# Thread: Proove that a + (b - c) = (a + b) - c.

1. ## Proove that a + (b - c) = (a + b) - c.

Hi there

I was reading through this webpage: Arithmetic Rules and I got to the author's attempt to prove the statement
$b + (-a) = b - a$ where $a, b \in \mathbb{R}$.

I am, however, not convinced by his argument.

Up to this point he had estabished the following to be true:

• $a + 0 = a$
• $a + b = b + a$
• $(a + b) + c = a + (b + c)$
• $a = b \implies a + c = b + c$
• $a + b = c \iff c - b = a$
• $-a = 0 - a$
• $(-a) + a = 0$
• $0 = -0$

He gives as a proof:

$-a = 0 - a$

$b + (-a) = b + 0 - a$

$b + (-a) = b - a$

Am I correct in thinking that $0-a$ should be in parentheses when $b$ is added to it, and since the author is yet to establish that $a + (b - c) = (a + b) - c$ the proof is invalid?

Would anyone be able to tell me whether or not I am correct in thinking this, and if not then explain what I am not understanding?

2. ## Re: Proove that a + (b - c) = (a + b) - c.

Originally Posted by userna
Hi there

I was reading through this webpage: Arithmetic Rules and I got to the author's attempt to prove the statement
$b + (-a) = b - a$ where $a, b \in \mathbb{R}$.

I am, however, not convinced by his argument.

Up to this point he had estabished the following to be true:

1. $a + 0 = a$
2. $a + b = b + a$
3. $(a + b) + c = a + (b + c)$
4. $a = b \implies a + c = b + c$
5. $a + b = c \iff c - b = a$
6. $-a = 0 - a$
7. $(-a) + a = 0$
8. $0 = -0$

He gives as a proof:
$-a = 0 - a$
$b + (-a) = b + 0 - a$
$b + (-a) = b - a$
Am I correct in thinking that $0-a$ should be in parentheses when $b$ is added to it, and since the author is yet to establish that $a + (b - c) = (a + b) - c$ the proof is invalid?
Would anyone be able to tell me whether or not I am correct in thinking this, and if not then explain what I am not understanding?
I numbered the list. Look at #6. It tells you that parentheses are not needed.
$\#5~\text{ and/or }\#7$ says $(-a)=-a$.

3. ## Re: Proove that a + (b - c) = (a + b) - c.

Originally Posted by userna
$b + (-a) = b - a$

Am I correct in thinking that $0-a$ should be in parentheses when $b$ is added to it, and since the author is yet to establish that $a + (b - c) = (a + b) - c$ the proof is invalid?
Let x = -a
b + x = b - a ?

4. ## Re: Proove that a + (b - c) = (a + b) - c.

Originally Posted by DenisB
Let x = -a
b + x = b - a ?
DenisB. In these proofs you need to give reasons for every little step.
There is no such step as "Let anything"
What justifies $b+x=b-a$ is the list of eight above?

5. ## Re: Proove that a + (b - c) = (a + b) - c.

Me no tryin' to prove nuttin' dere...
Just sumtin' for OP to look at...
Who da hell zinterested in provin' sumtin' so silleeee...

6. ## Re: Proove that a + (b - c) = (a + b) - c.

Originally Posted by DenisB
Me no tryin' to prove nuttin' dere...
Just sumtin' for OP to look at...
Who da hell zinterested in provin' sumtin' so silleeee...
You my dear friend would fail any analysis graduate course in any good university.
There are very good reasons for rigor in basic proofs.
Who are you to question a century of practice?
Have you done any graduate work in mathematics? If so where? I would like to show them there product.

7. ## Re: Proove that a + (b - c) = (a + b) - c.

Originally Posted by DenisB
Let x = -a
b + x = b - a ?
You have used what you are trying to prove in your proof!

8. ## Re: Proove that a + (b - c) = (a + b) - c.

Originally Posted by Plato
I numbered the list. Look at #6. It tells you that parentheses are not needed.
$\#5~\text{ and/or }\#7$ says $(-a)=-a$.