Here's a nice solution for this problem that a friend of mine came up with a while back. First, it's enough to do this in the unit interval
. Begin by doing it in the unit square,
; just consider all horizontal lines for instance. Next, choose a bijection
. For example, you could map
. (Of course, you have to define the decimal expansion in a unique way, but this isn't too tough.) Now just consider the images of all the pairwise disjoint and uncountable sets you started with; this will be another collection of pairwise disjoint uncountable sets, and their union will be
. So you're done.
By the way, for future reference, a good site for this type of question is math.stackexchange.