My task is to show that some particular class of sets $F$ is the class of sets of the form $A$ (I'm leaving out the details of the question).
My question is as follows:
To complete this task, must I show that if a set is of form $A$ then it belongs to the class $F$, and that if a set belongs to the class $F$ then it has the form $A$?
The reason I ask is that the writer of the textbook has solutions in the back, and seems to have intended only the former (if a set is in the class $F$, it has the form $A$).
To show that two (classes of) sets are the same, you have to show that the first contains the second and vice versa. In short - yes.