In Serge Lang's Algebra, Chapter 1, exercise 54, the problem statement begins with:
Let $G$ be a group and $\{G_{i}\}_{i\in I}$ be a family of subgroups generating $G$.
Does this mean that the union of all the $G_{i}$'s generates $G$? or that each $G_{i}$ on its own generates $G$?
Edit: Please correct me if this is wrong: (yes it is wrong)
So given that this means the union of the $G_{i}$'s generates $G$, I must in fact have that the union of the $G_{i}$'s is equal to $G$, since the union of subgroups is a subgroup.