# Is this correct proving function addition is associative?

1. Jan 30, 2012

### SMA_01

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. Jan 31, 2012

### SammyS

Staff Emeritus
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.
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.

3. Jan 31, 2012

### SMA_01

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?

4. Jan 31, 2012

### SMA_01

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

Also, should I mention associativity of addition after both results?

5. Jan 31, 2012

### HallsofIvy

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).

6. Jan 31, 2012

### SammyS

Staff Emeritus
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 .