### 数学代写|离散数学作业代写discrete mathematics代考|MATH 200

statistics-lab™ 为您的留学生涯保驾护航 在代写离散数学discrete mathematics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写离散数学discrete mathematics代写方面经验极为丰富，各种代写离散数学discrete mathematics相关的作业也就用不着说。

• 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|离散数学作业代写discrete mathematics代考|MATHEMATICAL INDUCTION

The Principle, or Axiom, or Postulate of Mathematical Induction, is one of the cornerstones of any mathematical reasoning and, in particular, of our course. It is claimed that the outstanding mathematician Leopold Kronecker (1823-1891) said that “God created natural numbers, all else is humans’ business.” As with any trivial truth, it can be wrong. Indeed, many thousand years ago, together with learning to talk, people started to count, and eventually, the names for small numbers had appeared, like “one,” “two,” “three,” etc., different for various languages. Moreover, we know, for example, from observations of Russian ethnographer and traveler Nikolas Miklukho-Maklai (1846-1888) over the indigenous people of Papua-New Guinea, that people had initially developed several specialized versions of the word “one,” such that the phrase “one tree” sounded initially differently than “one boat,” or than “one kid.” Only gradually, during millennia, those various versions of “one something” merged in the abstract word “one,” which means the natural number without any specific meaning attached.

During the centuries, arithmetic has been developed together with the human society, and now there is the highly sophisticated mathematical discipline, the Number Theory, which studies, in particular, the properties of the natural numbers ${ }^{1}{0,1,2,3, \ldots}$, of the (positive, negative, and zero) integers ${\ldots,-3,-2,-1,0,1,2,3,4, \ldots}$, of the prime numbers, etc. We accept as the known facts that the natural numbers and the integers satisfy the four standard arithmetic operations. In particular, if any three integers are connected as $a=b \times c$, which can be written as $a=b \cdot c$, then the integers $b$ and $c$ are called factors or multipliers, and $a$ is called the product. If we rewrite the equation as $a \div b=c$, then $a$ is called the dividend, $b$ the divisor, and $c$ the quotient. It is also said that $b$ (and $c$ as well) divides $a$, or that $a$ is divisible by $b$ and by $c$, or that $b($ and $c)$ divides into $a$.

## 数学代写|离散数学作业代写discrete mathematics代考|The Axiom of Mathematical Induction

The Axiom of Mathematical Induction. Consider a set of statements, maybe formulas $S_{n_{1}}, S_{n_{1}+1}, S_{n_{1}+2}, \ldots$, numbered by all sufficiently large integers $n \geq n_{1}$. Usually $n_{1}=1$ or $n_{1}=0$, but it can be any integer. The statement $S_{n}$ is called the Induction Hypothesis or Inductive Assumption.
(1) First, suppose that the statement $S_{n_{1}}$, called the basis step of induction, or just the base is valid. In applications of the method of mathematical induction, the verification of the basic step is an independent problem. In some problems, this step may be trivial, but it can never be skipped altogether.
(2) Second, suppose that for each integer $n \geq n_{1}$, that is, bigger than or equal to the basis value, we can prove a conditional statement $S_{n} \Rightarrow S_{n+1}$, that is, we can prove the validity of the hypothesis $S_{n+1}$ for each specified natural $n>n_{1}$ assuming the validity of $S_{n}$, and this conditional statement is valid for all natural $n \geq n_{1}$. This part of the method is called the inductive step.
(3) If we can independently show these two steps, then the Principle of Mathematical Induction claims that all infinitely many of the statements $S_{n}$, for all integer $n \geq n_{1}$ are valid.

This method of proof is accepted as an axiom because nobody can actually verify infinitely many statements $S_{n}, n \geq n_{1}$; the method cannot be justified without using some other, maybe even less intuitively obvious, properties of the set of natural numbers. Mathematicians have been using this principle for centuries and never arrived at a contradiction. Therefore, we accept the method of mathematical induction without a proof, as a postulate, and believe that this principle properly expresses a certain fundamental property of the infinite set $\mathcal{N}$ of natural numbers.

## 数学代写|离散数学作业代写discrete mathematics代考| The Axiom of Mathematical Induction

（1）首先，假设语句Sn1，称为归纳的基础步骤，或者只是基础是有效的。在数学归纳方法的应用中，基本步骤的验证是一个独立的问题。在某些问题中，此步骤可能微不足道，但永远不能完全跳过。
（2） 其次，假设对于每个整数n≥n1，即大于或等于基值，我们可以证明一个条件语句Sn⇒Sn+1，也就是说，我们可以证明假设的有效性Sn+1对于每个指定的自然n>n1假设Sn，并且此条件语句对所有自然有效n≥n1.该方法的这一部分称为归纳步骤。
（3）如果我们能独立地证明这两个步骤，那么数学归纳原理就声称，所有无限多的陈述Sn，对于所有整数n≥n1是有效的。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。