## 数学代写|离散数学作业代写discrete mathematics代考|CS3653

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## 数学代写|离散数学作业代写discrete mathematics代考|Constructive Proofs and Nonconstructive

Some theorems in mathematics are about establishing the existence of a particular object. A proof of a proposition of the form $\exists x P(x)$ is called an existence proof. There are two types of existence proofs. If we can find an object $a$ such that $P(a)$ is true, then such an existence proof is called a constructive existence proof. If we cannot fund an object a such that $P(a)$ is true but rather establish its existence by an indirect proof, usually using a proof by contradiction, then such an existence proof is called a nonconstructive existence proof.
Example 4.11
(a) Show that there is a set of three positive integers that the square of one of them is equal to the sum of the squares of the other two integers.
(b) Prove that there exists an even integer that can be written in two ways as a sum of two prime numbers.
(c) Given an integer $n$, there is an integer $m$ with $m>n$.
Solution
(a) We employ a constructive existence proof. This problem presents in a way a form of Pythagorean theorem. To this effect, there exist the set ${3,4,5}$ and the set ${5,12,13}$, which both can satisfy the requirement, as $5^2=4^2+3^2$ and $13^2=5^2+12^2$
(b) Using a constructive existence proof, the integer 24 is an even number and can be written as $24=7+17$ as well as $24=11+13$, where $7,11,13$, and 17 are all prime numbers.
(c) We employ a constructive existence proof. Suppose that $n$ is an integer. Let $m=n+1$. Then $m$ is an integer and $m>n$. The proof established the existence of the desired integer $m$ by showing that its value can be computed by adding 1 to the value of $n$.

## 数学代写|离散数学作业代写discrete mathematics代考|Definitions and Notation

A set is an unordered collection of distinct objects that are called elements or members of the set. It is essential to have a clear and rigorous definition of a set. For instance, “smart children in a town” does not form a set, as the word “smart” does not have a universally agreeable definition, and its membership is debatable, whereas “pregnant women in a town” does form a well-defined set.

It is common to use capital letters, such as $A$, to denote sets, and lowercase letters, such as $x$, to refer to set elements. If $x$ is an element of the set $A$ or equivalently $x$ belongs to $A$, we then use the notation $x \in A$, and if $x$ does not belong to the set $A$ or equivalently $x$ is not an element of $A$, we then write $x \notin A$. For instance, if $A$ is the set of all capital cities, then Tokyo, denoted by $x$, is an element of $A$, that is, $x \in A$, and if $B$ is the set of all European cities, then Tokyo, denoted by $x$, is not a member of $B$, because it is a city in Asia, we thus have $x \notin B$.

A set is generally represented by braces (curly brackets), that is, by {} . One way to specify a set with a finite number of elements is to use the set roster method, by which all the elements of the set are listed between curly brackets (i.e., within braces), such as ${3,6,9}$. The order of elements presented in a set is irrelevant, and a set remains the same if its elements are repeated or rearranged. Note that a set of a very large number of elements that follow a recognizable pattern is usually described by listing the first few elements, followed by ellipses “…,” which is read as “and so forth,” such as ${1,2,3,4, \ldots}$

Another way to specify a set is the set builder notation, through which some property held only by all members of the set is clearly and completely described, such as ${x \in N \mid x$ is a multiple of $3,0<x<10}$, where the vertical line (I) is read as “such that” and the comma (, ) as “and,” and $\boldsymbol{N}$ represents the set of all positive integers. Note that the general form ${x \in S \mid Q(x)}$, where $Q(x)$ is a predicate indicating the property that the object $x$ of the set $S$ has, is read as “the set of all $x$ in $S$ such that $x$ has the property $Q(x)$.”

A set usually presents a group of elements with common properties. However, it is possible for a set to contain any kind of elements whatsoever, and they are not required to be of the same type, such as the set {China, nose, baby, movie, ice cream, $\pi$, rainbow, stamp, soccer $}$.

# 离散数学代写

## 数学代写|离散数学作业代写discrete mathematics代考|Constructive Proofs and Nonconstructive

(a) 证明存在一组三个正整数，其中一个的平方等于另外 两个整数的平方和。
(b) 证明存在一个可以用两种方式写成两个质数之和的偶 数。
(c) 给定一个整数 $n$, 有一个整数 $m$ 和 $m>n$.

(a) 我们采用建设性的存在证明。这个问题以勾股定理的 形式出现。为此，存在集合 $3,4,5$ 和集合 $5,12,13$, 两者 都能满足要求, 如 $5^2=4^2+3^2$ 和 $13^2=5^2+12^2$
(b) 使用建设性存在证明，整数 24 是偶数，可以写成 $24=7+17$ 也 $24=11+13$ ，在哪里 $7,11,13,17$ 都是素数。
(c) 我们采用建设性的存在证明。假设 $n$ 是一个整数。让 $m=n+1$. 然后 $m$ 是一个整数并且 $m>n$. 证明建立 了所需整数的存在 $m$ 通过证明它的值可以通过将的值加 1 来计算 $n$.

## 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。