Cardinality of the Union of Two Sets that have Same Cardinality as Real Numbers

georgetown13

1. The problem statement, all variables and given/known data
Let U and V both have the same cardinality as R (the real numbers). Show that U[tex]\cup[/tex]V also has the same cardinality as R.


2. Relevant equations



3. The attempt at a solution
Because U and V both have the same cardinality as R, I that that this means
[tex]\exists[/tex] f: R[tex]\rightarrow[/tex]U that is one-to-one and onto.
[tex]\exists[/tex] g: R [tex]\rightarrow[/tex] V that is one-to-one and onto.

I think I need to show that [tex]\exists[/tex] h: R [tex]\rightarrow[/tex] U [tex]\cup[/tex] V.

But how do I get to that point? Please help! I would greatly appreciate any assistance.

Dick

Do you know that, for example, that (-infinity,0] and (0,infinity) both have the same cardinality as R?

georgetown13

Yes, but I can't just provide an example to prove the statement, right?

I understand the general concepts behind this proof but am having a difficult time putting it down in mathematical terms.

Dick

I didn't mean it to be an example. In the nonmessy case where U and V are disjoint, then R maps bijectively to (-infinity,0]U(0,infinity) via the obvious map 'identity' then map (-infinity,0] bijectively to U and (0,infinity) bijectively to V. Compose them and you have a bijective map from R to U union V.