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# 数学代写|实分析代写Real Analysis代考|MAT2400

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## 数学代写|实分析代写Real Analysis代考|The Real Numbers

In our development of the real number system, we shall assume that properties of the natural numbers, integers, and rational numbers are known. In order to agree on what the properties are, we summarize some of the more basic ones. Recall that the natural numbers are designated as
$$\mathbb{N}:={1,2, \ldots, k, \ldots} .$$
They form a well-ordered set when endowed with the usual ordering. The ordering on $\mathbb{N}$ satisfies the following properties:
(i) $x \leq x$ for every $x \in S$.
(ii) if $x \leq y$ and $y \leq x$, then $x=y$.
(iii) if $x \leq y$ and $y \leq z$, then $x \leq z$.
(iv) for all $x, y \in S$, either $x \leq y$ or $y \leq x$.
The four conditions above define a linear ordering on $S$, a topic that was introduced in Section 1.3 and will be discussed in greater detail in Section 2.3. The linear order $\leq$ of $\mathbb{N}$ is compatible with the addition and multiplication operations in $\mathbb{N}$. Furthermore, the following three conditions are satisfied:
(i) Every nonempty subset of $\mathbb{N}$ has a first element; i.e., if $\emptyset \neq S \subset \mathbb{N}$, there is an element $x \in S$ such that $x \leq y$ for any element $y \in S$. In particular, the set $\mathbb{N}$ itself has a first element that is unique, in view of (ii) above, and is denoted by the symbol 1 ,
(ii) Every element of $\mathbb{N}$, except the first, has an immediate predecessor. That is, if $x \in \mathbb{N}$ and $x \neq 1$, then there exists $y \in \mathbb{N}$ with the property that $y \leq x$ and $z \leq y$ whenever $z \leq x$.
(iii) $\mathbb{N}$ has no greatest element; i.e., for every $x \in \mathbb{N}$, there exists $y \in \mathbb{N}$ such that $x \neq y$ and $x \leq y$.

## 数学代写|实分析代写Real Analysis代考|Cardinal Numbers

25.1. Definition. Two sets $A$ and $B$ are said to be equivalent if there exists a bijection $f: A \rightarrow B$, and then we write $A \sim B$. In other words, there is a one-to-one correspondence between $A$ and $B$. It is not difficult to show that this notion of equivalence defines an equivalence relation as described in Definition 4.1 and therefore sets are partitioned into equivalence classes. Two sets in the same equivalence class are said to have the same cardinal number or to be of the same cardinality. The cardinal number of a set $A$ is denoted by $\operatorname{card} A$; that is, $\operatorname{card} A$ is the symbol we attach to the equivalence class containing $A$. There are some sets so frequently encountered that we use special symbols for their cardinal numbers. For example, the cardinal number of the set ${1,2, \ldots, n}$ is denoted by $n, \operatorname{card} \mathbb{N}=\aleph_0$, and $\operatorname{card} \mathbb{R}=c$.
25.2. Definition. A set $A$ is finite if $\operatorname{card} A=n$ for some nonnegative integer $n$. A set that is not finite is called infinite. Any set equivalent to the positive integers is said to be denumerable. A set that is either finite or denumerable is called countable; otherwise it is called uncountable.

One of the first observations concerning cardinality is that it is possible for two sets to have the same cardinality even though one is a proper subset of the other. For example, the formula $y=2 x, x \in[0,1]$ defines a bijection between the closed intervals $[0,1]$ and $[0,2]$. This also can be seen with the help of the figure below.

Another example, utilizing a two-step process, establishes the equivalence between points $x$ of $(-1,1)$ and $y$ of $\mathbb{R}$. The semicircle with endpoints omitted serves as an intermediary.

## 数学代写|实分析代写Real Analysis代考|The Real Numbers

$$\mathbb{N}:={1,2, \ldots, k, \ldots} .$$

(i) $x \leq x$对于每一个$x \in S$。
(ii)如果$x \leq y$和$y \leq x$，则$x=y$。
(iii)如果$x \leq y$和$y \leq z$，则$x \leq z$。
(iv)对于所有$x, y \in S$，请选择$x \leq y$或$y \leq x$。

(i) $\mathbb{N}$的每个非空子集都有第一个元素;例如，如果$\emptyset \neq S \subset \mathbb{N}$，有一个元素$x \in S$，这样$x \leq y$对应任何元素$y \in S$。特别地，集合$\mathbb{N}$本身有一个唯一的第一元素，根据上面的(ii)，用符号1表示。
$\mathbb{N}$的每一个要素，除了第一个要素外，都有一个直接的前导。也就是说，如果$x \in \mathbb{N}$和$x \neq 1$，那么只要$z \leq x$，就存在具有$y \leq x$和$z \leq y$属性的$y \in \mathbb{N}$。
(iii) $\mathbb{N}$没有最大元;也就是说，对于每一个$x \in \mathbb{N}$，存在$y \in \mathbb{N}$使得$x \neq y$和$x \leq y$。

## 数学代写|实分析代写Real Analysis代考|Cardinal Numbers

25.1. 定义。如果存在双射$f: A \rightarrow B$，则称两个集合$A$和$B$是等价的，然后写出$A \sim B$。换句话说，$A$和$B$之间存在一对一的对应关系。不难证明等价的概念定义了定义4.1中描述的等价关系，因此集合被划分为等价类。同一等价类中的两个集合称为具有相同的基数或具有相同的基数。集合$A$的基数用$\operatorname{card} A$表示;也就是说，$\operatorname{card} A$是我们附加到包含$A$的等价类上的符号。有一些集合是如此频繁地遇到，以至于我们使用特殊的符号来表示它们的基数。例如，集合${1,2, \ldots, n}$的基数用$n, \operatorname{card} \mathbb{N}=\aleph_0$和$\operatorname{card} \mathbb{R}=c$表示。
25.2. 定义。一个集合$A$是有限的，如果$\operatorname{card} A=n$对于某个非负整数$n$。非有限的集合称为无限集。任何等价于正整数的集合都是可数的。有限或可数的集合称为可数集合;否则称为不可数。

## 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。