triclino Posted July 24, 2009 Share Posted July 24, 2009 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 ?? Link to comment Share on other sites More sharing options...
ydoaPs Posted July 24, 2009 Share Posted July 24, 2009 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. Link to comment Share on other sites More sharing options...
timo Posted July 24, 2009 Share Posted July 24, 2009 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. Link to comment Share on other sites More sharing options...
triclino Posted July 27, 2009 Author Share Posted July 27, 2009 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?? Link to comment Share on other sites More sharing options...
timo Posted July 27, 2009 Share Posted July 27, 2009 (edited) 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, 2009 by timo ... edit, edit, edit .... Link to comment Share on other sites More sharing options...
ajb Posted July 27, 2009 Share Posted July 27, 2009 (edited) 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? Merged post follows: 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, 2009 by ajb slightly more detail added Link to comment Share on other sites More sharing options...
ydoaPs Posted July 29, 2009 Share Posted July 29, 2009 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. Link to comment Share on other sites More sharing options...
ajb Posted August 1, 2009 Share Posted August 1, 2009 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.) Link to comment Share on other sites More sharing options...
the tree Posted August 1, 2009 Share Posted August 1, 2009 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. Link to comment Share on other sites More sharing options...
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