ie Id like to show a(bc)=(ab)c but for any n operands eg abcdefg=gfdcabe etc
I can see this is very intuitive that this should be true for all n operands, but as a logical exercise I would like to see how this is formally shown.. Im interested in the reasoning.. and how to arrive at such things.
It seems there are infinite possible operands, so do we need to work backwards and use the inverse (division)?
thanks.
Associativity of a binary operation means that it naturally extends to $n$-ary operations for all $n=3,4,5,6,..$, uniquely in the sense that reordering brackets (i.e. the order of evaluation of the partial products) doesn't change the result (and it doesn't even change the order of members). So that we can uniquely define terms like $abcdefg$, which is, among many other possible brackettings, $=(abc(de)f)g$.
To prove this, you can use induction on $n$.
On the other hand, commutativity means that $ab=ba$ for all pairs of elements. In the example 'to show $abcdefg=gfdcabe$' you implicitly used that the operation is associative (because of the lack of any brackets, though originally product is said to be binary operation), and over that, the commutativity condition is needed to prove this.