Invertible Mapping

Definition (Invertible Mappings) A mapping is an inverse of if both and A mapping is said to be invertible if it has an inverse.

Proposition (Invertible Mappings)    

    (i) A mapping is invertible if and only if it is both one-to-one and onto.
    
    (ii) An inverse of an invertible mapping is invertible.
    
    (iii) If and are invertible, then is invertible.
    
    Proof. (i): Assume that is invertible with inverse Then being the identity mapping on is one-to-one; therefore, must be one-to-one. Also, being the identity mapping on is onto; whence is onto. Therefore, if is invertible it is also one-to-one and onto. Conversely, assume that is onto and one-to-one and Since is onto there exists such that But is also one-to-one, so this element must be unique. Using this , define This can be done for each element in and in this way a mapping is constructed such that and Thus is invertible.
    (ii):  Assume that is invertible with inverse Then being the identity mapping on is onto; therefore, must be onto. Also, being the identity mapping on is one-to-one; whence is one-to-one. Therefore β is invertible by part (i).
     (iii): If and are invertible then and are one-to-one and onto. So is one-to-one and onto; whence is invertible.  

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