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# 数学代写|复分析代写Complex analysis代考|MA8108 Sets

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## 数学代写|复分析代写Complex analysis代考|Sets

What is a set? Don’t we already have an idea of what a set is? The following informal definition relies on our intuitive idea of a set while making precise some notation and vocabulary of membership.

Definition 2.1.1. A set is a collection of objects. These objects are called members or elements of that set. If $m$ is a member of a set $A$, we write $\mathbf{m} \in \mathbf{A}$, and also say that $A$ contains $m$. If $m$ is not a member of a set $A$, we write $\mathbf{m} \notin \mathbf{A}$.

The set of all polygons contains triangles, squares, rectangles, pentagons, and so on. The set of all polygons does not contain circles or disks. The set of all functions contains the trigonometric, logarithmic, exponential, constant functions, and so on.
Examples and notation 2.1.2.
(1) Intervals are sets:
$(0,1)$ is the interval from 0 to 1 that does not include 0,1 . From the context you should be able to distinguish between the interval $(0,1)$ and a point $(0,1)$ in the plane.

## 数学代写|复分析代写Complex analysis代考|Cartesian product

The set ${a, b}$ is the same as the set ${b, a}$, as any element of either set is also the element of the other set. Thus, the order of the listing of elements does not matter. But sometimes we want the order to matter. We can then simply make another new notation for ordered pairs, but in general it is not a good idea to be inventing many new notations and concepts; it is better if we can reuse and recycle old ones. We do this next:
Definition 2.2.1. An ordered pair $(a, b)$ is defined as the set ${{a},{a, b}}$.
So here we defined $(a, b)$ in terms of already known constructions: $(a, b)$ is the set one of whose elements is the set ${a}$ with exactly one element $a$, and the remaining element of $(a, b)$ is the set ${a, b}$ that has exactly two elements $a, b$ if $a \neq b$ and has exactly one element otherwise. Thus for example the familiar ordered pair $(2,3)$ really stands for ${{2},{2,3}}$, $(3,2)$ stands for ${{3},{2,3}}$, and $(2,2)$ stands for ${{2},{2,2}}={{2},{2}}={{2}}$.

Theorem 2.2.2. $(a, b)=(c, d)$ if and only if $a=c$ and $b=d$.
Proof. [ReCall that $P \Leftrightarrow Q$ IS THE SAmE AS $P \Rightarrow Q$ AND $P \Leftarrow Q$. Thus the PROOF CONSISTS OF TWO PARTS.]

Proof of $\Rightarrow$ : Suppose that $(a, b)=(c, d)$. Then by the definition of ordered pairs, ${{a},{a, b}}={{c},{c, d}}$. If $a=b$, this says that ${{a}}={{c},{c, d}}$, so that ${{c},{c, d}}$ has only one element, so that ${c}={c, d}$, so that $c=d$. But then ${{a},{a, b}}={{c},{c, d}}$ is saying that ${{a}}={{c}}$, so that ${a}={c}$, so that $a=c$. Furthermore, $b=a=c=d$, which proves the consequent in case $a=b$. Now suppose that $a \neq b$. Then ${{a},{a, b}}={{c},{c, d}}$ has two elements, and so $c \neq d$. Note that ${a}$ is an element of ${{a},{a, b}}$, hence of ${{c},{c, d}}$. Thus necessarily either ${a}={c}$ or ${a}={c, d}$. But ${a}$ has only one element and ${c, d}$ has two (since $c \neq d$ ), it follows that ${a}={c}$, so that $a=c$. But then ${a, b}={c, d}$, and since $a=c$, it follows that $b=d$. This proves the consequent in the remaining cases.

Proof of $\Leftarrow$ : If $a=c$ and $b=d$, then ${a}={c}$ and ${a, b}={c, d}$, so that ${{a},{a, b}}={{c},{c, d}}$
Note that by our definition an ordered pair is a set of one or two sets.

## 数学代写|复分析代写Complex analysis代考|Sets

(1) 区间是集合:
$(0,1)$ 是从 0 到 1 的区间，不包括 0,1 。从上下文中您应该能够区分区间 $(0,1)$ 和一点 $(0,1)$ 在飞机上。

## 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。