"So, once you have found all the cosets whos union is the orginal group, you are done. Correct?" Well, yes...obviously!
The question here is to understand that a subgroup of a group defines an equivalence relation on the group
whose equivalence classes are precisely the left (or right) cosets of the group wrt that sbgp.
Not only that: if , so you could check easily whether
two of the sets you were given are or not different cosets.
For example, . But the cosets