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# 数学代写|实分析代写Real Analysis代考|YSC3206

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## 数学代写|实分析代写Real Analysis代考|Functions

If $X$ and $Y$ are sets, the Cartesian product of $X$ and $Y$ is
$$X \times Y={\text { all ordered pairs }(x, y): x \in X, y \in Y} .$$
The ordered pair $(x, y)$ is thus to be distinguished from $(y, x)$. We will discuss the Cartesian product of an arbitrary family of sets later in this section.

A relation from $X$ to $Y$ is a subset of $X \times Y$. If $f$ is a relation, then the domain and range of $f$ are
$$\begin{array}{r} \operatorname{dom} f=X \cap{x:(x, y) \in f \text { for some } y \in Y} \ \operatorname{rng} f=Y \cap{y:(x, y) \in f \text { for some } x \in X} . \end{array}$$
Frequently symbols such as $\sim$ or $\leq$ are used to designate a relation. In these cases the notation $x \sim y$ or $x \leq y$ will mean that the element $(x, y)$ is a member of the relation $\sim$ or $\leq$, respectively.

A relation $f$ is said to be single-valued if $y=z$ whenever $(x, y)$ and $(x, z) \in$ $f$. A single-valued relation is called a function. The terms mapping, map, transformation are frequently used interchangeably with function, although the term function is usually reserved for the case when the range of $f$ is a subset of $\mathbb{R}$. If $f$ is a mapping and $(x, y) \in f$, we let $f(x)$ denote $y$. We call $f(x)$ the image of $x$ under $f$. We will also use the notation $x \mapsto f(x)$, which indicates that $x$ is mapped to $f(x)$ by $f$. If $A \subset X$, then the image of $A$ under $f$ is
$$f(A)={y: y=f(x), \text { for some } x \in \operatorname{dom} f \cap A} .$$
Also, the inverse image of $B$ under $f$ is
$$f^{-1}(B)={x: x \in \operatorname{dom} f, f(x) \in B} .$$

## 数学代写|实分析代写Real Analysis代考|Set Theory

A fundamental question that arises in the definition of the Cartesian product of an arbitrary family of sets is the existence of choice mappings. This is an example of a question that cannot be answered within the context of elementary set theory. In the beginning of the $20^{\text {th }}$ century, Ernst Zermelo formulated an axiom of set theory called the Axiom of Choice, which asserts that the Cartesian product of an arbitrary family of nonempty sets exists and is nonempty. The formal statement is as follows.
7.2. The Axiom of Choice. If $X_\alpha$ is a nonempty set for each element $\alpha$ of an index set $I$, then
$$\prod_{\alpha \in I} X_\alpha$$
is nonempty.
7.3. Proposition. The following statement is equivalent to the Axiom of Choice: If $\left{X_\alpha\right}_{\alpha \in A}$ is a disjoint family of nonempty sets, then there is a set $S \subset \cup_{\alpha \in A} X_\alpha$ such that $S \cap X_\alpha$ consists of precisely one element for every $\alpha \in A$.
Proof. The Axiom of Choice states that there exists $f: A \rightarrow \cup_{\alpha \in A} X_\alpha$ such that $f(\alpha) \in X_\alpha$ for each $\alpha \in A$. The set $S:=f(A)$ satisfies the conclusion of the statement. Conversely, if such a set $S$ exists, then the mapping $A \stackrel{f}{\longrightarrow} \cup_{\alpha \in A} X_\alpha$ defined by assigning the point $S \cap X_\alpha$ the value of $f(\alpha)$ implies the validity of the Axiom of Choice.

7.4. Definition. Given a set $S$ and a relation $\leq$ on $\mathrm{S}$, we say that $\leq$ is a partial ordering if the following three conditions are satisfied:
(i) $x \leq x$ for every $x \in S$ (reflexive)
(ii) if $x \leq y$ and $y \leq x$, then $x=y$, (antisymmetric)
(iii) if $x \leq y$ and $y \leq z$, then $x \leq z$. (transitive) If, in addition,
(iv) either $x \leq y$ or $y \leq x$, for all $x, y \in S$, (trichotomy) then $\leq$ is called a linear or total ordering.

For example, $\mathbb{Z}$ is linearly ordered with its usual ordering, whereas the family of all subsets of a given set $X$ is partially ordered (but not linearly ordered) by $\subset$. If a set $X$ is endowed with a linear ordering, then each subset $A$ of $X$ inherits the ordering of $X$. That is, the restriction to $A$ of the linear ordering on $X$ induces a linear ordering on $A$. The following two statements are known to be equivalent to the Axiom of Choice.

## 数学代写|实分析代写Real Analysis代考|Functions

$$X \times Y={\text { all ordered pairs }(x, y): x \in X, y \in Y} .$$

$$\begin{array}{r} \operatorname{dom} f=X \cap{x:(x, y) \in f \text { for some } y \in Y} \ \operatorname{rng} f=Y \cap{y:(x, y) \in f \text { for some } x \in X} . \end{array}$$

## 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。