Example

Let G be the cyclic (additive) group Z/nZ of order n = pq, and H the subgroup generated by the residue class q Z/nZ.

Then H has order p, and r ~ s if and only if q divides r - s.

In particular, the equivalence class of r consists of all residue classes in Z/nZ. of s Z such that s = r (mod q).

Let's take the specific values

n = 15, p = 5, q = 3.

Then H = {0, 3, 6, 9, 12} and the equivalence classes are:

H = {0, 3, 6, 9, 12},
1+H = {1, 4, 7, 10, 13},
2+H = {2, 5, 8, 11, 14}.