
cardinality
Show that the sets [2,4] u [6,10] and (2,2) have the same cardinality.(using the SchroederBerstein Theorem) so I have to define f: [2,4] u [6,10] > (2,2) by f(x)= any f(x) I get that works... when you inverse it so that I define f: (2,2) > [2,4] u [6,10] by g(x)= I can't get to work... can anyone help me?

Quote:
when you inverse it...
I am not sure if you mean inversing the first function. If you use Cantor–Bernstein–Schroeder theorem, you don't have to construct a bijection (which has to have an inverse). You just have to provide two injections: one from $\displaystyle [2,4]\cup [6,10]$ to $\displaystyle (2,2)$ and one back.
Well, that should be easy. You can use linear functions $\displaystyle f(x)=ax+b$ for some $\displaystyle a,b$ to map $\displaystyle [2,4]$ inside $\displaystyle (2,2)$, say, into $\displaystyle [1.5,0,5]$, and $\displaystyle [6,10]$ into, say, $\displaystyle [0.5,1.5]$. So, use make two linear functions and then glue them into one using a declaration like $\displaystyle f(x)=f_1(x)$ if ..., and $\displaystyle f(x)=f_2(x)$ if ... .
Conversely, you can inject $\displaystyle (2,2)$ into, say, $\displaystyle (6,10)\subseteq [2,4]\cup [6,10]$.