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