The point of the exercise seems to be to expose and exploit the following properties of bijections:
The identity map is a bijection.
The inverse of a bijection is a bijection.
The composition of two bijections is a bijection.
These properties are easy to prove and correspond to reflexivity, symmetry, and transitivity.
However, there is an important technical detail:
an equivalence relation is defined on a set. You cannot use the set of all sets because that is not a set. You need to fix a universe.
Bottom line, the relation given is an equivalence relation on (any subset of) the set of all subsets of a fixed set $U$.