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Math Help - A substitution

  1. #1
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    A substitution

    Given the following axiomatic system :

    For all A,B,C:

    1) A+B= B+A .................................................. ......AB= BA

    2) A+(B+C)= (A+B)+C........................................... ....(AB)C= A(BC)

    3) A+0 =A................................................ .................1A=A

    4) For all A ,there exists B : A+B=0.....................................For all A =/=0 ,there exists B: AB=1

    5) A(B+C)= AB+AC

    6) 1=/=0

    We want to prove:

    For all A : A+(-A) = 0

    Proof:

    From axiom (4) if we put A = -B we have : (-B)+B =0 => B+(-B) =0 ,by axiom (1)

    Hence by changing the variables we have :

    For all A : A+(-A) = 0

    I s that proof correct??
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  2. #2
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    Re: A substitution

    The symbol "-A" is not mentioned in your axioms so you can't prove "A+ (-A)= 0"! How are you defining "-A"? Do you not mean that you want to prove that "for all A there exist B such that B+ A= 0"? In that case, yes, it follows from (4) and (1).
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    Re: A substitution

    Quote Originally Posted by HallsofIvy View Post
    The symbol "-A" is not mentioned in your axioms so you can't prove "A+ (-A)= 0"!
    Where do you base that.

    On the other hand i can claim the opposite supported by the following axiom of the predicate logic:


    \forall uP\Longrightarrow P(t/u) , where P is a formula, t is a term and u is a variable

    And in our case we have: \forall AP\Longrightarrow P(-B/A), where -B is a term and A a variable and .

    Hence (-B) + B = B+(-B)=0
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  4. #4
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    Re: A substitution

    You cannot prove anything about "-A" because you have not defined "-A". I was not clear whether "A", "B", and "C" were integers and "+" addition or if "A", "B", and "C" were "statements" and "+" is "and". In either case, you have to define "-A" explicitly.
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    Re: A substitution

    Quote Originally Posted by HallsofIvy View Post
    You cannot prove anything about "-A" because you have not defined "-A". I was not clear whether "A", "B", and "C" were integers and "+" addition or if "A", "B", and "C" were "statements" and "+" is "and". In either case, you have to define "-A" explicitly.
    A,B,C are variables.

    "+", "." are two place operation symbols

    0,1 are constants
    Last edited by psolaki; July 25th 2014 at 10:44 PM.
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