers and the set of all pairs of natural numbers How can the below data be one-to-one ...
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### On the bijection between the naturals and a countable number of countable sets.

I am assuming we are utilizing the axiom of choice because I read in the suggested questions that may have my answer that it's needed for the following bijection to work. Suppose I am given a ...
Ok so here is a combinatorial problem that I thought of. Suppose N is in $\mathbb N$ such that $N>1$, then there is a way to count (set an index) to all pairs \$(i,j) \in \{1,\dots,\mathbb N\}\...