Is there a construction of pairwise disjoint, countable subsets of $\mathbb N$

Question

I am looking to construct a collection of countable subsets whose elements are $\forall n \in \mathbb N$, $X_n \subset \mathbb N$. Consider the set $\{X_n\}$:

Each $\{X_n\}$ is countable, i.e., it has the same cardinality as $\mathbb N$
The subsets are pairwise disjoint, meaning that for any two distinct subsets $X_i$ and $X_j$, we have $X_i \cap X_j = \emptyset.$

What would be an example of such a construction? How can one define these sets explicitly to satisfy both conditions?

I may not understand your idea. Could you explain it more clearly? — UsedT0Be, Feb 21 '25 at 16:56
@Randall That only gives two sets, and the OP (I think) wants infinitely many. Of course it's not hard to modify that example to still work, and I'm sure this has been asked here before. — Noah Schweber, Feb 21 '25 at 16:58
Additionally, meanwhile, the set ${X_n}$ itself is also a set with cardinality of $\mathbb N$ — UsedT0Be, Feb 21 '25 at 16:58
If OP wants restrictions on the size of the collection, this should be made clear. That was my point. — Randall, Feb 21 '25 at 16:59
@UsedT0Be Can you generalize the idea with even and odd numbers? — chirico, Feb 21 '25 at 17:03
It seems hard to do so because the set ${X_n}$ is a infinite set. — UsedT0Be, Feb 21 '25 at 17:06
Take $X_{n}$ to be the set of naturals that are not divisible by the first $n-1$ primes, but divisible by $n^{\text{th}}$ — chirico, Feb 21 '25 at 17:07
Wait, there may be a problem with this idea. For instance, take 6 which is both divisible by 2 and 3. It fails to guarantee any two sets' intersection is empty. @chirico — UsedT0Be, Feb 21 '25 at 17:15
@Randall Indeed, even ${\mathbb N}$ would work (a one element collection). — amrsa, Feb 21 '25 at 17:49

score 2 · Accepted Answer · answered Feb 21 '25 at 17:13

2

$1,3,5,\ldots$

$2\times1,2\times 3,2\times 5,\ldots$

$2^2\times 1, 2^2\times 3, 2^2\times 5,\ldots$

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answered Feb 21 '25 at 17:13

jjagmath

22,582

Yes, it seems great. Thx. – UsedT0Be Feb 21 '25 at 17:19
1

I like this one. My father used to say “every number is a power of two times an odd number” :) – Malady Feb 21 '25 at 18:21

Is there a construction of pairwise disjoint, countable subsets of $\mathbb N$

1 Answers1