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Math Help - Constructing injection questions

  1. #1
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    Constructing injection questions

    Let S_2 = {0,1}^N, S_2^* = { (a_n) \in S_2; (there exists k\in N) s.t ( for all n>=k) a_n = 1}. Recall that the function f: S_2-S_2^*--> [0,1), given by f((a_n)) = \displaystyle \sum_{n = 1}^{\infty}a_n/2^n is a bijection. Denote by \varphi: [0,1)-->S_2-S_2^* the inverse function f^{-1}. Note that for x \in [0,1),
    \varphi (x) is the sequence \varphi(x) = (\varphi_n(x))_{n = 1}^{\infty} \in S_2-S_2^* with f(\varphi(x)) = \displaystyle \sum_{n = 1}^{\infty} \varphi_n(x)/2^n = x.

    a) Construct an injection g: S_2-S_2^* --> [0,1) x [0,1).

    b) Construct an injection h: [0,1) x [0,1) --> S_2-S_2^*. (Hint: Interlace decimals)

    The question is kind of confusing with all these notations, any help will be useful.
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  2. #2
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    Re: Constructing injection questions

    I am truly not sure what you are asking for?
    Do you understand that S_2 is the set of infinite bit strings?
    That S_2^* is the set of infinite bit strings with at a finite collection of zeros?
    Every number in [0,1) has binary representation.
    The set S_2\setminus S_2^* represent the binary representation that are unique.

    Does that help?
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  3. #3
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    Re: Constructing injection questions

    sorry, I do not understand those terms, I think this question has things to do with decimal reprsentation
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