Is $\mathbb{R}$ a subset of $\mathbb{R}^2$? Is it correct to say that $\mathbb{R}$ is a subset of $\mathbb{R}^2$? Or, put more generally, given $n,m\in\mathbb{N}$, $n<m$, is $\mathbb{R}^n$ a subset of $\mathbb{R}^m$?
Also, strictly related to that: what is then the "relationship" between the set $\{(x,0)\in\mathbb{R}^2,x\in\mathbb{R}\}\subset\mathbb{R}^2$ and $\mathbb{R}$? Do they coincide (I would say no)? As vector spaces, do they have the same dimension (I would say yes)?
If you could give me a reference book for this kind of stuff, I would really appreciate it.
Thank you very much in advance.
(Please correct the tags if they are not appropriate)

I wouldn't say so, even though every onedimensional subspace of $\mathbb{R}^n$ is isomorphic to $\mathbb{R}$, but there is no natural embedding. 
But a more or less funny is, that even thought nearly everyone say that $\mathbb{R}\not\subset\mathbb{R}^2$ many mathematicans say that $\mathbb{R}\subset\mathbb{C}$
even though there is a canocial transformation from $\mathbb{C}$ to $\mathbb{R}^2$.
I guess sometime one stops to distinguish often between things which are isomorphic but not the same.
