# 数学代写|离散数学作业代写discrete mathematics代考|Proof by Induction

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## 数学代写|离散数学作业代写discrete mathematics代考|Proof by Induction

The logical validity of the method of proof by induction is intimately bound up with the construction of the natural numbers, with ordinal arithmetic, and with the so-called well-ordering principle. We shall not treat those logical niceties here, but shall instead concentrate on the technique. As with any good idea in mathematics, we shall be able to make it intuitively clear that the method is a valid and useful one. So no confusion should result.

Consider a statement $P(n)$ about the natural numbers. For example, the statement might be “The quantity $n^2+5 n+6$ is always even.” If we wish to prove this statement, we might proceed as follows:

Prove the statement $P(1)$.

Prove that $P(k) \Rightarrow P(k+1)$ for every $k \in{1,2, \ldots}$.

Let us apply the syllogism modus ponendo ponens from the end of Sec. 1.5 to determine what we will have accomplished. We know $P(1)$ and, from Step (2) with $k=1$, that $P(1) \Rightarrow P(2)$. We may therefore conclude $P(2)$. Now Step (2) with $k=2$ says that $P(2) \Rightarrow P(3)$. We may then conclude $P(3)$. Continuing in this fashion, we may establish $P(n)$ for every natural number $n$.

Notice that this reasoning applies to any statement $P(n)$ for which we can establish Steps (1) and (2) above. Thus Steps (1) and (2) taken together constitute a method of proof. It is a method of establishing a statement $P(n)$ for every natural number $n$. The method is known as proof by induction.

## 数学代写|离散数学作业代写discrete mathematics代考|Elements of Set Theory

Beginning in this section, we will be doing mathematics in the way that it is usually done. That is, we shall define terms and we shall state and prove properties that they satisfy. In earlier chapters we were careful, but we were less mathematical. Sometimes we even had to say “This is the way we do it; don’t worry.” Many of the topics in Chaps. 1 and 2 are really only best understood from the advanced perspectives of mathematical logic. Now, and for the rest of this book, it is time to show how mathematics is done in practice.

We use theorems, propositions, and lemmas to formulate our ideas. The device of proofs is used to validate those ideas. Another formal ingredient of mathematical exposition is the “definition.” A definition usually introduces a new piece of terminology or a new idea and explains what it means in terms of ideas and terminology that have already been presented. As you read this chapter, pause frequently to check that we are following this paradigm.

Definition 3.2 Let $S$ and $T$ be sets. We say that $S$ is a subset of $T$, and we write $S \subset T$ or $T \supset S$, if
$$x \in S \Rightarrow x \in T$$
We do not prove our definitions. There is nothing to prove. A definition introduces you to a new idea, or piece of terminology, or piece of notation.

# 离散数学代写

## 数学代写|离散数学作业代写discrete mathematics代考|Elements of Set Theory

$$x \in S \Rightarrow x \in T$$

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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。

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