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Prove Set of all onto mappings from A->A is closed

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  1. Feb 14, 2015 #1
    1. The problem statement, all variables and given/known data
    Prove that set of all onto mappings of A->A is closed under composition of mappings:

    2. Relevant equations
    Definition of onto and closure on sets.

    3. The attempt at a solution
    Say, ##f## and ##g## are onto mappings from A to A.
    Now, say I have a set S(A) = {all onto mappings of A to A }, so ##f## and ##g \in S##
    Then,

    ## f o g(a_0) = f(g(a_0)) = f(a_i) , a_i, a_0 \in A; a_i ## represents all elements in A that are being hitted.
    Now, the domain of ##f## is set A, since ##g## is onto. And now all elements in the domain of ##f## will hit all elements in the codomain of f.
    Therefore, ##fog \in S(A) ##

    Is this correct? Or is it weak, illogical, flawed ... ?
     
  2. jcsd
  3. Feb 14, 2015 #2

    PeroK

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    It's not correct. Both the general approach and the details are not right.

    First, there's no need to introduce S(A) when you did. Stick with f and g being onto. You need to show that f o g is onto. Now, can you write down the definition of onto for f o g? Try writing it as:

    "We need to show that ..."
     
  4. Feb 14, 2015 #3
    Thanks.
    We need to show that ##fog## is onto or in other words
    fog = z = f(g(x)) =A?

    f: A-> A : f(a) = A
    g: A ->A: g(a) = A

    Is this correct?
     
  5. Feb 14, 2015 #4

    SammyS

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    You need to show that for every z in A, there is an x in A such that ##\ (f \circ g) \, (x)=z \ ## .
     
  6. Feb 14, 2015 #5
    Still not very sure on how to start.
    BTW, is there anything rescuable from my first approach?
     
  7. Feb 14, 2015 #6

    SammyS

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    Sure.

    Having f & g be chosen from set S is fine.

    Then show that ##\ f\circ g\ ## is onto.
     
  8. Feb 15, 2015 #7

    PeroK

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    If you don't know the definition of onto, how can you prove a function is onto? Formal definitions are essential for rigorous mathematics.

    Does the formal definition that Sammy gave you make sense?

    I'd do two things to start. First, I'd write down what you know about f and g being onto.

    Then, I'd find an example. Perhaps f, g: [0, 1] -> [0, 1], where f(x) = x^2 and g(x) = 1 - x. Check that these are onto, then show that f o g is onto. Why is f o g onto? Then, try to generalise this for any functions.
     
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