Because of the properties of an equivalence relation, naming all the equivalence classes is equivalent to naming the relation. For example, say
for some equivalence class of the relation
Then the equivalence relation on any set is merely being in the same equivalence class. The two notions of an equivalence relation, listing all the pairs and listing the equivalence class, are equivalent.
Proof: (i)Since a is in its own equivalence class, (a,a) would be in the relation, which shows that being in an equivalence class is reflexive.
(ii)if a and b are in the same equivalence class, both of the pairs (a,b) and (b,a) are in the relation, so its symmetric
(iii) if all of a, b, and c are in the same class then (a,b),(b,c) and (a,c) are all in the same equivalence class, so transitivity is also shown.
And if any two relations yield the same equivalence classes, then they're the same relation, so listing the classes is just as good (and much more efficient) than listing all the pairs of the relation.