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Math Help - Equinumerous Sets

  1. #1
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    Equinumerous Sets

    Please help on the followings.

    1. Show that the following sets are denumerable (countably infinite)

    (a) Q x Q (Q is the set of rational numbers) (This one is ok, but I am stuck at (b)

    (b) Q( \sqrt(2)) = {a+b \sqrt(2) | a \in Q and b \in Q}



    2. Show that (0,1) ~ R (R is the set of reals)

    3. Show that (0,1] ~ (0,1)


    For 2 I simply could not think of the function from (0,1) to R that is 1-1 onto. For 3, the existence of {1} is a problem. I could not find a 1-1 function that can deal with {1}.

    Thanks!
    Last edited by armeros; May 6th 2009 at 08:05 AM.
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  2. #2
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    For 1b use what you know from 1a.
    \varphi :\mathbb{Q} \times \mathbb{Q} \mapsto \mathbb{Q}\left( {\sqrt 2 } \right)\;,\;\varphi (a,b) = a + b\sqrt 2 .
    Show that \varphi is a bijection.

    For #2 consider the function \tan \left( {\pi x - \frac{\pi }{2}} \right) on (0,1).

    For #3 define F = \left\{ {\frac{1}{n}:n \in \mathbb{Z}^ +  } \right\} for ease of notation.
    Define a function 0,1] \mapsto (0,1)\;,\;\Phi (x) = \left\{ {\begin{array}{rl}
    {\frac{1}
    {{n + 1}},} & {x = \frac{1}
    {n} \in F} \\
    x, & {\text{else}} \\

    \end{array} } \right." alt="\Phi 0,1] \mapsto (0,1)\;,\;\Phi (x) = \left\{ {\begin{array}{rl}
    {\frac{1}
    {{n + 1}},} & {x = \frac{1}
    {n} \in F} \\
    x, & {\text{else}} \\

    \end{array} } \right." />
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  3. #3
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    Thanks

    Thank you very much.

    For question 1, I proceeded that way too,but I am stucked at showing f(a, b) is a 1-1 function. To show it, we have to set f(a1, b1) = f(a2,b2), but then we have only one equation, which I have no idea to show a1 = a2, and b1 = b2.
    Last edited by armeros; May 6th 2009 at 10:39 PM.
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