A theorem on equivalence relation states that for any set S, the set of equivalence classes of S under an equivalence relation R constitutes a partition of a set. Moreover, given any partition of a set, one can define an equivalence relation on the set. What allows you to "create" a partition of a set, say the set of its equivalence classes? If the set is finite, then it is intuitively easy to see that a partition can be created since the elements of the set must eventually be exhausted. But if the set is infinite, say Z, then what guarantees we can create a partition, of say, cosets of some subgroup of Z.