$T = \{(i,j,k,z) \mid i,j,k,z ∈ \mathbb N\}$

How to prove that this set is countable? I understand how to do it with triples by finding one-to-one function from $T$ to $\mathbb N$, but I cannot understand how to use this approach in case of quadruples.

  • 2
    $\begingroup$ Can you do it for doubles and then for doubles again? $\endgroup$
    – Sorfosh
    Commented Apr 3, 2023 at 15:11

1 Answer 1


An easy 1-1 function $\mathbb N^4 \to \mathbb N$: $$ \alpha(i,j,k,z) = 2^i 3^j 5^k 7^z $$ For more than $4$ arguments, use more primes: $11, 13, 17, \dots$.


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