Given an arbitrary equivalence
relation R in an arbitrary set X, R divides X into mutually disjoint subsets Ai called
partitions or subdivisions  of X satisfying:
(i)
all elements of Ai are related to each other, for all i (ii)
no element of Ai is related to any element of Aj , i ≠ j (iii)
∪ Aj = X and Ai ∩ Aj = φ, i ≠ j The subsets Ai are called equivalence classes