While I was reading about Isomorphism and sets, I came across with the following question:
Find a subset of $\mathbb{R}$ that is Isomorphism to $\mathbb{N} \oplus \mathbb{N}$.
Firstly I tried to think of a subset like this, but my creativity has failed me. Later I tried to find some information about it through the Internet, but hasn't yet found. How hard is it really to think of a simple solution?
Find a subset of $\mathbb{R}$ that is
Isomorphismto $\mathbb{N} \oplus \mathbb{N}$.This should say
Find a subset of $\mathbb{R}$ that is isomorphic to $\mathbb{N} \oplus \mathbb{N}$.
The set $\{ a+ b\sqrt 2 : a,b\in \mathbb N \}$ will serve, because the correspondence between $a+b\sqrt 2$ and $(a,b)$ is an isomorphism.
However $\{a+br : a,b\in\mathbb N\}$ cannot serve if $r$ is rational. For example, suppose $r=22/7.$ Then $0 + 7r = 22 + 0r,$ so only one number, $22,$ would correspond to two different pairts of members of $\mathbb N,$ namely $(0,7)$ and $(22,0).$