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

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$$\cup$$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
$$\exists$$ f: R$$\rightarrow$$U that is one-to-one and onto.
$$\exists$$ g: R $$\rightarrow$$ V that is one-to-one and onto.

I think I need to show that $$\exists$$ h: R $$\rightarrow$$ U $$\cup$$ V.

But how do I get to that point? Please help! I would greatly appreciate any assistance.
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 Recognitions: Homework Help Science Advisor Do you know that, for example, that (-infinity,0] and (0,infinity) both have the same cardinality as R?

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

Recognitions:
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