A simple question about functions of sets

Question

Given $A=\{a,b,c\}, \; B=\{1,2,3\}$ and $f:A \to B \;$ given by $f(a)=1, \; f(b)=2$, does $f(A)=f(\{a,b\})=\{1,2\}$?

Answer 1

No, as $f(c)$ is unspecified, you can just say that $$\{1,2\} \subseteq f[A]$$

and no more on the current data. Of course, $f[\{a,b\}]=\{1,2\}$ is true, we don't need $f(c)$ to compute it.