Composition of Mappings

Definition (Composition of Mappings) If and then the composition (denoted by ), is a mapping from to and is defined by for each

Proposition (Composition Mappings) Assume and
    (i) If and are onto, then is onto.
    (ii) If is onto, then is onto.
    (iii) If and are one-to-one, then is one-to-one.
    (iv) If is one-to-one, then is one-to-one.
    
    Proof.  (i): Assume that and are onto and Because is onto, there exists such that Since is onto there exists such that So which means there exists such that and so is onto.
    (ii): Assume that is onto and Then there exists such that But then with Hence is onto.
    (iii): Assume that both and are one-to-one and Then because is one-to-one; and because is one-to-one. Therefore, is one-to-one.
    (iv): Assume that is one-to-one, and then Since is one-to-one, and thus is one-to-one.  

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