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## Homework Statement

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.

## Homework Equations

## 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.