A relation partitions a set if and only if it is an equivalence relation. The partitions are known as equivalence classes. If you've worked with congruence classes (mod n), then you will have a good idea of this concept from experience.

Wikipedia states it thus: "For any equivalence relation on a set X, the set of its equivalence classes is a partition of X. Conversely, from any partition P of X, we can define an equivalence relation on X by setting x ~ y precisely when x and y are in the same part in P. Thus the notions of equivalence relation and partition are essentially equivalent." (

source)

So we know beforehand from the problem statement that we are dealing with an equivalence relation, but we could verify it if we wanted.

But anyway, I'd like to consider a larger example to better illustrate how this applies.

A = {1,2,3,4,5,6}

B = {1,2,3}

Now notice the powerset of B has 2^3 elements, and each of these sets defines an equivalence class. So for example, {1,2} is in the powerset of B. Consider

C = {1,2,5}

D = {1,2,5,6}

E = {1,2,4}

F = {1,2}

These are all equivalent under the relation R.

Does that make sense?