1. ## 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??

2. ## 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).

3. ## Re: A substitution

Originally Posted by HallsofIvy
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:

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

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

Hence (-B) + B = B+(-B)=0

4. ## 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.

5. ## Re: A substitution

Originally Posted by HallsofIvy
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