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# 数学代写|现代代数代考Modern Algebra代写|Congruence Classes

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## 数学代写|现代代数代考Modern Algebra代写|Congruence Classes

In connection with the relation of congruence modulo $n$, we have observed that there are $n$ distinct congruence classes. Let $\mathbf{Z}_n$ denote this set of classes:
$$\mathbf{Z}_n={[0],[1],[2], \ldots,[n-1]} .$$
When addition and multiplication are defined in a natural and appropriate manner in $\mathbf{Z}_n$, these sets provide useful examples for our work in later chapters.
Addition in $\mathbf{Z}_n$
Consider the rule given by
$$[a]+[b]=[a+b]$$
a. This rule defines an addition that is a binary operation on $\mathbf{Z}_n$.
b. Addition is associative in $\mathbf{Z}_n$ :
$$[a]+([b]+[c])=([a]+[b])+[c] .$$
c. Addition is commutative in $\mathbf{Z}_n$ :
$$[a]+[b]=[b]+[a] .$$
d. $\mathbf{Z}_n$ has the additive identity $[0]$.
e. Each $[a]$ in $\mathbf{Z}_n$ has $[-a]$ as its additive inverse in $\mathbf{Z}_n$.
Proof
a. It is clear that the rule $[a]+[b]=[a+b]$ yields an element of $\mathbf{Z}_n$, but the uniqueness of this result needs to be verified. In other words, closure is obvious, but we need to show that the operation is well-defined. To do this, suppose that $[a]=[x]$ and $[b]=[y]$. Then
$$[a]=[x] \Rightarrow a \equiv x(\bmod n)$$
and
$$[b]=[y] \Rightarrow b \equiv y(\bmod n)$$
By Theorem 2.24,
$$a+b \equiv x+y(\bmod n),$$
and therefore $[a+b]=[x+y]$.
b. The associative property follows from
\begin{aligned} {[a]+([b]+[c]) } & =[a]+[b+c] \ & =[a+(b+c)] \ & =[(a+b)+c] \ & =[a+b]+[c] \ & =([a]+[b])+[c] . \end{aligned}
Note that the key step here is the fact that addition is associative in $\mathbf{Z}$ :
$$a+(b+c)=(a+b)+c .$$
c. The commutative property follows from
\begin{aligned} {[a]+[b] } & =[a+b] \ & =[b+a] \ & =[b]+[a] . \end{aligned}
d. $[0]$ is the additive identity, since addition is commutative in $\mathbf{Z}_n$ and
$$[a]+[0]=[a+0]=[a] .$$
e. $[-a]=[n-a]$ is the additive inverse of $[a]$, since addition is commutative in $\mathbf{Z}_n$ and
$$[-a]+[a]=[-a+a]=[0]$$

## 数学代写|现代代数代考Modern Algebra代写|Introduction to Coding Theory (Optional)

In this section, we present some applications of congruence modulo $n$ found in basic coding theory. When information is transmitted from one satellite to another or stored and retrieved in a computer or on a compact disc, the information is usually expressed in some sort of code. The ASCII code (American Standard Code for Information Interchange) of 256 characters used in computers is one example. However, errors can occur during the transmission or retrieval processes. The detection and correction of such errors are the fundamental goals of coding theory.

In binary coding theory, we omit the brackets on the elements in $\mathbf{Z}_2$ and call ${0,1}$ the binary alphabet. $\mathrm{A} \mathrm{bit}^{\dagger}$ is an element of the binary alphabet. A word (or block) is a sequence of bits, where all words in a message have the same length; that is, they contain the same number of bits. Thus a 2-bit word is an element of $\mathbf{Z}_2 \times \mathbf{Z}_2$. For notational convenience, we omit the comma and parentheses in the 2-bit word $(a, b)$ and write $a b$, where $a \in{0,1}$ and $b \in{0,1}$. Thus
$\begin{array}{llll}000 & 010 & 001 & 011 \ 100 & 110 & 101 & 111\end{array}$
are all eight possible 3-bit words using the binary alphabet. There are thirty-two 5-bit words, so 5-bit words are frequently used to represent the 26 letters of our alphabet, along with 6 punctuation marks.

During the process of sending a message using $k$-bit words, one or more bits may be received incorrectly. It is essential that errors be detected and, if possible, corrected. The general idea is to generate a code, send the coded message, and then decode the coded message, as illustrated here:
$$\text { message } \stackrel{\text { encode }}{\longrightarrow} \text { coded message } \stackrel{\text { send }}{\longrightarrow} \text { received message } \stackrel{\text { decode }}{\longrightarrow} \text { message. }$$
Ideally, the code is devised in such a way as to detect and/or correct any errors in the received message. Most codes require appending extra bits to each $k$-bit word, forming an $n$-bit code word. The next example illustrates an error-detecting scheme.

# 现代代数代写

## 数学代写|现代代数代考Modern Algebra代写|Congruence Classes

$$\mathbf{Z}_n={[0],[1],[2], \ldots,[n-1]} .$$

$$[a]+[b]=[a+b]$$
a.该规则定义了一个对$\mathbf{Z}_n$的二进制加法操作。
b.在$\mathbf{Z}_n$中，加法是关联的:
$$[a]+([b]+[c])=([a]+[b])+[c] .$$
c.在$\mathbf{Z}_n$中加法是可交换的:
$$[a]+[b]=[b]+[a] .$$
D. $\mathbf{Z}_n$具有加性特性$[0]$。
e. $\mathbf{Z}_n$中的每个$[a]$在$\mathbf{Z}_n$中都有$[-a]$作为其相加逆。

a.很明显，规则$[a]+[b]=[a+b]$产生了一个元素$\mathbf{Z}_n$，但是这个结果的唯一性需要被验证。换句话说，闭包是显而易见的，但我们需要表明该操作是定义良好的。为此，假设$[a]=[x]$和$[b]=[y]$。然后
$$[a]=[x] \Rightarrow a \equiv x(\bmod n)$$

$$[b]=[y] \Rightarrow b \equiv y(\bmod n)$$

$$a+b \equiv x+y(\bmod n),$$

b.结合律由
\begin{aligned} {[a]+([b]+[c]) } & =[a]+[b+c] \ & =[a+(b+c)] \ & =[(a+b)+c] \ & =[a+b]+[c] \ & =([a]+[b])+[c] . \end{aligned}

$$a+(b+c)=(a+b)+c .$$
c.可交换性由
\begin{aligned} {[a]+[b] } & =[a+b] \ & =[b+a] \ & =[b]+[a] . \end{aligned}
D. $[0]$是可加性恒等式，因为在$\mathbf{Z}_n$和
$$[a]+[0]=[a+0]=[a] .$$
E. $[-a]=[n-a]$是$[a]$的可加逆，因为在$\mathbf{Z}_n$和中加法是可交换的
$$[-a]+[a]=[-a+a]=[0]$$

## 数学代写|现代代数代考Modern Algebra代写|Introduction to Coding Theory (Optional)

$\begin{array}{llll}000 & 010 & 001 & 011 \ 100 & 110 & 101 & 111\end{array}$

$$\text { message } \stackrel{\text { encode }}{\longrightarrow} \text { coded message } \stackrel{\text { send }}{\longrightarrow} \text { received message } \stackrel{\text { decode }}{\longrightarrow} \text { message. }$$

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