小結:
https://baike.baidu.com/item/同余關系
https://en.wikipedia.org/wiki/Congruence_relation
https://en.wikipedia.org/wiki/Equivalence_relation
https://baike.baidu.com/item/等價關系
等價關系定義為:設R是非空集合A上的二元關系,若R是自反的、對稱的、傳遞的,則稱R是A上的等價關系。研究等價關系的目的在於將集合中的元素進行分類,選取每類的代表元素來降低問題的復雜度,如軟件測試時,可利用等價類來選擇測試用例。
The prototypical example of a congruence relation is congruence modulo on the set of integers. For a given positive integer
, two integers
and
are called congruent modulo
, written
if is divisible by
(or equivalently if
and
have the same remainder when divided by
).
for example, and
are congruent modulo
,
since is a multiple of 10, or equivalently since both
and
have a remainder of
when divided by
.
Congruence modulo (for a fixed
) is compatible with both addition and multiplication on the integers. That is,
if
-
and
then
-
and
The corresponding addition and multiplication of equivalence classes is known as modular arithmetic. From the point of view of abstract algebra, congruence modulo is a congruence relation on the ringof integers, and arithmetic modulo
occurs on the corresponding quotient ring.