An easy question about cartesian product

1. Apr 12, 2005

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

2. Apr 12, 2005

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).

3. Apr 12, 2005

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.

4. Apr 13, 2005

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

5. Apr 13, 2005

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.:)

6. Apr 13, 2005

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)