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Problems about numbers and bijective functions

  1. Feb 1, 2012 #1
    1. The problem statement, all variables and given/known data

    1. Prove that any finite sequence of digits is a starting sequence of digits for some power of 2.


    2. Relevant equations


    3. The attempt at a solution
    This one seems easy, i first thought about binary system ( sum of power of 2)
    then: suppose the random finite sequence is ai (sorry about the underline, i dont know how to make bar over letters)
    need to prove that for every ai there exist bi and n such that aibi = 2n
    or aix10k + bi = 2n
    we need 0 < bi < 10k
    the problem becomes, prove that for every ai, there exist n and k such that
    aix10k < 2n < (ai+1)x10k
    i'm still working on it...
    perhaps give me a hint on how to prove this or maybe there's a better way :redface:

    1. The problem statement, all variables and given/known data
    2. Prove that for any function [itex]f(x): \mathbb{Q} \to \mathbb{Q} [/itex] there exist three bijective functions
    [itex]\phi_1: \mathbb{Q} \to \mathbb{Q} [/itex] ,[itex]\phi_2: \mathbb{Q} \to \mathbb{Q} [/itex] and [itex]\phi_3: \mathbb{Q} \to \mathbb{Q} [/itex] satisfying [itex]f(x) = \phi_1 +\phi_2 + \phi_3 [/itex]


    2. Relevant equations



    3. The attempt at a solution
    I don't even know where or how to start with this one :cry:
    should i use contradiction or something ?
     
  2. jcsd
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