What is the comparable number modulo 2 I know it when a ≡ b (mod n)
if n divides (a – b)
What is the set of residue classes modulo n?
with the sources I realized that oboznachaetsya like Zn, where n – this is the module
.
Example Zn = {[0], [1], [2]}, each equivalence class єlement nazyvaetsja?
And what is the residue class?
it is when in every class єkvivalentnosti there are many in which every pair of elements 2 are congruent modulo?
example:
[0] = […. 6, 3.0, 3.6 …..] then modulo n = 3.
And that topic comparability means the symbol “⊕” ???. an operation with the equivalence classes?
Answer 1
A lot of residue classes modulo m
– it is a set of numbers 0
, 1
, …, m1
. Designated
On the set Z / mZ
defined operations of addition, subtraction, multiplication and division. I will outline them [+]
, []
, [*]
and [/]
, to distinguish it from obchno operations of addition and multiplication of integers:

a [+] b = (a + b)% m
– remainder of division of the normal amounta
andb
form
, 
a [*] b = (a * b)% m
– remainder of division of the usual producta
andb
form
, 
[] a = m  a
, and[] 0 = 0
. Accordinglya [] b = a [+] ([] b)
The division – the trickiest operation in a number of deductions. a [/] b
– it is the number of c
, that b [*] c = a
. So, unlike integers, which are divided into each other rarely, residue classes fall into each other almost always. For this it is necessary and sufficient a
and b
have no common divisors with m
. If m
– prime, then all nonzero pair a
and b
can be divided. For dividing Z / mZ
used extended Euclidean algorithm .
As it is connected with the equivalence classes. Take your example Z3 = {[0], [1], [2]}
The equivalence class [0]
indicates all those integers which when divided by 3, a remainder of 0 will: [0] == {0, ± 3, ± 6, ± 9, ...}
The equivalence class [1]
indicates all those integers which when divided by 3 give residue 1: [1] == {1, 2, 4, 5, 7 , 8, 10, ...}
The equivalence class [2]
indicates all those integers which when divided by 3 give residue 2: [2] == {2, 1, 5, 4, 8 , 7, 11, ...}
Operations with the equivalence classes are defined as follows (for definiteness, we consider the sum): Let’s get out of each class of one representative, add them, and as a result take the equivalence class of the resulting amount. For example [1] [+] [2]
– take from [1]
number 1, and of [2]
vozomem 2. 1 + 2 = 3
. Equivalence class to which it belongs 3 – is [0]
. Consequently, [1] [+] [2] [0]
It is easy to prove that these operations are reduced to the modulo operations, as I wrote above, except for the division. Therefore, in real life, no one fools his head ekvivaletnosti classes, but just consider the number of modulo.