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Math Help - bijective

  1. #1
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    bijective

    If  f: A \to B is bijective show that  f^{-1} is unique.

    Suppose that  f^{-1} and  h are inverses of  f . Let  y = f(x) . We need to show that  f^{-1}(y) = h(y) for all  y \in B . So  f^{-1}(f(x)) = h(f(x)) = x for all  y \in B , thus  f^{-1} is unique?
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  2. #2
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    Quote Originally Posted by particlejohn View Post
    If  f: A \to B is bijective show that  f^{-1} is unique?
    Let g,h: B\mapsto A such that g(f(a)) =h(f(a))= a for all a\in A. To show that g=h we need to show to things, (i) they have the same domain (ii) g(x)=h(x) for all x in domain. The first part is straightforward, it is given that g,h have domain B. Now let x\in B. Then since f is bijective it means x=f(a) for some a\in A. Thus, g(x) = g(f(a)) = a = h(f(a)) = h(x). And thus these are the same functors.
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