### 数学代写|现代代数代写Modern Algebra代考|MATH 355

statistics-lab™ 为您的留学生涯保驾护航 在代写现代代数Modern Algebra方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写现代代数Modern Algebra代写方面经验极为丰富，各种代写现代代数Modern Algebra相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|现代代数代写Modern Algebra代考|Introductory Concepts

The concept of ‘set’ is very important in all branches of mathematics. We come across certain terms or concepts whose meanings need no explanation. Such terms are called undefined terms and are considered as primitive concepts. If one defines the term ‘set’ as ‘a set is a well defined collection of objects’, then the meaning of collection is not clear. One may define ‘a collection’ as ‘an aggregate’ of objects. What is the meaning of ‘aggregate’? As our language is finite, other synonyms, such as ‘class’, ‘family’ etc., will exhaust. Mathematicians accept that there are undefined terms and ‘set’ shall be such an undefined term. But we accept the familiar expressions, such as ‘set of all integers’, ‘set of all natural numbers’, ‘set of all rational numbers’, ‘set of all real numbers’ etc.

We shall neither attempt to give a formal definition of a set nor try to lay the groundwork for an axiomatic theory of sets. Instead we shall take the operational and intuitive approach to define a set. A set is a well defined collection of distinguishable objects.

The term ‘well defined’ specifies that it can be determined whether or not certain objects belong to the set in question. In most of our applications we deal with rather specific objects, and the nebulous notion of a set, in these, emerge as something quite recognizable. We usually denote sets by capital letters, such as $A, B, C, \ldots$. The objects of a set are called the elements or members of the set and are usually denoted by small letters, such as $a, b, c, \ldots$. Given a set $A$ we use the notation throughout ‘ $a \in A^{\prime}$ to indicate that an element $a$ is a member of $A$ and this is read as ‘ $a$ is an element of $A^{\prime}$ or ‘ $a$ belongs to $A$ ‘; and ‘ $a \notin A$ ‘ to indicate that the element $a$ is not a member of $A$ and this is read as ‘ $a$ is not an element of $A$ ‘ or ‘ $a$ does not belong to $A^{\prime}$. Since a set is uniquely determined by its elements, we may describe a set either by a characterizing property of the elements or by listing the elements. The standard way to describe a set by listing elements is to list elements of the set separated by commas, in braces. Thus a set $A={a, b, c}$ indicates that $a$, $b, c$ are the only elements of $A$ and nothing else. If $B$ is a set which consists of $a$, $b, c$ and possibly more, then notationally, $B={a, b, c, \ldots}$. On the other hand, a set consisting of a single element $x$ is sometimes called singleton $x$, denoted by ${x}$. By a statement, we mean a sentence about specific objects such that it has a truth value of either true or false but not both. If a set $A$ is described by a characterizing property $P(x)$ of its elements $x$, the brace notation ${x: P(x)}$ or ${x \mid P(x)}$ is also often used, and is read as ‘the set of all $x$ such that the statement $P(x)$ about $x$ is true.’ For example, $A={x: x$ is an even positive integer $<10}$.

## 数学代写|现代代数代写Modern Algebra代考|Relations on Sets

In mathematics, two types of very important relations, such as equivalence relation and ordered relations arise frequently. Sometimes, we need study decompositions of a non-empty set $X$ into disjoint subsets whose union is the entire set $X$ (i.e., $X$ is filled up by these subsets). Equivalence relations on $X$ provide tools to generate such decompositions of $X$ and produce new sets bearing a natural connection with the original set $X$.

A binary relation $R$ on a non-empty set $A$ is a mathematical concept and intuitively is a proposition such that for each ordered pair $(a, b)$ of elements of $A$, we can determine whether $a R b$ (read as $a$ is in relation $R$ to $b$ ) is true or false. We define it formally in terms of the set concept.

Definition 1.2.1 A binary relation $R$ on a non-empty set $A$ is a subset $R \subseteq A \times A$ and a binary relation $S$ from $A$ to $B$ is a subset $S$ of $A \times B$. The pair $(a, b) \in R$ is also denoted as $a R b$.

A binary relation $S$ from $A$ to $B$ is sometimes written as $S: A \rightarrow B$. Instead of writing a binary relation on $A$, we write only a relation on $A$, unless there is any confusion.

Example 1.2.1 For any set $A$, the diagonal $\Delta={(a, a): a \in A} \subseteq A \times A$ is the relation of equality.

In a binary relation $R$ on $A$, each pair of elements of $A$ need not be related i.e., $(a, b)$ may not belong to $R$ for all pairs $(a, b) \in A \times A$.
For example, if $a \neq b$, then $(a, b) \notin \Delta$ and also $(b, a) \notin \Delta$.
Example 1.2.2 The relation of inclusion on $\mathcal{P}(A)$ is ${(A, B) \in \mathcal{P}(A) \times \mathcal{P}(A): A \subseteq$ $B} \subseteq \mathcal{P}(A) \times \mathcal{P}(A)$

## 数学代写|现代代数代写Modern Algebra代考|Introductory Concepts

“集合”的概念在所有数学分支中都非常重要。我们遇到了一些不需要解释的术语或概念。此类术语称为末定义术 语并被视为原始概念。如果将术语“集合”定义为”集合是定义明确的对象集合”，那么集合的含义就不清楚了。可 以将“集合”定义为对象的“集合”。“聚合”是什么意思? 由于我们的语言是有限的，其他的同义词，如“阶级”、“家 庭”等，将会用尽。数学家接受有末定义的术语，“集合”应该是这样一个末定义的术语。但我们接受熟悉的表达方 式，例如“所有整数的集合”、“所有自然数的集合”、“所有有理数的集合”、“所有实数的集合”等。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。