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Homework Help: Function composition

  1. Apr 30, 2010 #1
    1. The problem statement, all variables and given/known data

    Make [; f: A \rightarrow B ;], [; g: C \rightarrow D ;], [; h: E \rightarrow F ;] functions in which [; \text{Im} f \subseteq C;] and [; \text{Im} g \subseteq E;]. Show that [; f \circ ( g \circ h ) ;] and [; h \circ ( g \circ f ) ;] are valid if, and only if, [; f \circ ( g \circ h ) = h \circ ( g \circ f) ;].

    2. Relevant equations

    ...

    3. The attempt at a solution

    Though the proof seems to be very trivial, I couldn't see very deeply.

    I set the propositions necessary for the functions to exist, but I couldn't find a relation in the images, domains and codomains to make them equal.

    Thanks
     
  2. jcsd
  3. Apr 30, 2010 #2
    You probably meant:

    Make [itex] f: A \rightarrow B [/itex], [itex] g: C \rightarrow D [/itex], [itex] h: E \rightarrow F [/itex] functions in which [itex] \text{Im} f \subseteq C[/itex] and [itex] \text{Im} g \subseteq E[/itex]. Show that [itex] f \circ ( g \circ h ) [/itex] and [itex] h \circ ( g \circ f ) [/itex] are valid if, and only if, [itex] f \circ ( g \circ h ) = h \circ ( g \circ f) [/itex].

    But it looks wrong at first sight.
     
    Last edited: Apr 30, 2010
  4. Apr 30, 2010 #3
    Yeah, that's exactly what I wrote
     
  5. Apr 30, 2010 #4
    Came out as:

    Make [; f: A \rightarrow B ;], [; g: C \rightarrow D ;], [; h: E \rightarrow F ;] functions in which [; \text{Im} f \subseteq C;] and [; \text{Im} g \subseteq E;]. Show that [; f \circ ( g \circ h ) ;] and [; h \circ ( g \circ f ) ;] are valid if, and only if, [; f \circ ( g \circ h ) = h \circ ( g \circ f) ;].

    on my screen. But there are some welly strange things happening with the Latex processing.
     
  6. Apr 30, 2010 #5
    Suppose [itex]f:\mathbb{N}\rightarrow \mathbb{N}[/itex] is [itex]f:n\mapsto n+1[/itex], [itex]g=f[/itex] and [itex]h:\mathbb{N}\rightarrow \mathbb{N}[/itex] is [itex]h:n\mapsto max(n-2,0)[/itex].

    Are both [itex]f\circ(g\circ h)[/itex] and [itex]h\circ(g\circ f)[/itex] defined? If so, are they equal?
     
  7. May 1, 2010 #6

    HallsofIvy

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    Science Advisor

    Martin, when you edit and it doesn't work correctly (and editing LaTex often gives that problem), try clicking on the "refresh" button. That often clears up the problem. Why it doesn't "refresh" automatically, I don't know!
     
  8. May 1, 2010 #7
    Thanks. With luck that should save me some work.

    But in this instance it was Hobold's entry that was garbled and I hadn't edited it. In fact it still looks garbled on my screen (even after refresh).
     
  9. May 1, 2010 #8
    When I said, "But it looks wrong", I was referring to the content rather than the typesetting.
     
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