What can be the Cardinality of the union of disjoint sets, each of which have a cardinality of reals? How should this be proved.
I know using Schroder bernstein theorem, it is easy to see that the cardinality of the union must be equal to the cardinality of reals. But without using it, how should this be proved?
I have an idea:
[0,1) has the same cardinality as that of reals. Similarly, [1,2) has the same cardinality as reals. Similarly a semi-open interval [n,n+1) has the same cardinality and they are all disjoint. So, can I combine them and simply say that their cardinality is equal to reals? Is this correct. I am looking at a more formal proof along this line.
Best Answer
Map one of the sets to $(-\infty, 0)$, one to $[0, 1]$, and one to $(1, \infty)$.
(added as requested)
Therefore, each element of each of these sets gets mapped into a unique real, so the cardinality of their union is that of the reals.