Firstly, I want to show $(\Bbb R , +)$ is isomorphic to $\Bbb R^2$ (under component wise addition).
In this I used the mapping $\phi_1$ : $\Bbb R^2 \rightarrow \Bbb R$
Where, $\phi_1(a,b) = 2^a3^b$
And then , show $\Bbb R^2$ is isomorphic to $(\Bbb C , +)$.
In this I used the mapping
$\phi_2$ : $\Bbb R^2 \rightarrow \Bbb C$
Where, $\phi_2(a,b) = a+ib$
And then, use the transitive property of isomorphic groups to show $(\Bbb R , +)$ is isomorphic to $(\Bbb C , +)$
Now ,my doubts are that, whether the mappings i used are correct or not. Does this mappings makes the two groups isomorphic?
Please, correct me wherever I am wrong?
Edit: I see that $\phi_1$ is not bijective. So , their is no way to prove ,$\phi_1$ : $\Bbb R^2 \rightarrow \Bbb R$ is an isomorphism by just defining a mapping or without using the concept of metric space!