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Math Help - Cardinality Question

  1. #1
    Junior Member
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    Cardinality Question

    Let S be the set of real numbers r s.t.  \cos (r) \in Q\sqrt 2.

     Q\sqrt 2 be the set of real numbers of the form  a+b\sqrt 2 , where a,b is rational.

    Find the cardinality of S.

    I know 0,2pi,4pi.. etc are in S, so S is infinite
    and  Q\sqrt 2 is countably infinite (dont know if it matters)
    then im not sure how to go on...any hints would be appreciated.

    Thanks in advance
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  2. #2
    MHF Contributor
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    For each number x\in\mathbb{R}, the set \{r\in\mathbb{R}\mid\cos(r)=x\} is at most countable. Picture the (co)sine wave that is crossed by a horizontal line; there are at most countably many intersection points.

    Now, \{r\in\mathbb{R}\mid\cos(r)\in\mathbb{Q}(\sqrt{2})  \}=\bigcup_{x\in\mathbb{Q}(\sqrt{2})}\{r\in\mathbb  {R}\mid\cos(r)=x\}, so...
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