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# 数学代写|代数拓扑代考Algebraic Topology代考|Set Theory

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## 数学代写|代数拓扑代考Algebraic Topology代考|Set Theory

This section conveys some basic concepts of set theory (naive) initiated around 1870 by the German mathematician Georg Cantor (1845-1918) which are used throughout the book. Set theory occupies an important position in mathematics. Many concrete concepts and examples are based on it. It is assumed that the readers are familiar with the sets
$\mathbf{N}$ (set of natural numbers/positive integers)
$\mathbf{Z}$ (set of integers)
$\mathbf{Q}$ (set of rational numbers)
$\mathbf{R}$ (set of real numbers)
$\mathbf{C}$ (set of complex numbers)
For precise description of many concepts of mathematics and also for mathematical reasoning the concepts of relations(functions) and cardinality of sets are very important, which are discussed first.

A binary relation $\rho$ on a nonempty set $X$ is a subset of $X \times X$, which is said to be an equivalence relation if $\rho$ is reflexive, i.e., $(x, x) \in \rho$ for each $x \in X$; symmetric, i.e., $(x, y) \in \rho$ implies $(y, x) \in \rho$ and transitive i.e., $(x, y) \in \rho$ and $(y, z) \in \rho$ imply $(x, z) \in \rho$ for $x, y, z \in X$.

Definition 1.1.1 Let $X$ be a nonempty set and $\rho$ be an equivalence relation on $X$. The disjoint classes $[x]$ into which the set $X$ is partitioned by $\rho$ constitute a set, called the quotient set of $X$ by $\rho$, denoted by $X / \rho$, where $[x]$ denotes the class (determined by $\rho$ ) containing the element $x$ of $X$. Each element $x$ of the class $[x]$ is called a representative of $[x]$.

Example 1.1.2 Given a positive integer $n$, the quotient set $\mathbf{Z}n$ consists of all $n$ distinct classes $[0],[1], \ldots,[n-1]$. The set $\mathbf{Z}_n$ is called the residue classes of $\mathbf{Z}$ modulo $n$. Remark 1.1.3 The set $\mathbf{Z}_n$ provides very strong different algebraic structures (depending on $n$ ). The visual description of $\mathbf{Z}{12}$ is a 12-h clock.

## 数学代写|代数拓扑代考Algebraic Topology代考|Groups and Fundamental Homomorphism Theorem

This section conveys some basic results of group theory which are used throughout the book. Originally, a group was defined as the set of permutations (i.e., bijections) on a nonempty set $X$ with the property that combination (called composition) of two permutations is also a permutation on $X$. Earlier definition of a group is generalized to the present concept of an abstract group by a set of axioms.

Definition 1.2.1 A group $G$ is a nonempty set $G$ together with a binary operation (called composition), that is, a rule that assigns to each ordered pair $(a, b)$ in $G \times G$, an element of $G$, denoted by $a b$ (or $a \cdot b$ called a multiplication) such that
G(1) $a b(c)=a(b c)$ for all $a, b, c$ in $G$ (associative law);
$\mathbf{G ( 2 )}$ there exists an element $e$ in $G$ such that $a e=e a=a$ for all $a$ in $G$ (existence of identity);
$\mathbf{G ( 3 )}$ for each $a$ in $G$, there is an element $a^{\prime}$ in $G$ such that $a a^{\prime}=a^{\prime} a=e$ (existence of inverse).

Remark 1.2.2 In a group $G, e$ is unique and for each $a$ in $G, a^{\prime}$ is also unique. The element $a^{\prime}$ denoted by $a^{-1}$, is called the inverse of $a$ for each $a \in G$. In additive notation, $a b$ is written as $a+b ; e$ is as 0 (zero) and $a^{-1}$ as $-a$.

A group $G$ is said to be commutative (or abelian) if $a b=b a$ for all $a, b$ in $G$. We usually use the term ‘abelian group’ when the composition law is in additive notation. A group $G$ is said to be finite if its underlying set $G$ is finite; otherwise, it is said to be infinite.

## 数学代写|代数拓扑代考Algebraic Topology代考|Set Theory

$\mathbf{N}$(自然数/正整数的集合)
$\mathbf{Z}$(一组整数)
$\mathbf{Q}$(一组有理数)
$\mathbf{R}$(实数集)
$\mathbf{C}$(复数集合)

1.1.1设$X$为非空集合，$\rho$为$X$上的等价关系。集合$x$被$\rho$分割成的不相交类$[x]$构成一个集合，称为$x$除以$\rho$的商集，表示为$x / \rho$，其中$[x]$表示包含$x$的元素$x$的类(由$\rho$决定)。类$[x]$中的每个元素$x$被称为$[x]$的代表。

## 数学代写|代数拓扑代考Algebraic Topology代考|Groups and Fundamental Homomorphism Theorem

$\mathbf{G(2)}$在$G$中存在一个元素$e$，使得$a e=e a=a$对于$G$中的所有$a$(单位的存在性);
$\mathbf{G(3)}$对于$G$中的每一个$a$，在$G$中存在一个$a^{\素数}$使得$a a^{\素数}=a^{\素数}a=e$(逆的存在性)。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。