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Math Help - cardinality

  1. #1
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    Unhappy 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?
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  2. #2
    MHF Contributor
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    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 [2,4]\cup [6,10] to (-2,2) and one back.

    Well, that should be easy. You can use linear functions f(x)=ax+b for some a,b to map [2,4] inside (-2,2), say, into [-1.5,-0,5], and [6,10] into, say, [0.5,1.5]. So, use make two linear functions and then glue them into one using a declaration like f(x)=f_1(x) if ..., and f(x)=f_2(x) if ... .

    Conversely, you can inject (-2,2) into, say, (6,10)\subseteq [2,4]\cup [6,10].
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