Home computickets What is the set of residue classes modulo n?

# What is the set of residue classes modulo n?

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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

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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?

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.

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