TOPICS
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, ..., 
.
See also
Congruence, Coset, Equivalence RelationExplore with Wolfram|Alpha
References
Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 56-57, 1993.Referenced on Wolfram|Alpha
Equivalence ClassCite this as:
Weisstein, Eric W. "Equivalence Class." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EquivalenceClass.html