0

I turned it into a three-dimensional array where I have values (i,j,k) but I need to find a function that enumerates $\mathbb N$ x $\mathbb N$ x $\mathbb N$. I found the function for the enumeration of $\mathbb N^2$ which was $(((i+j)(i+j+1))/2) +j$

kdd
  • 21
  • Welcome to Math.SE. Maybe you want to take the tour (http://math.stackexchange.com/tour) What did you try so far? Where did you get stuck? Also, using MathJax would be nice (https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference). Maybe visualizing the construction for $\mathbb N^2$ helps in finding one for $\mathbb N^3$. Or you just iterate the one you have found. Also, what's the connection to cardinals? Please read the tag description before using it. – martin.koeberl Mar 24 '17 at 13:28
  • What "enumeration of $N^2$" means? – Arnaldo Mar 24 '17 at 13:29
  • Let's say you have a bijection $\mathbb{N}^2 \rightarrow \mathbb{N}$ : $f(i,j)$, then to have a bijection $\mathbb{N}^3 \rightarrow \mathbb{N}$ : $g(i,j,k)$ simply define $g(i,j,k) = f(f(i,j),k)$ – Zubzub Mar 24 '17 at 13:30
  • What if you just found an injection $f: \mathbb{N}^3 \to \mathbb{N} , ; , (a,b,c) \mapsto 2^a3^b5^c$? – joeb Mar 24 '17 at 13:38

1 Answers1

1

If an enumeration of $\mathbb{N}\times\mathbb{N}$ is given by $$ \varphi:(i,j)\mapsto \binom{j}{1}+\binom{i+j+1}{2} $$ then an enumeration of $\mathbb{N}\times\mathbb{N}\times\mathbb{N}$ is given by $$ \psi:(i,j,k)\mapsto \binom{k}{1}+\binom{k+j+1}{2}+\binom{k+j+i+2}{3} $$ or by $$ \eta:(i,j,k)\mapsto \varphi(i,\varphi(j,k)) $$ for instance.

Jack D'Aurizio
  • 353,855