同余关系 等价关系 同余关系的原型
小结:
https://baike.baidu.com/item/同余关系
https://en.wikipedia.org/wiki/Congruence_relation
https://en.wikipedia.org/wiki/Equivalence_relation
https://baike.baidu.com/item/等价关系
等价关系定义为:设R是非空集合A上的二元关系,若R是自反的、对称的、传递的,则称R是A上的等价关系。研究等价关系的目的在于将集合中的元素进行分类,选取每类的代表元素来降低问题的复杂度,如软件测试时,可利用等价类来选择测试用例。
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.