Conjunction of Sets: x,y in Rm+n vs Vector z in R(m+n)

[krcmd1]

If x $$\epsilon$$ A $$\subset$$ R$$\overline{}n$$ and y $$\epsilon$$ B $$\subset$$ R$$\overline{}m$$,

then A X B $$\subset$$ R$$\overline{}m+n$$.

Is there any difference between the connotation of (x,y) $$\epsilon$$ R$$\overline{}m+n$$ and a vector z in R$$\overline{}(m+n)$$?

thanks!

(can't get the (m+n) above the line yet)

 

[mrandersdk]

the short answer is no. But this is because when the notation is like that, it is clearly isomorphic. That is to every z there is a unique (x,y) and the other way around.

Don't know why you distinguish between $R^{m+n}$ and $R^{(m+n)}$, have never seen that notation.