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Set Theory? question

  1. Mar 13, 2016 #1
    1. The problem statement, all variables and given/known data
    Let

    f: X ----> Y and g: Y ----> Z

    be functions and let

    h = g o f: X ----> Z

    2. Relevant equations

    a. If h is surjective then g is surjective

    b. If h is surjective then f is surjective.

    3. The attempt at a solution

    Here

    h: X ----> Z

    a.
    Suppose h: x ---> z is surjective for ∈ Z. Since h is surjective ∃a ∈ X such that
    h(a) = g(f(a)) = k

    Now let y = f(a) ∈ Y so...
    g(y) = g(f(a)) = k; as declared QED.

    b.
    Suppose h: x ---> z is surjective for y.........I'm not even sure how to start.

    PS: To be honest, I really need to find a good textbook on proofs because my lecturer is outright atrocious. If anyone knows of any texts, do post me some recommendations as well.
     
  2. jcsd
  3. Mar 13, 2016 #2

    PeroK

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    Your proof of a) looks quite good. You didn't say what k is, but it's fairly obvious.

    Why do you think b) is true?
     
  4. Mar 13, 2016 #3
    k is just an integer that belongs to set Z.

    As for b, I think f is surjective because h is. Since f is an input of g, I'm not exactly sure how I can re-utilise my proof from part a.
     
  5. Mar 13, 2016 #4

    PeroK

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    If I can't see how to prove something, I usually try to disprove it and see what happens.
     
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