is a subset of if every element of is an element of .
is a subgroup of if is a subset of and is a group.
The subgroup generated by is the smallest subgroup of containing . Informally, you need to throw as little "stuff" as possible into until you get a group. More formally, take the intersection of all subgroups of containing . Since arbitrary intersections of a group form a group, you get the smallest subgroup of that contains .