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Retraction function -> Onto proof

  1. Mar 3, 2014 #1
    Wrong forum! sorry please move to math!


    1. The problem statement, all variables and given/known data
    Prove that if ##r : X \rightarrow A## is a retraction, then r is onto.

    Please let me know if I did this correctly!

    2. Relevant equations
    ##i_A(x)=x## is the identity function of A


    3. The attempt at a solution

    If 'r' is a retraction function then there is a function 's' such that ##A \rightarrow X## and ##r \circ s=i_A##

    Then regardless of the input we put into ##r(x)## we will always get x out. Meaning that there is one, and only one, result for each element in 'r'. Then if 'r' is a function in ##\mathbb{R}## it will result in one output for each element in ##\mathbb{R}##.
     
  2. jcsd
  3. Mar 3, 2014 #2

    PeroK

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    I'm not sure about your proof. To show r is onto, you need to show that for every element a of A, there exists an element x of X, such that r(x) = a.

    Note that r need not be 1-1.
     
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