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Is this correct proving function addition is associative?

  1. Jan 30, 2012 #1
    Is this correct...proving function addition is associative???

    1. The problem statement, all variables and given/known data

    Let F be the set of all real-valued functions having as domain the set ℝ of all real number. Prove that function addition + on F is associative.

    2. Relevant equations




    3. The attempt at a solution

    I'm not sure if I approached this correctly, but here is how I did it:

    For all f,g,h in F:

    (f+g)(x)+h(x)
    =f(x)+g(x)+h(x) [by definition of function addition]

    and

    f(x)+(g+h)(x)
    =f(x)+g(x)+h(x) [by definition of function addition]

    so that, (f+g)(x)+h(x)=f(x)+(g+h)(x) for all x in ℝ

    Did I do this correctly? Any help is appreciated, thanks.
     
  2. jcsd
  3. Jan 31, 2012 #2

    SammyS

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    Re: Is this correct...proving function addition is associative???

    In my opinion, you skipped some steps.

    I would say that:
    (f+g)(x)+h(x)
    =(f(x)+g(x))+h(x) [by definition of function addition]​
    Anything beyond that result requires the mention of or convention for "order of operations".
    I admit this is being a little picky.

    The following is not being picky.
    Your other statement should say:
    f(x)+(g+h)(x)
    =f(x)+(g(x)+h(x)) [by definition of function addition] ​
    To go beyond this requires you to call on the associativity of ordinary addition.
     
  4. Jan 31, 2012 #3
    Re: Is this correct...proving function addition is associative???

    SammyS- Thank you, I am not fully understanding what you mean by order of operations though, can you please clarify?
     
  5. Jan 31, 2012 #4
    Re: Is this correct...proving function addition is associative???

    Also, should I mention associativity of addition after both results?
     
  6. Jan 31, 2012 #5

    HallsofIvy

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    Re: Is this correct...proving function addition is associative???

    For functions f and g, the sum, f+ g, is defined by (f+ g)(x)= f(x)+ g(x). Do you understand the notation? f and g are the functions, f(x) and g(x) are the values of the functions. f+ g is a sum of functions, f(x)+ g(x) is a sum of numbers.

    (f+ g)+ h is the function such that ((f+ g)+ h)(x)= (f(x)+ g(x))+ h(x). Now, because addition of numbers is associative, (f(x)+ g(x))+ h(x)= f(x)+ (g(x)+ h(x))= (f+ (g+h))(x).
     
  7. Jan 31, 2012 #6

    SammyS

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    Re: Is this correct...proving function addition is associative???

    What I meant by "order of operations" was that: if all operations are the same, in this case they're all addition, then a + b + c is the same as (a + b) + c .
     
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