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 , …, m-1 . 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 amount a and b for m ,

• a [*] b = (a * b)% m – remainder of division of the usual product a and b for m ,

• [-] a = m - a , and [-] 0 = 0 . Accordingly a [-] 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.