Indexed Families I'm trying to find an indexed family {An:n∈N} that satisfies:


*

*each An is an infinite subset of the naturals N

*the intersection of any two arbitrary sets is empty

*the union of all the subsets is is the naturals N


Let $f\colon \Bbb N \times \Bbb N \to \Bbb N$ be any bijection, for example
$$
f(a,b) = \frac {(a+b)(a+b+1)} 2 + b.
$$
Define
$$
A_n = \{f(n, k) \mid k \in \Bbb N\}, \quad \text{for $n\in\Bbb N$}.
$$
Then the $A_n$ are pairwise disjoint and their union is $\Bbb N$. Every $A_n$ is infinite, because for each $n$, $k \mapsto f(n,k)\colon \Bbb N \to \Bbb N$ is an injection.
