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Homework Help: Explicit formula

  1. Feb 13, 2013 #1
    1. The problem statement, all variables and given/known data
    Give an explicit formula for a function f : ℕ ⇒ ℕ that is

    a) onto, but not one-to-one.
    b) neither one-to-one nor onto.

    1. The attempt at a solution
    a) The formula f[itex]_{2}[/itex](n)= ⌊[itex]\frac{n}{3}[/itex]⌋. it's onto cause f[itex]_{2}[/itex](3n)= n for every n. but, it's not one-to-one, cause f[itex]_{2}[/itex](1)= ⌊[itex]\frac{1}{3}[/itex]⌋ = 0 = f[itex]_{2}[/itex](0)

    b) f[itex]_{4}[/itex](n)=4 ⌊[itex]\frac{n}{4}[/itex]⌋. This is not one-to-one, cause it's counterexample f[itex]_{4}[/itex](0) = f(1) = 0. Nor is it onto cause there is no odd number in the range of f[itex]_{4}[/itex]
    Last edited: Feb 13, 2013
  2. jcsd
  3. Feb 13, 2013 #2


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    1/3 isn't equal to 0. You have to explain what f_2 means a lot better than that. Do you mean something like the floor function? Oh, I see you do. Guess I should wait till you finish posting.
  4. Feb 13, 2013 #3
    Sorry for that, it wasn't completed yet, now it's.
  5. Feb 13, 2013 #4


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    Mmm. ##f_2(1)=0##. 0 isn't in N is it? You might have to modify it a bit.
  6. Feb 13, 2013 #5
    natural numbers { 0, 1, 2, 3, ...}

    Yes it's.
  7. Feb 14, 2013 #6


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    Not to me it isn't. In some places natural numbers do not include zero {1, 2, 3, ...}. We use another term, whole numbers, to indicate {0, 1, 2, 3, ...}. Why is there no agreement on the definition of natural numbers I have no idea.
  8. Feb 14, 2013 #7
    i agree.
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