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Math Help - Role of Rule in Prooving a Function? Help!

  1. #1
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    Role of Rule in Proving a Function? Help!

    Problem:
    Define h: Z \rightarrow Z by the rule
    h(n) = 4n-1 for all n \in Z
    Is h onto?

    Solution:
    I know that h is not onto because if we suppose 4n-1 = m \in Z then we have to search an n \in Z such that h(n) = m, and for m=0 \in Z there doesn't exist any integer n.

    As A Counter Example
    h(n) = 0
    4n-1 = 0
    4n = 1
    n = 1/4 \notin Z
    It is all right.

    Role of Rule in Proving a Function? Help!
    But, what is the purpose of the rule h(n) = 4n-1 ? because if we stuck on this rule then 4n-1 will never be equal to 0.

    For Example:

    According to h(n) = 4n-1 For All  n \in Z

    n = -1 Then 4(-1) - 1
    n = 0 Then 4(0) - 1
    n = 1 Then 4(1) - 1
    n = 2 Then 4(2) - 1

    ...So on, and in this case 4n-1 will always be an integer thus h will become onto.
    Any help??
    Last edited by mrsenim; March 30th 2010 at 12:32 PM.
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  2. #2
    MHF Contributor
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    Not sure what your question is.

    I know that h is not onto because if we suppose then we have to search an such that h(n) = m, and for there doesn't exist any integer n.
    Correct.

    But, what is the purpose of the rule h(n) = 4n-1 ?
    You need the definition of a function to prove that it is onto, don't you? You can't prove that some unknown function is a surjection because not all functions are.

    So on, and in this case 4n-1 will always be an integer thus h will become onto.
    How does it follow that h is onto if its values are always integers? For h to be onto, its values must be all integers.
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