Main Question or Discussion Point
So if A is an infinite set, we know that |A|+|A|=|A|. But are we allowed to go backwards, i.e. divide A into two disjoint subsets B and C such that A = B U C, and |B|=|C|=|A|. For the integers and for the reals, this is clear, e.g. R = (-infinity,0) U [0, infinity), each with cardinality c. But are we allowed to do this for any infinite set of larger cardinality? I'm sure the answer is yes, but how do we go about proving this? Do we have to use the Axiom of Choice and transfinite induction? How complicated will the proof be?