Find $Ker(\phi)$ and $\phi(-3,2)$ for the given homomorphism $\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ where $\phi(1,0) = 3$ and $\phi(0,-1) = -5$
I have no idea why how they came up with the mapping $\phi(m,n)$ themselves nor do I fully understand why the kernel is correct. My thinking for this problem was
$Ker(\phi) = \{(m,n) \in \mathbb{Z} \times \mathbb{Z}: \phi(m,n)=0\}$ so the problem now lies in finding $m$ and $n$.
What is the thinking behind the solution? I really don't understand how they come up with the function themselves
EDIT: okay I understand they are basically setting the function to 0 and that's how they came up with that kernel, but still no explanation on how they came up with the function