Equivalence Class

An equivalence class is defined as a subset of the form , where  is an element of  and the notation "" is used to mean that there is an equivalence relation between  and . It can be shown that any two equivalence classes are either equal or disjoint, hence the collection of equivalence classes forms a partition of . For all , we have  iff  and  belong to the same equivalence class.

A set of class representatives is a subset of  which contains exactly one element from each equivalence class.

For  a positive integer, and  integers, consider the congruence , then the equivalence classes are the sets ,  etc. The standard class representatives are taken to be 0, 1, 2, ..., .

REFERENCES:

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 56-57, 1993.