Equivalence Class

An equivalence class is defined as a Subset of the form $\{x \in X: xRa\}$, where $a$ is an element of $X$ and the Notation ``$xRy$'' is used to mean that there is an Equivalence Relation between $x$ and $y$. It can be shown that any two equivalence classes are either equal or disjoint, hence the collection of equivalence classes forms a partition of $X$. For all $a,b \in X$, we have $aRb$ Iff $a$ and $b$ belong to the same equivalence class.

A set of Class Representatives is a Subset of $X$ which contains Exactly One element from each equivalence class.

For $n$ a Positive Integer, and $a,b$ Integers, consider the Congruence $a\equiv b\ \left({{\rm mod\ } {n}}\right)$, then the equivalence classes are the sets $\{\ldots, -2n, -n, 0, n, 2n, \ldots\}$, $\{\ldots, 1-2n, 1-n, 1, 1+n, 1+2n, \ldots\}$ etc. The standard Class Representatives are taken to be 0, 1, 2, ..., $n-1$.

See also Congruence, Coset

References

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