### 数学代写|实分析作业代写Real analysis代考|MATHS 2100

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|实分析作业代写Real analysis代考|Functions

We begin this section with the fundamental concept of a function. In many texts a function or a mapping $f$ from a set $A$ to a set $B$ is described as a rule that assigns to each element $x \in A$ a unique element $y \in B$. This is generally expressed by writing $y=f(x)$ to denote the value of the function $f$ at $x$. The difficulty with this “definition” is that the terms “rule” and “assigns” are vague and difficult to define. Consequently we will define “function” strictly in terms of sets, using the notation and concepts introduced in the preceding section.

The motivation for the following definition is to think of the graph of a function; namely the set of ordered pairs $(x, y)$ where $y$ is given by the “rule” that defines the function.

DEFINITION 1.2.1 Let $A$ and $B$ be any two sets. A function $f$ from $A$ into $B$ is a subset of $A \times B$ with the property that each $x \in A$ is the first component of precisely one ordered pair $(x, y) \in f$; that is, for every $x \in A$ there exists $y \in B$ such that $(x, y) \in f$, and if $(x, y)$ and $\left(x, y^{\prime}\right)$ are elements of $f$, then $y=y^{\prime}$. The set $A$ is called the domain of $f$, denoted Dom $f$. The range of $f$, denoted Range $f$, is defined by
$$\text { Range } f-{y \in B:(x, y) \in f \text { for some } x \in A} \text {. }$$
If Range $f=B$, then the function $f$ is said to be onto $B$. (See Figure 1.3)
If $f$ is a function from $A$ to $B$ and $(x, y) \in f$, then the element $y$ is called the value of the function $f$ at $x$ and we write
$$y=f(x) \quad \text { or } \quad f: x \rightarrow y .$$

## 数学代写|实分析作业代写Real analysis代考|Mathematical Induction

Throughout the text we will on occasion need to prove a statement, identity, or inequality involving the positive integer $n$. As an example, consider the following identity. For each $n \in \mathbb{N}$,
$$r+r^{2}+\cdots r^{n}=\frac{r-r^{n+1}}{1-r}, \quad r \neq 1 .$$
Mathematical induction is a very useful tool in establishing that such an identity is valid for all positive integers $n$.

THEOREM 1.3.1 (Principle of Mathematical Induction) For each $n \in \mathbb{N}$, let $P(n)$ be a statement about the positive integer $n$. If
(a) $P(1)$ is true, and
(b) $P(k+1)$ is true whenever $P(k)$ is true,
then $P(n)$ is true for all $n \in \mathbb{N}$.
The proof of this theorem depends on the fact that the positive integers are well-ordered; namely, every nonempty subset of $\mathbb{N}$ has a smallest element. This statement is usually taken as a postulate or axiom for the positive integers: we do so in this text. Since it will be used on several other occasions, we state it both for completeness and emphasis.

WELL-ORDERING PRINCIPLE Every nonempty subset of $\mathbb{N}$ has a smallest element.

The well-ordering principle can be restated as follows: If $A \subset \mathbb{N}, A \neq \emptyset$, then there exists $n \in A$ such that $n \leq k$ for all $k \in A$.

To prove Theorem 1.3.1 we will use the method of proof by contradiction. Most theorems involve showing that a statement $P$ implies the statement $Q$; namely, if $P$ is true, then $Q$ is true. In a proof by contradiction one assumes that $P$ is true and $Q$ is false, and then shows that these two assumptions lead to a logical contradiction; namely show that some statement $R$ is both true and false.

## 数学代写|实分析作业代写Real analysis代考|Functions

Range $f-y \in B:(x, y) \in f$ for some $x \in A$.

$$y=f(x) \quad \text { or } \quad f: x \rightarrow y .$$

## 数学代写|实分析作业代写Real analysis代考|Mathematical Induction

$$r+r^{2}+\cdots r^{n}=\frac{r-r^{n+1}}{1-r}, \quad r \neq 1 .$$

(b) $P(k+1)$ 任何时候都是真的 $P(k)$ 是真的，

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## MATLAB代写

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