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# 数学代写代考| Abstract Algebra作为 离散数学 的基础

## Abstract Algebra作为 离散数学 的基础

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## 离散数学代写

$6.6$ Abstract Algebra|
One of the important features of modern mathematics is the power of the abstract approach. This has opened up whole new areas of mathematics, and it has led to a large body of new results and problems. The term ‘ $a b s t r a c t$ ‘ is subjective, as what is abstract to one person may be quite concrete to another. We shall introduce some important algebraic structures in this section including monoids, groups, rings, fields and vector spaces.
$6.6$ Abstract Algebra
113
6.6.1 Monoids and Groups
A non-empty set $M$ together with a binary operation ‘ ${ }^{\prime}$ ‘ is called a monoid if for all elements $a, b, c \in M$ the following properties hold:
\begin{tabular}{l|l}
(1) $a * b \in M$ & (Closure property) \
\hline (2) $a *(b * c)=(a * b) * c$ & (Associative property) \
\hline (3) $\exists u \in M$ such that $a * u=u * a=a(\forall a \in M) .$ & (Identity Element)
\end{tabular}
A monoid is commutative if $a * b=b * a$ for all $a, b \in M .$ A $\operatorname{semin}$ group $(M, )$ is a set with a binary operation ” such that the closure and associativity properties hold (but it may not have an identity element).
Example 6.1 (Monoids)
(i) The set of sequences $\Sigma *$ under concatenation with the empty sequence $\Lambda$ is the identity element.
(ii) The set of integers under addition forms an infinite monoid in which 0 is the identity element.

A non-empty set $G$ together with a binary operation ‘*’ is called a group if for all elements $a, b, c \in G$, the following properties hold:
\begin{tabular}{l|l}
\hline (i) $a * b \in G$ & (Closure property) \
\hline (ii) $a *(b * c)=(a * b) * c$ & (Associative property) \
\hline (iii) $\exists e \in G$ such that $a * e=e * a=a(\forall a \in G)$ & (Identity Element) \
\hline (iv) For every $a \in G, \exists a^{-1} \in G$, such that $a * a^{-1}=a^{-1} * a=e .$ & (Inverse Element)
\end{tabular}
The identity element is unique, and the inverse $a^{-1}$ of an element $a$ is unique (see exercise 5). A commutative group has the additional property that $a * b=b * a$ for all $a, b \in G$. The order of a finite group $G$ is the number of elements in $G$, and is denoted by $o(G)$.
Example 6.2 (Groups)
(i) The set of integers under addition $(\mathbb{Z},+)$ forms an infinite group in which 0 is the identity element.
(ii) The set of $2 \times 2$ integer matrices under addition, where the identity element is $\left(\begin{array}{ll}0 & 0 \ 0 & 0\end{array}\right)$.

##### 抽象代数作为离散数学的基础

\开始{表格}{l|l}
(1) $a * b \in M$ & (闭包属性) \
\hline (2) $a *(b * c)=(a * b) * c$ & (关联属性) \
\hline (3) $\exists u \in M$ 使得 $a * u=u * a=a(\forall a \in M) .$ & (Identity Element)
\end{表格}

(i) 与空序列$\Lambda$ 串联的序列集$\Sigma *$ 是恒等元素。
(ii) 加法下的整数集合形成一个无限幺半群，其中 0 是单位元素。

(i) 加法 $(\mathbb{Z},+)$ 下的整数集合形成一个无限群，其中 0 是单位元。
(ii) 加法下的 $2 \times 2$ 整数矩阵的集合，其中单位元为 $\left(\begin{array}{ll}0 & 0 \ 0 & 0\end{array}\right)$ .

## 图论代考

(a) 假设一个操作有 $m$ 个可能的结果，而第二个操作有 $n$ 个可能的结果，那么执行第一个操作后执行第二个操作时可能结果的总数是 $m \times n$ (Product Rule )。
(b) 假设一个操作有 $m$ 个可能的结果，而第二个操作有 $n$ 个可能的结果，那么第一个操作或第二个操作的可能结果总数由 $m+n$ 给出（求和规则） .

$(1, \mathrm{H}),(2, \mathrm{H}),(3, \mathrm{H}),(4, \mathrm{H}),(5, \mathrm{H}) ,(6, \mathrm{H}),(1, \mathrm{~T}),(2, \mathrm{~T}),(3, \mathrm{~T}),(4, \mathrm{ ~T}),(5, \mathrm{~T}),(6, \mathrm{~T})$

5.7 排列组合
97

(a) 假设有一组 367 人，那么必须至少有两个人的生日相同。

(b) 假设有 102 名学生参加了一次考试（考试的结果是 0 到 100 之间的分数）。然后，至少有两名学生获得相同的分数。

## 密码学代考

• Cryptosystem
• A system that describes how to encrypt or decrypt messages
• Plaintext
• Message in its original form
• Ciphertext
• Message in its encrypted form
• Cryptographer
• Invents encryption algorithms
• Cryptanalyst
• Breaks encryption algorithms or implementations

## 编码理论代写

1. 数据压缩（或信源编码
2. 前向错误更正（或信道编码
3. 加密编码
4. 线路码