What does the quotient group $(A+B)/B$ actually mean? I understand that $A+B$ is the set containing all elements of the form $a+b$, wit $a\in A, b\in B$. 
When you do the quotient group, that's like forming equivalence classes modulo $B$. All elements in $B$ should be congruent to 0 modulo $B$, as far as I understand. So wouldn't $(A+B)/B$ be the same as $A/B$?
Where is the error in my reasoning?

The problem here is, since $B \not\subseteq A$,  how to even make sense of $A/B$.
There are two reasonable ways:


*

*We can still make sense of the congruence relation $x \sim y$ when $x - y \in B$, and take the quotient of $A$ by this equivalence relation. This is the same thing as $A / (A \cap B)$.

*Since everything in $B$ should vanish anyways, adding them in shouldn't affect the result but gives a group that does contain $B$. This is the calculation $(A + B) / B$.


By the second isomorphism theorem, both approaches give isomorphic results.
