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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|图论作业代写Graph Theory代考|Proof Techniques

Graph Theory as a mathematical discipline straddles the distinction between applied and theoretical mathematics. As we’ve already seen in this chapter, graphs can be used to model various scenarios, especially some complicated modern applications. Alternatively, isomorphisms comprise a very theoretical aspect of graph theory. Within each of these areas, however, we use proofs to deepen and demonstrate our understanding of graphs and their properties.
While this book relies on a basic understanding of logic, proof structure, and proof techniques, it is by no means expected that the reader is a proficient writer of proofs. This section is meant to review the basics of mathematical proof, and introduce some early graph results that can be proven with little intuition about graphs and their structure. For a more complete introduction to logic and proofs, see Discrete Mathematics by Susanna Epp [31].

Most mathematical statements have an underlying conditional form; that is, they can be written as “If …, then….” For example, we may say “The sum of two odd integers is even” but we are, in fact, making the conditional statement “If $x$ and $y$ are odd integers, then $x+y$ is even.” Writing a statement in the standard if-then form allows the logical structure to stand out and provides guidance into the format of the argument.

In logical symbols, conditional statements are given as $p \rightarrow q$. A direct proof begins by assuming the premise of the conditional $(p)$ and uses logic, definitions, and previously proven theorems to show the conclusion $(q)$ is true. The example below uses the definition of odd, even, and the assumption that the sum of two integers is still an integer.

## 数学代写|图论作业代写Graph Theory代考|Indirect Proofs

Direct proofs can be considered the preferable method of proof as their structure models the statement they are proving. However, some statements are either impossible or much more difficult to prove in this way and a different technique is needed. Classic examples of this include proving there are infinitely many primes or that $\sqrt{2}$ is irrational. While these are great examples, better examples exist in graph theory for the usefulness of indirect proofs.
There are two main types of indirect proofs: contradiction and contraposition. For a Proof by Contradiction, we assume the negation of the statement is true. Through logic, definitions, and previous results, we show a contradiction must be occurring, thus proving the original statement must be true. An example from elementary number theory is shown below.
Proposition 1.23 For any integer $n$, if $n^{2}$ is odd then $n$ is odd.
Proof: Suppose for a contradiction that $n^{2}$ is odd but $n$ is even. Then $n=2 k$ for some integer $k$ and $n^{2}=(2 k)^{2}=4 k^{2}=2 j$ where $j$ is the integer $2 k^{2}$. Thus $n^{2}$ is both even and odd, a contradiction. Therefore if $n^{2}$ is odd then $n$ is also odd.

For a Proof by Contraposition, we use a direct proof on the contrapositive $(\sim q \rightarrow \sim p)$ of the original conditional statement $(p \rightarrow q)$. Since the contrapositive is logically equivalent to the original statement, this shows the intended result to be true. The statement ahove can also be proven using the contrapositive, as shown below.

# 图论代考

## 有限元方法代写

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## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师