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Is it Defined, Or Can It Be Proven?

  1. Aug 19, 2014 #1
    Can the equality a-b = a + (-b) be proven, or are a - b and a + (-b) defined to be the same?
     
  2. jcsd
  3. Aug 19, 2014 #2

    LCKurtz

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    It can be proven. I will use ##^-b## for the additive inverse of ##b## so not to confuse it with subtraction. So you want to know whether ##a +^-b = a-b##. Remembering that ##a-b## is the number you can add to ##b## to get ##a##, let's check whether it works:$$
    b +(a + ^-b) = b +(^-b +a) = (b + ^-b) + a = 0 + a = a$$ so it works. Can you fill in the reason for each step?
     
  4. Aug 19, 2014 #3
    I understand each of the properties you appealed to, to justify each step; but I do not see how this shows that
    a - b and a + (-b) are equivalent.
     
  5. Aug 19, 2014 #4

    LCKurtz

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    I just showed that if you add ##a+^-b## to ##b## you get ##a##. That is the definition of ##a-b## since ##a-b## is the number you can add to ##b## to get ##a##. So ##a+^-b## is ##a-b##.
     
  6. Aug 19, 2014 #5

    Fredrik

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    I would take it as the definition of subtraction. If you want to prove it, you have to specify what other definition of subtraction that you're using. LCKurtz is defining a-b (for arbitrary a and b) as the unique number x such that b+x=a. If you add -b to both sides of this equality, you see that x=a+(-b). I wouldn't say that this is the definition of subtraction. It's just a definition of subtraction.
     
  7. Aug 19, 2014 #6

    D H

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    Funny: The wikipedia article on subtraction uses LCKurtz's definition, but the wikipedia article on the integers uses Fredrik's definition.

    With LCKurtz's definition, that subtraction is the inverse function of addition is axiomatic, but that subtraction is equivalent to adding with the additive inverse is a theorem. With Fredrik's definition, it's the other way around.
     
  8. Aug 19, 2014 #7
    A little bit of both. The issue concerns whether (-b) is a unique element.

    In abstract algebra the ring axiom says that for every ##a## there is some ##x## such that ##a + x = 0_R##. It does not claim how many ##x## exist for a given ##a##. So, this is what you have to prove.

    If ##-b## isn't unique you might end up getting multiple answers when doing a subtraction!
     
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