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Thread: Surjections and a true/false statement.

  1. #1
    Super Member Showcase_22's Avatar
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    Surjections and a true/false statement.

    This is a question I have on a test paper:

    Is the following statement always true? Justify your answer.

    If $\displaystyle f:A \rightarrow B$ and $\displaystyle g:C \rightarrow D$ are surjections where $\displaystyle Im(f) \subset C$, then g o f: $\displaystyle A\rightarrow D$ is a surjection.
    (I wrote "g o f" when I mean the "composite function of g and f").

    I would say this statement is true because the domain of g is contained in the image of f (which is a necessary criterion for g o f to exist).

    The composite function of two surjections is also a surjection.

    Both these facts mean that the statement is always true.

    I think this is right, I just wanted to see if I had overlooked anything.
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi,

    Take a look at this diagram :

    Surjections and a true/false statement.-surjection.png

    $\displaystyle f$ maps $\displaystyle A$ onto $\displaystyle B$, $\displaystyle g$ maps $\displaystyle C$ onto $\displaystyle D$ and $\displaystyle \mathrm{Im}(f)=B\subset C$ so, according to you, $\displaystyle g\circ f$ should map $\displaystyle A$ onto $\displaystyle D$. Is that true ?

    Edit: That's my 666th post.
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  3. #3
    Super Member Showcase_22's Avatar
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    Right, i'd say that the statement is false then.

    The third point in the set C (the point that is in C but not in B) isn't connected to a point in A and since $\displaystyle B \subset C$ there will always be a point that is not connected to a point in a.

    Therefore it cannot be a surjection.

    I think i've got it.
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  4. #4
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by Showcase_22 View Post
    The third point in the set C (the point that is in C but not in B) isn't connected to a point in A and since $\displaystyle B \subset C$ there will always be a point that is not connected to a point in a.

    Therefore it cannot be a surjection.
    That's it!
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