A set $A$ is equipotent to a set $B$ $(A\sim B)$, if a bijection $f: A \rightarrow B$ exists.
How to prove, that $\sim$ is a equivalence relation?
EDIT: I understand the concept of reflexivity, symmetry and transitivity. I just don't know how to write it formally as a proof.
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 $U$.