cardinality

• Dec 8th 2009, 03:43 PM
Mathgirl2008
cardinality
Show that the sets [2,4] u [6,10] and (-2,2) have the same cardinality.(using the Schroeder-Berstein 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?
• Dec 9th 2009, 12:37 PM
emakarov
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]\$.