This feels like a simple question but at the moment I am a little lost with how to think.
Let $\phi : A \rightarrow A/B$ be a homomorphism of groups, is $B \subset \ker$ $\phi $ ?
I know this is true if $\phi(x) = xB$ which is the only example that our lecture notes gives. But I was curious about the general case, and how to prove it, if true.
Not necessarily. Let $A=\mathbb Z^2$ and let $B=\langle(0,1)\rangle$. Then we have a homomorphism $A\to A/B$ given by $(a,b)\mapsto (b,a)+B$. Its kernel is not $B$, but rather $\langle(1,0)\rangle$.