Let $X$ and $Y$ denote magmas, and suppose $f : X \rightarrow Y$ is homomorphism. Then I think that for all $A,B \subseteq X$, we have $f(A+B)=f(A)+f(B).$ However, I'm not happy with my:
Proof. The following are equivalent.
$y \in f(A+B)$
$y = f(a+b)$ for some $a \in A$ and $b \in B$
$y = f(a) + f(b)$ for some $a \in A$ and $b \in B$.
$y \in f(A)+f(B)$.
Its the final step that's bothering me. How do we know that $3 \leftrightarrow 4$?
Methinks you most likely failed to break it down into enough little pieces to wrap your mind around.
Suppose that there exist $a \in A$ and $b \in B$ such that $y = f(a) + f(b)$. Then $f(a) \in f[A]$ and $f(b) \in f[B]$, so $y\in f[A]+f[B]$.
Suppose that $y \in f[A]+f[B]$. Then there exist $p\in f[A]$ and $q\in f[B]$ such that $y = p + q$. By the definitions of $f[A]$ and $f[B]$, there exist $a\in A$ and $b\in B$ such that $f(a)=p$ and $f(b)=q$. But then $y = f(a) + f(b)$ as desired.