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, ..., 
.
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