What theorem or axiom allow us to do the substitutions :

 

x= a + b ,or y= a + b + c, or x= 2y + z^2 e.t.c , e.t.c

 

some times in a mathematical proof ??

What theorem or axiom allow us to do the substitutions :

 

x= a + b ,or y= a + b + c, or x= 2y + z^2 e.t.c , e.t.c

 

some times in a mathematical proof ??

 

iirc, it's the symmetric property.

 

If a=b, then b=a.

Closure of a group under the operation(s)? E.g. if a is a real and b is a real then a+b is just another real which you can call it y if you like.

Thanks .

 

But in a system where the primitive symbols are: +,=, and

 

the constants are : 0 ,and

 

the only axioms are :

 

1) a+ (b+c) = (a+b) + c ,for all a,b,c

 

2) a+b = b +a ,for all a,b

 

3) a +0 =a ,for all a

 

the substitution : x= a+b+c ,would be allowed??

I'd say if there is no guarantee that a+b exists (are you sure that it is not given in your system?) then it makes no sense to substitute it with y or at least not to substitute it with y and assume that y exists (in the system). I don't know what system you are talking about but note that e.g. mathematical groups do have closure under the operation as an axiom .

Edited July 27, 200916 yr by timo
... edit, edit, edit ....

The structure looks like a commutative monoid, that is a semigroup with an identity.

 

The "+" looks like an associative binary operation. However, we do not know that the set (I assume we have an underlying set) is closed under "+". You could have a partial function and not a binary opertaion.

 

 

If your structure is a monoid then x = a+ b + c is fine, with x in your monoid.

 

If your "+" is just a partial function then you have a category. If it is just a category, then you have to specify if a+b (etc.) exists.

 

It is a small category if we have an underlying set.

 

So the question is

 

"2) a+b = b +a ,for all a,b"

 

is that for all a,b or just for a, b if a+b is allowed?

 

 

 

 

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Consecutive posts merged

I don't know what system you are talking about but note that e.g. mathematical groups do have closure under the operation as an axiom .

 

As an aside, it is possible to have structures that do not allow the product (composition) of all elements, that is we have a partial function and not a binary operation. The "group like" structure that comes from relaxing the binary operation of groups is called a groupoid. Groupoids are very important in geometry and physics.

Edited July 27, 200916 yr by ajb
slightly more detail added

Thanks .

 

But in a system where the primitive symbols are: +,=, and

 

the constants are : 0 ,and

 

the only axioms are :

 

1) a+ (b+c) = (a+b) + c ,for all a,b,c

 

2) a+b = b +a ,for all a,b

 

3) a +0 =a ,for all a

 

the substitution : x= a+b+c ,would be allowed??

 

Surely if a=b and b=c, then a=c by the transitive property.

Surely if a=b and b=c, then a=c by the transitive property.

 

That is absolutely true. More formally, equals is an equivalence relation.

 

However, the question is one of composing all elements or just some.

 

[math]x = a+ b+ c[/math]

 

may not exist. So, I think the question is if the formal system is a monoid or a just a category? (Monoids are themselves categories, but I won't worry about that possible confusion for now.)

I'd say that for any Magma with + as an operation, if it contains a,b and c then a+b+c should be contained.

 

(a+b) is contained under closure, and (a+b)+c is contained under closure, which generally means a+b+c.