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Inverse functions

  1. Feb 2, 2008 #1
    1. The problem statement, all variables and given/known data
    My textbook states that the inverse of a bijection is also a bijection and is unique. I understand how to show that the inverse would be a bijection and intuitively I understand that it would be unique, but I'm not sure how to show that part.

    2. Relevant equations

    3. The attempt at a solution

    My idea is to somehow say that if the inverse function is bijective and maps S -> T such that f-1(f(x))=x, then any other function that produces the same result must be the same function, but I can't quite figure out how to make this statement mathematically...
  2. jcsd
  3. Feb 2, 2008 #2
    Well, about showing that the inverse is unique, try to prove it by using a contradiction. That is suppose that there is another function call it g that is different from f^-1 ( the inverse of f) but that is also the inverse of f, ( suppose that also g is the inverse of f) and try to derive a contradiction, in other words try to show that indeed f^-1=g.

    This is the ide, the rest are details.
  4. Feb 2, 2008 #3
    Suppose f:X->Y is a bijection. Let g and h be inverses of f. Show that g(y)=h(y) for all y in Y. To do this express y as f(x) (possible since f is a surjection), and use the definition of g and h.
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