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Invertible <=> F is 1-1 and onto

  1. Apr 2, 2009 #1
    invertible <==> F is "1-1" and "onto"

    Never mind, haha- found out

    Does anyone know why the following is true:

    F is invertible <==> F is "1-1" and "onto"?
    Last edited: Apr 2, 2009
  2. jcsd
  3. Apr 2, 2009 #2
    Re: Invertibility2

    Yes it is true.

    Say F maps from A to B then:

    1) As F is onto every element in B corresponds to at least one element in A i.e. for all [itex] b \in B [/itex] there is at least one [itex] a \in A [/itex] such that [itex] f(a) = b [/itex].

    2) As F is 1-1 every element in B can correspond to at most one element in A ie. [itex] f(a) = f(a') \Rightarrow a=a' [/itex].

    This is enough to be able to construct the inverse
    [tex] f^{-1}(b) = a \mbox{ if and only if }f(a) = b [/tex]
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