1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    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

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    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

    eumyang

    User Avatar
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook