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Onto and on to one functoins, if a f is onto must g be onto?

  1. Nov 6, 2006 #1
    Hello everyone. I've tried for awhile to find a counter example for the following, it says either prove or give a counter example:

    If f; x->y and g: y -> z are functions and g o f is onto, must both f and g be onto? Prove or give a counter example.

    for somthing to be one to one, that means all the domain in x must only point to 1 range in Y.

    For somthing to be onto every Z in the range must have a co-domain. Meaning, every thing in the range must be used, you can't have any left overs. So every domain must point to a range.

    My counter examples look like this:
    heres my onto f:
    X = {1,2,3}
    Y = {1,2,3}
    the domain of 1 points to 1 in the range
    the domain of 2 points to 2 in the range
    the domain of 3 poitns to 3 in the range
    ( i could have mixed it up but i'm trying to make it simple)

    heres my g:eek:ne-to-one function:
    Y = {1,2,3}
    Z = {3,4,5}
    the domain of 1 points to 3
    the domain of 2 points to 4
    the domain of 3 points to 5

    But when I do this, they both look like onto functions, because I can't make g o f onto, if f is isn't onto.

    can you find any counter examples or shall I attempt to prove it?
  2. jcsd
  3. Nov 6, 2006 #2


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    Science Advisor

    No, you're on the right track finding counterexamples. Just play around some more. What could you do to the functions that you have defined to make one of them not onto?
  4. Nov 6, 2006 #3
    If i make 2 of the domains point to 1 of the same ranges that will make it not onto, but it seems to not work out as shown below

    onto function g:
    y = {1,2,3}
    z = {1,2,3}

    other function f:
    x = {1,2,3}
    y = {1,2,3}

    g o f (1) = g(f(1)) = g(1) = 3
    g o f (2) = g(f(2)) = g(2) = 2
    g o f (3) = g(f(3)) = g(2) = 2

    So one of them are onto, but the g o f is not onto

    Am I allowed to make one of the ranges bigger than the others? like:
    function f:
    x = {1,2,3,4}
    y = {1, 2, 3, 4}

    Then let

    but don't use domain 4 for the g function?
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