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

  1. #1
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    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?

    Thanks in advance for your help.
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  2. #2
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    Re: Proove that a + (b - c) = (a + b) - c.

    Quote Originally Posted by userna View Post
    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 $(-a)=-a$.
    Last edited by Plato; Jul 10th 2017 at 01:07 PM.
    Thanks from topsquark and userna
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    Re: Proove that a + (b - c) = (a + b) - c.

    Quote Originally Posted by userna View Post
    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 ?
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    Re: Proove that a + (b - c) = (a + b) - c.

    Quote Originally Posted by DenisB View Post
    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?
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  5. #5
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    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...
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    Re: Proove that a + (b - c) = (a + b) - c.

    Quote Originally Posted by DenisB View Post
    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.
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    Re: Proove that a + (b - c) = (a + b) - c.

    Quote Originally Posted by DenisB View Post
    Let x = -a
    b + x = b - a ?
    You have used what you are trying to prove in your proof!
    Last edited by userna; Jul 11th 2017 at 02:42 AM.
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    Re: Proove that a + (b - c) = (a + b) - c.

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

    The point that I don't understand is that when we add b according to #4, do we not need to treat 0-a as a 'single object', and therefore write it in parenthesis?
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    Re: Proove that a + (b - c) = (a + b) - c.

    Quote Originally Posted by Plato View Post
    You my dear friend...
    WRONG...me, I no have no friends...
    Have you done any graduate work in mathematics?
    Seriously no. High School only (grade 13 back in 50's).
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