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# 数学代写|实变函数作业代写Real analysis代考|Review of Set Theory

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## 数学代写|实变函数作业代写Real analysis代考|Introduction and Notations

We do not attempt to define a “set”. All of us agree to consider a “set” as undefined. A set can be “described” as a collection of objects considered as a single entity. The objects of this collection are called elements or members of the set. These objects are said to belong to the set or are contained in the set. The set is said to contain its objects or is composed of its objects.

Sets are usually denoted by capital letters A, B, C, . . . or X, Y, Z, . . .,and elements of the set are denoted by lower-case letters a, b, c, . . . or x, y, z, . . ..
If x is an element of a set X, we write x ∈ X. In such a case, we also say that ” $x$ belongs to $X$ “. If $x$ does not belong to $X$, we write $x \notin X$. Often, we denote sets by displaying their elements within braces. For example, the set of all positive integers less than 12 is denoted by y {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} or by {x : x is an integer, 0 < x < 12}. is an integer, $0<x<12}$. More generally, if $X$ is the set of all elements $x$ satisfying a property $P$, we denoted it by
$$X={x: x \text { satisfies } P} .$$

## 数学代写|实变函数作业代写Real analysis代考|Ordered Pairs and Cartesian Product

We note that ${a, b}={b, a}$ as sets. Here the question of the order in which the elements $a, b$ occur does not arise. However, there are situations where the order in which the elements occur is of paramount importance. For example, in analytic geometry of the plane, the point $(1,2)$ is different from the point $(2,1)$. Again, if we consider the complex number $a+i b$ as the ordered pair $(a, b)$, then the pair $(3,4)$ is different from the pair $(4,3)$. So, if we wish to consider a set of two elements $a, b$ as being ordered, a occurring first from the left and then $b$, we shall denote it by the symbol $(a, b)$.

Given two ordered pair of elements, $(a, b)$ and $(c, d)$, we define $(a, b)=(c, d)$ if and only $a=c$ and $b=d$.

Given two sets $A, B$, we define their Cartesian product, denoted by $A \times B$, by
$$A \times B={(a, b): a \in A, b \in B}$$
If $\mathbb{R}$ denotes the set of all real numbers and $\mathbb{C}$ denotes the set of all complex numbers, then
$$\mathbb{C}=\mathbb{R} \times \mathbb{R}$$

## 数学代写|实变函数作业代写Real analysis代考|Relations and Functions

Any set of ordered pairs is called a relation. If $R$ is a relation, the set ${x:(x, y) \in R}$ is called the domain of $R$, denoted by $d(R)$ and the set ${y \in(x, y) \in R}$ is called the range of $R$, denoted by $r(R)$.

Let $R$ be a relation with domain $d(R)$. The relation $R$ is said to be:
(a) reflexive if $(a, a) \in R$ for every $a \in d(R)$;
(b) symmetric if $(a, b) \in R$ implies that $(b, a) \in R$;
(c) transitive if $(a, b) \in R$ and $(b, c) \in R$ imply that $(a, c) \in R$.

X=X:X 满足 磷.

C=R×R

## 数学代写|实变函数作业代写REAL ANALYSIS代考|RELATIONS AND FUNCTIONS

b对称如果(一种,b)∈R暗示(b,一种)∈R;
C传递如果(一种,b)∈R和(b,C)∈R暗示(一种,C)∈R.

## 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。