In my practice problem I have:

$\phi=\mathbb{Z}\times \mathbb{Z} \rightarrow (\mathbb{Z}/2\mathbb{Z})\times \mathbb{Z}$ by homomorphism

$\phi(x,y)=([x+y]\_2, 2x-y)$

Find $\ker(\phi)$ and show it is isomorphic to $\mathbb{Z}$

My answer:

$\ker(\phi)=\{ (x,2x) \in \mathbb{Z}^2 | x \in 2\mathbb{Z} \}$ as this ensures $[x+y]=[0]$ and $2x=y$

Let $f=x$ for $x \in 2\mathbb{Z}$. It is injective because $f=(0)$ iff $x=0$

Is my answer correct and how do I show it is surjective? It seems strange that $\mathbb{Z} \cong 2\mathbb{Z}$ because what would the preimage of $13 \in \mathbb{Z}$ be in $2\mathbb{Z}$