What $\mathbb Z/\ker f$ mean if $f$ is injective?

Question

Here, I am trying to really understand the concept of cosets. From my understanding, given groups A and B, then A/B would mean the set of all the cosets of B in A. In that case, if I have $\mathbb Z/\ker f$, and $f$ is injective, does that mean I have infinitely many cosets with only one element, or the entire $\mathbb Z$? Please help me understand this.

Thanks.

Answer 1

$f$ is injective $\Rightarrow$ $Ker(f)=0$, and $a+\{0\}=a$ since $a+B:=\{a+b\,|\, b\in B\}$.