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Homework Help: Injective proof

  1. Jun 8, 2010 #1
    Prove that:

    If f : X → Y is injective, g, h : W → X, and f ∘ g = f ∘ h, then g = h.
     
  2. jcsd
  3. Jun 8, 2010 #2
    What is your progress on this problem?
     
  4. Jun 8, 2010 #3
    Definition of injective for If f : X → Y :
    For all y [tex]\in[/tex] Y, there exists at most one x [tex]\in[/tex] X such that f(x) = y

    Because f : X → Y and g, h : W → X,

    f ∘ g : W → X → Y and f ∘ h : W → X → Y

    so f ∘ g, f ∘ h : W → Y


    that's where I get stuck.
     
  5. Jun 8, 2010 #4

    lanedance

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    note that an injective function is a function that preserves distinctness... so you can consider the inverse of f
     
  6. Jun 8, 2010 #5
    Then f^(-1) : Y → X is also injective.. but I don't see what I can do from that.
     
  7. Jun 8, 2010 #6
    It may be easier to see via contradiction. If there is w in W such that g(w) is not equal to h(w), what happens to (f ∘ g)(w) and (f ∘ h)(w)?
     
  8. Jun 8, 2010 #7

    lanedance

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    or if f o g = f o h, then apply f-1
     
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