Results 1 to 7 of 7

Math Help - proofs with functions 1-1 and onto

  1. #1
    Junior Member
    Joined
    Jan 2008
    From
    Waipahu, HI
    Posts
    37

    proofs with functions 1-1 and onto

    The problem is:
    Suppose that f:A-->B, g:B-->C are functions. In the following cases, answer yes or no. If your answer is yes, prove it. If it is no, give a counterexample and say what additional hypotheses are needed to make the statement true, then prove.
    1. if gof is 1-1, must f be 1-1?
    2. if gof is 1-1, must g be 1-1?
    3. if gof is onto, must f be onto?
    4. if gof is onto, must g be onto?

    I'm fine with 1 and 4, and for 2, I think the answer is yes, but I'm not sure how to go about proving it. I understand it, and I could explain it in English (if g isn't 1-1, then two elements in B go to the same element in C, and the two elements in A go to the two elements in B, which go to the same element in C) but I'm not sure how to say it in a proof. For 3, I'm just confused. The element that is not in the range of f could still be in the domain of g, so I think I have to say no, and add something to it, but I'm not sure what.

    Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Jan 2008
    From
    Waipahu, HI
    Posts
    37
    I'm not sure how to delete this post, but I figured it out.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor kalagota's Avatar
    Joined
    Oct 2007
    From
    Taguig City, Philippines
    Posts
    1,026
    you dont have to delete it..
    anyways, you think that 2 is false?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Jan 2008
    From
    Waipahu, HI
    Posts
    37
    No, I thought that at first, but I drew lots of little pictures until I figured out that g doesn't need to be 1-1. My answers ended up being yes, no, no, yes.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor kalagota's Avatar
    Joined
    Oct 2007
    From
    Taguig City, Philippines
    Posts
    1,026
    consider f(x) = \sqrt{x}.. thus, domain is nonnegative reals.
    suppose that g(x) = x^2..
    therefore (g o f)(x) = (\sqrt{x})^2 = |x| = x where x is nonnegative reals (since the domain takes the domain of f for which g is defined), which makes g o f to be 1-1..

    however, g(x) = x^2 is not 1-1..
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Jan 2008
    From
    Waipahu, HI
    Posts
    37
    Yeah, so for number 2, g doesn't need to be 1-1 for gof to be 1-1, so the answer is no, right?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor kalagota's Avatar
    Joined
    Oct 2007
    From
    Taguig City, Philippines
    Posts
    1,026
    yup. the answer is no.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 9
    Last Post: December 6th 2011, 04:02 PM
  2. Replies: 4
    Last Post: August 12th 2010, 11:37 AM
  3. Proofs: Domain of composite functions
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: April 20th 2009, 03:49 PM
  4. Proofs: pigeonhole principle and functions
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: February 25th 2009, 09:37 PM
  5. Help w/ some proofs even/odd functions
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 16th 2006, 12:25 PM

Search Tags


/mathhelpforum @mathhelpforum