Number of partitions of countable and uncountable set

Question

For a countably infinite set (say N), we can find a partition into countably infinite number of countably infinite subsets with each disjoint with other. But how to find how many such partitions possible. I am a beginer and please explain me in layman language.

Also how to find number of partions of uncountable set?

Answer 1

Pick any such partition. Now reove the first of these sets and augment the second by an arbitrary subset of it and augment the third by the rest. This gives us (number of subsets of countably infinite set=) $2^{\aleph_0}$ partitions. On the other hand, any such partition can be viewed as a map $\Bbb N\to\Bbb N$, and there are $2^{\aleph_0}$ such maps. Hence the desired number of partitions is $2^{\aleph_0}$.