R in a set X. Given an arbitrary equivalence relation R in an arbitrary set X, R divides X into mutually disjoint subsets A i called partitions or subdivisions of X satisfying: (i) all elements of A i are related to each other, for all i . (ii) no element of A i is related to any element of A j , i ≠ j . (iii) ∪ A j = X and A i ∩ A j = φ , i ≠ j .
The subsets A i are called equivalence classes . The interesting part of the situation is that we can go reverse also. For example, consider a subdivision of the set Z given by three mutually disjoint subsets A , A and A whose union is Z with A = { x ∈ Z : x is a multiple of } = {..., – , – , , , , ...} A = { x ∈ Z : x – is a multiple of } = {..., – , – , , , , ...} A = { x ∈ Z : x – is a multiple of } = {..., – , – , , , , ...} Define a relation R in Z given by R = {( a , b ) : divides a – b }. Following the arguments similar to those used in Example , we can show that R is an equivalence relation.
Also, A coincides with the set of all integers in Z which are related to zero, A coincides with the set of all integers which are related to and A coincides with the set of all integers in Z which are related to . Thus, A = [ ], A = [ ] and A = [ ]. In fact, A = [ r ], A = [ r + ] and A = [ r + ], for all r ∈ Z . Example Let R be the relation