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Math Help - How do you prove that [3,4)~[3,4)\{pi} ?

  1. #1
    jfk
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    Question How do you prove that [3,4)~[3,4)\{pi} ?

    Can anyone provide a proof, plz? I need to understand how the proof works, 10x in advance...
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    Re: How do you prove that [3,4)~[3,4)\{pi} ?

    Let a = \pi-3 and let A = \left\{3+\frac{a}{2^n}\;\middle\vert\; n\in\mathbb{N}\right\} where \mathbb{N}=\{0,1,2,\dots\}. In particular, \pi\in A. Then we can construct f:A\to A such that \pi is not in the image of f. Namely, define f\left(3+\frac{a}{2^n}\right)=3+\frac{a}{2^{n+1}}. Then it is easy to extend f to [3, 4) so that \pi is still not in the image of f.
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    jfk
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    Re: How do you prove that [3,4)~[3,4)\{pi} ?

    thank you very much, can I ask you what is the procedure to deal with this kind of questions?
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    Re: How do you prove that [3,4)~[3,4)\{pi} ?

    Quote Originally Posted by jfk View Post
    can I ask you what is the procedure to deal with this kind of questions?
    That is known as subset shift. You are proving an uncountable set equivalent to a subset of itself formed by removing one point.
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    Re: How do you prove that [3,4)~[3,4)\{pi} ?

    See also Hilbert's paradox of the Grand Hotel.

    In general, when we are dealing with infinite sets, the cardinality of many set operations depends only on the cardinality of the larger set. In particular, finite set don't matter compared to countably infinite sets, and those don't matter compared to continuum. So, if B is infinite and |A|\le |B| (the cardinality of A is less than or equal to the cardinality of B), then |A\cup B| = |B| and |A\times B|=|B|.
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    jfk
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    Re: How do you prove that [3,4)~[3,4)\{pi} ?

    Thank you for the reference and the explanation, it was very helpful 😃
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