Posted on Categories:Discrete Mathematics, 数学代写, 离散数学

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|离散数学代写Discrete Mathematics代考|The Euclidean Algorithm

The greatest common divisor of two integers $a$ and $b$ is the largest integer that divides both $a$ and $b$. For example, the greatest common divisor of 12 and 30 is 6 . The Euclidean algorithm provides a very efficient way to compute the greatest common divisor of two integers.
Definition
Let $a$ and $b$ be integers that are not both zero. The greatest common divisor of $a$ and $b$, denoted $\operatorname{gcd}(\boldsymbol{a}, \boldsymbol{b})$, is that integer $d$ with the following properties:

1. $d$ is a common divisor of both $a$ and $b$. In other words,
$$d \mid a \quad \text { and } \quad d \mid b$$
2. For every integer $c$, if $c$ is a common divisor of both $a$ and $b$, then $c$ is less than or equal to $d$. In other words,
for every integer $c$, if $c \mid a$ and $c \mid b$ then $c \leq d$.

## 数学代写|离散数学代写Discrete Mathematics代考|Calculating Some gcd’s

a. Find $\operatorname{gcd}(72,63)$.
b. Find $\operatorname{gcd}\left(10^{20}, 6^{30}\right)$.
c. In the definition of greatest common divisor, $\operatorname{gcd}(0,0)$ is not allowed. Why not? What would $\operatorname{gcd}(0,0)$ equal if it were found in the same way as the greatest common divisors for other pairs of numbers?
Solution
a. $72=9 \cdot 8$ and $63=9 \cdot 7$. So $9 \mid 72$ and $9 \mid 63$, and no integer larger than 9 divides both 72 and 63. Hence $\operatorname{gcd}(72,63)=9$.
b. By the laws of exponents, $10^{20}=2^{20} \cdot 5^{20}$ and $6^{30}=2^{30} \cdot 3^{30}=2^{20} \cdot 2^{10} \cdot 3^{30}$. It follows that
$$2^{20} \mid 10^{20} \text { and } 2^{20} \mid 6^{30}$$
and by the unique factorization of integers theorem, no integer larger than $2^{20}$ divides both $10^{20}$ and $6^{30}$ (because no more than twenty 2’s divide $10^{20}$, no 3 ‘s divide $10^{20}$, and no 5 ‘s divide $\left.6^{30}\right)$. Hence $\operatorname{gcd}\left(10^{20}, 6^{30}\right)=2^{20}$.
c. Suppose $\operatorname{gcd}(0,0)$ were defined to be the largest common factor that divides 0 and 0 . The problem is that every positive integer divides 0 and there is no largest integer. So there is no largest common divisor!

## 数学代写|离散数学代写Discrete Mathematics代考|The Euclidean Algorithm

1. $d$ 是两者的公约数 $a$ 和 $b$. 换句话说，
$$d \mid a \quad \text { and } \quad d \mid b$$
2. 对于每个整数 $c$ ，如果 $c$ 是两者的公约数 $a$ 和 $b$ ，然后 $c$ 小于或等于 $d$. 换句话说， 对于每个整数 $c$ ，如果 $c \mid a$ 和 $c \mid b$ 然后 $c \leq d$.

## 数学代写|离散数学代写Discrete Mathematics代考|Calculating Some ged’s

A。寻找 $\operatorname{gcd}(72,63)$.
b. 寻找 $\operatorname{gcd}\left(10^{20}, 6^{30}\right)$.
C。在最大公约数的定义中， $\operatorname{gcd}(0,0)$ 不允许。为什么不? 什么会 $\operatorname{gcd}(0,0)$ 如果以与其他数字对的最大公因 数相同的方式找到它是否相等? 解决

$72=9 \cdot 8$ 和 $63=9 \cdot 7$. 所以 $9 \mid 72$ 和 $9 \mid 63$ ，并且没有大于 9 的整数可以同时除以 72 和 63 。因此 $\operatorname{gcd}(72,63)=9$.
b. 根据指数定律, $10^{20}=2^{20} \cdot 5^{20}$ 和 $6^{30}=2^{30} \cdot 3^{30}=2^{20} \cdot 2^{10} \cdot 3^{30}$. 它遧循
$2^{20} \mid 10^{20}$ and $2^{20} \mid 6^{30}$

C。认为 $\operatorname{gcd}(0,0)$ 被定义为除以 0 和 0 的最大公因数。问题是每个正整数都被 0 整除，没有最大的整数。所以 没有最大公约数!

## 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。

Posted on Categories:Discrete Mathematics, 数学代写, 离散数学

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|离散数学代写Discrete Mathematics代考|Special Graphs

One important class of graphs consists of those that do not have any loops or parallel edges. Such graphs are called simple. In a simple graph, no two edges share the same set of endpoints, so specifying two endpoints is sufficient to determine an edge.
Definition and Notation
A simple graph is a graph that does not have any loops or parallel edges. In a simple graph, an edge with endpoints $v$ and $w$ is denoted ${v, w}$.
Some Simple Graphs
Draw all simple graphs with the four vertices ${u, v, w, x}$ and two edges, one of which is ${u, v}$
Solution
Since one edge of the graph is specified to be ${u, v}$, the possibilities for the other edge are ${u, w},{u, x},{v, w},{v, x}$, and ${w, x}$. The resulting graphs are shown below.

## 数学代写|离散数学代写Discrete Mathematics代考|Determining whether Certain Simple Graphs Exist

Draw a graph with the specified properties or show that no such graph exists.
a. A simple graph with six vertices and sixteen edges.
b. A simple graph with four vertices of degrees $1,1,3$, and 3 .
Solution
a. There is no simple graph with six vertices and sixteen edges.
Proof (by contradiction): Suppose there is a graph $G$ with six vertices and sixteen edges. According to the handshake theorem (Theorem 4.9.1), since $G$ has sixteen edges its total degree is $2 \cdot 16=32$, and because $G$ has six vertices,
$$\text { the average degree of each vertex }=\frac{\text { the total degree }}{\text { the number of vertices }}=\frac{32}{6}=5 \frac{1}{3} \text {. }$$
The only way this can happen is for at least one vertex-say $v$, to have degree greater than five. But since $G$ has only six vertices, there are at most five other vertices to which $v$ can be connected. Thus, in order for $v$ to have degree greater than five, either there are at least two edges connecting $v$ to one of the other vertices or there is a loop incident on $v$. In either case $G$ would not be simple. Thus there is no simple graph that satisfies the given conditions.
b. There is no simple graph with four vertices of degrees $1,1,3$, and 3 .
You might first try the same approach as in the solution for part (a): Assume such a graph exists and divide its total degree by the number of edges. Since the total degree is $1+1+3+3=8$ and there are four vertices, the result is $8 / 4=2$. But this, by itself, is not a problem. You can easily find examples of simple graphs with four vertices where the average number of edges per vertex is two. Nonetheless, as the following argument shows, you will not be able to find a simple graph with total degree of eight and four vertices of degrees $1,1,3$, and 3 .
Proof (by contradiction): Suppose there is a simple graph $G$ with four vertices of degrees $1,1,3$, and 3. Call $a$ and $b$ the vertices of degree 1 , and call $c$ and $d$ the vertices of degree 3. Since $\operatorname{deg}(c)=3$ and $G$ does not have any loops or parallel edges (because it is simple), there must be edges that connect $c$ to $a, b$, and $d$.
By the same reasoning, there must be edges connecting $d$ to $a, b$, and $c$.

## 数学代写|离散数学代写Discrete Mathematics代考|Special Graphs

Some Simple Graphs

## 数学代写|离散数学代写Discrete Mathematics代考|Determining whether Certain Simple Graphs Exist

A。具有六个顶点和十六条边的简单图形。
b. 具有四个度数顶点的简单图 $1,1,3$, 和 3 。解决

the average degree of each vertex $=\frac{\text { the total degree }}{\text { the number of vertices }}=\frac{32}{6}=5 \frac{1}{3}$.

b. 没有四个顶点度数的简单图 $1,1,3$, 和 3 。

## 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。

Posted on Categories:Discrete Mathematics, 数学代写, 离散数学

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|离散数学代写Discrete Mathematics代考|Matrix Inversion

A square matrix $\boldsymbol{A}$ is said to be invertible if there exists a square matrix $\boldsymbol{B}$ such that
$$A B=B A=I \rightarrow B=A^{-1} \& A=B^{-1}$$
where the matrix $B$ is called the inverse of the matrix $A$ and denoted by $A^{-1}$. If $B$ is the inverse of $\boldsymbol{A}$, then $\boldsymbol{A}$ is the inverse of $\boldsymbol{B}$. A square matrix that is not invertible is called singular. Note that if a matrix is not square, then it has no inverse. In addition, a product of invertible matrices is always invertible, and the inverse of the product is the product of the inverses in the reverse order, that is, we have $(A B)^{-1}=B^{-1} A^{-1}$. The inverse of the matrix $A$ plays much the same role in matrix algebra that the reciprocal of a number plays in the arithmetic of real numbers.

An effective method to find the inverse of a matrix is to employ the elementary row operations. The elementary row operations consist of the following suboperations:

• Interchange two rows.
• Multiply all entries in a row by a nonzero number.
• Add a multiple of a row to another row.

## 数学代写|离散数学代写Discrete Mathematics代考|Matrix Inversion

A square matrix $\boldsymbol{A}$ is said to be invertible if there exists a square matrix $\boldsymbol{B}$ such that
$$A B=B A=I \rightarrow B=A^{-1} \& A=B^{-1}$$
where the matrix $B$ is called the inverse of the matrix $A$ and denoted by $A^{-1}$. If $B$ is the inverse of $\boldsymbol{A}$, then $\boldsymbol{A}$ is the inverse of $\boldsymbol{B}$. A square matrix that is not invertible is called singular. Note that if a matrix is not square, then it has no inverse. In addition, a product of invertible matrices is always invertible, and the inverse of the product is the product of the inverses in the reverse order, that is, we have $(A B)^{-1}=B^{-1} A^{-1}$. The inverse of the matrix $A$ plays much the same role in matrix algebra that the reciprocal of a number plays in the arithmetic of real numbers.

An effective method to find the inverse of a matrix is to employ the elementary row operations. The elementary row operations consist of the following suboperations:

• Interchange two rows.
• Multiply all entries in a row by a nonzero number.
• Add a multiple of a row to another row.

## 数学代写|离散数学代写Discrete Mathematics代考|Matrix Inversion

$$A B=B A=I \rightarrow B=A^{-1} \& A=B^{-1}$$

• 交换两行。
• 将一行中的所有条目乘以一个非零数。
• 将一行的倍数添加到另一行。

## 数学代写|离散数学代写Discrete Mathematics代考|Matrix Inversion

$$A B=B A=I \rightarrow B=A^{-1} \& A=B^{-1}$$

• 交换两行。
• 将一行中的所有条目乘以一个非零数。
• 将一行的倍数添加到另一行。

## 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。