Is (A²)×B the same as A×A×B?

[C0nfused]

Hi everybody,
When we have two sets A and B , we define the cartesian product of A and B as the set A*B={(x,y): (x element of A) and (y element of B)}. We also define A*A*...*A (n factors)=A^n. So when we write (A^2)*B, this is the same as A*A*B? I mean, for example (R^2)*R is the same as R*R*R, or ((1,2),3)=(1,2,3) ?
Thanks

 

[Palindrom]

They're isomorphic as vector spaces, i.e. they are the same. (unless you want to get hyper formal, in which case I think you need to stick with just isomorphic).

 

[Palindrom]

Hum, I answered for R as a vector space, sorry. Given any 2 sets, then you have a 1-1 and onto mapping between (A^2)xB and AxAxB.

 

[C0nfused]

Thanks for your answer. Actually i am asking this question because I read somewhere that the graph of a function f (if it's called this way) is the set G={(x,f(x)):x element of f's domain}. So if f: (R^2) -> R then the vectors (x,y,f(x,y) that are points of f's 3D representation must be the same as ((x,y),f(x,y)). ( I hope u understood what i am asking).

Thanks again

 

[Palindrom]

Yes, I think I do.
And yes, they are the same. Eventually, it's just 2 different ways of looking at it- just like thinking of f(x,y) as a function of 2 scalar variables or of 1 vector variable.
I hope you understand what I'm trying to say.:)

 

[matt grime]

To put in "proper" terms (ie ones that might help you search for other things on it) you're getting towards the idea that the cartesian product is associative: that is there are natural isomorphisms from (AxB)xC to Ax(BxC)