# Pair Permutation of the set of Natural Numbers

Given the set of natural numbers $N$ is it possible to preform a series of operations that would result in a set with all of the different permutations of pairs? Something like: $$\{\{1,1\},\{1,2\},\{1,3\},...\{2,1\},\{2,2\},...\}$$

Thanks for any help!

• What do you mean by "a series of operations"? Are you looking for a function that takes a number $n$ and gives the $n^{th}$ sequential pair? – Omnomnomnom Jul 26 '13 at 18:36
• Yes, exactly a function. – Chris Jul 26 '13 at 18:41

You can define a new set $S$ such that
$S=\left \{ \left \{ a, b \right \}\mid a, b \in \mathbb{N} \right \}$