Lets say I have [itex] \aleph_1 [/itex] numbers of sets that each have [itex] \aleph_1 [/itex] number of elements and I want to show that the union of all of these sets has [itex] \aleph_1 [/itex] number of elements. I start with the line segment [0,1] and shift this line segment up by all the reals from 0 to 1. So now we have the unit square. Now we want to show that this unit square can be mapped to [0,1]. So can we use trick where you take the decimal form of a point and expand it to 2 dimensions. [itex] (.x_1x_2x_3x_4......)\rightarrow (.x_1x_3...),(x_2x_4....) [/itex] or another thought I had was to take the cantor set and move it around with a set of reals and map each set to a cantor set that was shifted across the real line.