$$\mathbb N =\bigcup_{j\in \mathbb N}\Delta_j $$ where each $\Delta_j$ is an infinite subset of $\mathbb N$ and $\Delta_j\cap \Delta_i=\Phi \ for\ i\neq j.$
Now what I need is a few examples of such decompositions. The only one I can think of now is the collection of the odd numbers and the even numbers. That satisfies it.
Another possibility I was considering was like this ::
$$\Delta_1=2\mathbb N\\ \Delta_2=3\mathbb N \backslash \Delta_1\\ \Delta_3=5\mathbb N\backslash (\Delta_1\cup\Delta_2)\\.\\.\\.\\.\\.\\.\\so\ \ on.$$ The technique here is for any arbitrary $k$ , $\Delta_k=p\mathbb N\backslash \left(\bigcup_{i=1}^{k-1}\Delta_i\right)$. Clearly I can see all these sets are mutually disjoint and also infinite since there are infinitely many prime numbers.
So , are there any other construction of this kind possible $?$ If so please let me know . Also , if there is any fault in my above construction point it out .
Thank you.
For every $j\ge 1$, let $\Delta_j$ consist of all natural numbers of the form $2^{j-1} m$, where $m$ is odd.
Remarks: $1.$ This can be used to produce a simple bijection between $\mathbb{N}$ and $\mathbb{N}\times \mathbb{N}$.
$2.$ You can get an interesting example of a different character by using the Cantor Pairing Function. Note that the linked article includes $0$ among the natural numbers, so you will have to modify it a little if you want $\mathbb{N}$ to exclude $0$.