Properties of Mappings

Proposition (Properties of Mappings) Let     

    (i) Then, is onto if and only if there exists a mapping such that
    
    (ii) Then, is one-to-one if and only if for every set and all mappings and if then
    
    (iii) Then, is onto if and only if there exists a mapping such that
    
    (iv) Then, is one-to-one if and only if for every set and all mappings and if then
    
    Proof. (i): Assume that is onto and let For each there exists such that (Axiom of Choice). For each pick one such that and let be the set of all such chosen pairs Then each is the first component of one ordered pair in so is a mapping. Let if and only if Then and so Conversely, assume that and there is a mapping with Given let Then so is onto.
    (ii): Assume that is onto and let Then there exists with If then Since for all it follow Assume that is not onto and let such that Define by for all and define by for all and Then since On the other hand and since Since and have the same domain and same codomain,
    (iii): Assume that is one-to-one and let Pick (Axiom of Choice) Let Since is one-to-one, each element of is the first entry in one and only one ordered pair in Hence, is a mapping. Let be where Then and since has domain and codomain we have Conversely, assume that for there exists with and that Then, if then and so is one-to-one.
    (iv): Assume that is one-to-one, and Then and so Since is one-to-one, Since for all and and have the same domain and same codomain, Conversely, assume that is not one-to-one. Then there exists in with Let and define by and define by and Then but since and   

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Cite this as:
Properties Of Mappings
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/properties-of-mappings.html