Mappings on Power Sets

Proposition (Mappings on Power Sets) Let and let and be the sets of all subsets of and respectively. Then

    (i) induces a mapping defined by for
    
    (ii) is one-to-one if and only if is one-to-one,
    
    (iii)   is onto if and only if is onto, and
    
    (iv) if and only if
    
    Proof. (i): Suppose that such that and So there exists such that and Clearly impossible. Therefore, for any
    (ii): Assume where W.L.O.G. there exists and Then because is one-to-one, but So and therefore, is onto-to-one. Conversely, let in Since is one-to-one, and so Therefore, is one-to-one.
    (iii): Let and and consider the subset Then and Further, since is onto Conversely, let Then, since is onto, there exists such that By definition of there exists such that and so is onto.
    (iv): If then and so   Conversely, if   then and so   If then for all and so for all Conversely, if then for all and so in particular, for all Therefore, for all as desired.

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Mappings On Power Sets
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/mappings-on-power-sets.html