Equivalence Class
المؤلف:
Shanks, D
المصدر:
Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea
الجزء والصفحة:
pp. 56-57
9-1-2022
1443
Equivalence Class
An equivalence class is defined as a subset of the form ![<span style=]()
{x in X:xRa}" src="https://mathworld.wolfram.com/images/equations/EquivalenceClass/Inline1.gif" style="height:16px; width:80px" />, 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 ![<span style=]()
{...,-2n,-n,0,n,2n,...}" src="https://mathworld.wolfram.com/images/equations/EquivalenceClass/Inline16.gif" style="height:16px; width:155px" />, ![<span style=]()
{...,1-2n,1-n,1,1+n,1+2n,...}" src="https://mathworld.wolfram.com/images/equations/EquivalenceClass/Inline17.gif" style="height:16px; width:221px" /> 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.
الاكثر قراءة في نظرية المجموعات
اخر الاخبار
اخبار العتبة العباسية المقدسة
