Question

any infinite set can be written as the countably infinite union of pairwise disjoint infinite subsets?

Answer 1

We now use the result of the last question I answered.

If a set X is infinite (by which I mean it has cardinality aleph-0; if it is another sort of infinite and a different cardinality, then you need to modify this argument very slightly), then: there is an isomorphic function
\[f : \mathbb{N} \rightarrow X\]

Now consider the subsets R_p (or whatever I called them) we had in the last problem.

I'll let you figure out what to do from here.

Answer 2

You'll aslo see the the original subset we have won't do the trick, but you'll need a new collection of subsets: prime numbers will be very helpful still.

Answer 3

thank you again. i will start working on it