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|Basic Definitions

A Boolean algebra is a mathematical system. A Boolean algebra consists of a nonempty set $B$ together with two binary operations of addition ” + ” and multiplication “.”, where they map elements of $B \times B$ to elements of $B$ (i.e., if $x, y \in B$, then $x+y$ and $x \cdot y$ are also in $B$ ), a unary operation of complementation “،,” where it maps elements of $B$ to elements of $B$, two distinct elements ” 1 ” and ” 0 “, and the axioms for all elements $x, y$, and $z$ in $B$, as summarized in Table 8.1 .

In symbols, a Boolean algebra is designated by its six parts $\left{B,+, \cdot,{ }^{\prime}, 0,1\right}$. The operators “+”, “.”, and “،’, are called sum, product, and complement, respectively. Note that “+”” and “.” are not the usual arithmetic operators “plus” and “times.” The symbols ” 0 ” and ” 1 “, which are called the zero element and the unit element, respectively, do not represent numbers on the real number line. However, the names “plus,” “times,” “complement,” “zero,” and “one” are commonly used informally when discussing Boolean algebras. For convenience, ” $x \cdot y$ ” is shown as ” $x y$ “, and it is also common to replace the complement operation “،,” by a bar “”, (e.g., $\left.x^{\prime}=\bar{x}\right)$.

## 数学代写|离散数学代写Discrete Mathematics代考|Boolean Expressions and Boolean Functions

Let $B={0,1}$. The variable $x$ is called a Boolean variable if it assumes values only from $B$ (i.e., if its only possible values are 0 and 1). Then, $B^n=$ $\left{x_1, x_2, \ldots, x_n \mid x_i \in B\right.$ for $\left.1 \leq i \leq n\right}$ is the set of all possible $n$-tuples of 0 s and $1 \mathrm{~s}$ and has $2^n$ elements. A function from $B^n$ to $B$ is called a Boolean function of degree $\boldsymbol{n}$. Boolean functions can be defined by Boolean tables. Because a Boolean function is an assignment of 0 or 1 to each of these $2^n$ different $n$-tuples, there are $(2)^{\left(2^n\right)}$ different Boolean functions of degree $n$, labeled $F_1, F_2, \ldots, F_{2^2}$. Table 8.2 presents all Boolean functions of degree two (i.e., $n=2$ ), labeled $F_1, F_2, \ldots, F_{16}$.

A Boolean expression consists of Boolean variables and Boolean operators. The Boolean expression in the Boolean variables $x_1, x_2, \ldots, x_n$ are defined recursively through the basic clause that states that $0,1, x_1, x_2, \ldots, x_n$ are Boolean expressions, and the recursive clause that states that the sum and the product of any two Boolean expressions as well as the complement of any Boolean expression are also Boolean expressions. Each Boolean expression represents a Boolean function. The values of a Boolean function are obtained by substituting 0 and 1 for the Boolean variables in the Boolean expression.

## 数学代写|离散数学代写Discrete Mathematics代考|Boolean Expressions and Boolean Functions

(即，如果它的唯一可能值是 0 和 1)。然后， $B^n=$ 合 $n$-元组 0 和 $1 \mathrm{~s}$ 并且有 $2^n$ 元素。一个函数来自 $B^n$ 到 $B$ 被称为度数的布尔函数 $\boldsymbol{n}$. 布尔函数可以由 布尔表定义。因为布尔函数是对这些中的每一个赋值 0 或 $12^n$ 不同的 $n$-元组，有 $(2)^{\left(2^n\right)}$ 度的不同布 尔函数 $n$ ，标记 $F_1, F_2, \ldots, F_{2^2}$. 表 8.2 列出了所有二阶布尔函数 (即， $n=2$ ), 标记 $F_1, F_2, \ldots, F_{16}$

## 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|Product Notation

The notation for the product of a sequence of numbers is analogous to the notation for their sum. The Greek capital letter pi, П, denotes a product. For example,
$$\prod_{k=1}^5 a_k=a_1 a_2 a_3 a_4 a_5 .$$
Definition
If $m$ and $n$ are integers and $m \leq n$, the symbol $\prod_{k=m}^n a_k$, read the product from $k$ equals $\boldsymbol{m}$ to $\boldsymbol{n}$ of $\boldsymbol{a}$-sub- $\boldsymbol{k}$, is the product of all the terms $a_m, a_{m+1}, a_{m+2}, \ldots, a_n$. We write
$$\prod_{k=m}^n a_k=a_m \cdot a_{m+1} \cdot a_{m+2} \cdots a_n$$

## 数学代写|离散数学代写Discrete Mathematics代考|Properties of Summations and Products

The following theorem states general properties of summations and products. The proof of the theorem is discussed in Section 5.6.
Theorem 5.1.1
If $a_m, a_{m+1}, a_{m+2}, \ldots$ and $b_m, b_{m+1}, b_{m+2}, \ldots$ are sequences of real numbers and $c$ is any real number, then the following equations hold for any integer $n \geq m$ :

1. $\sum_{k=m}^n a_k+\sum_{k=m}^n b_k=\sum_{k=m}^n\left(a_k+b_k\right)$
2. $c \cdot \sum_{k=m}^n a_k=\sum_{k=m}^n c \cdot a_k \quad$ generalized distributive law
3. $\left(\prod_{k=m}^n a_k\right) \cdot\left(\prod_{k=m}^n b_k\right)=\prod_{k=m}^n\left(a_k \cdot b_k\right)$

## 数学代写|离散数学代写Discrete Mathematics代考|Product Notation

$$\prod_{k=1}^5 a_k=a_1 a_2 a_3 a_4 a_5 .$$

$$\prod_{k=m}^n a_k=a_m \cdot a_{m+1} \cdot a_{m+2} \cdots a_n$$

## 数学代写|离散数学代写Discrete Mathematics代考|Properties of Summations and Products

1. $\sum_{k=m}^n a_k+\sum_{k=m}^n b_k=\sum_{k=m}^n\left(a_k+b_k\right)$
2. $c \cdot \sum_{k=m}^n a_k=\sum_{k=m}^n c \cdot a_k$ 广义分配律
3. $\left(\prod_{k=m}^n a_k\right) \cdot\left(\prod_{k=m}^n b_k\right)=\prod_{k=m}^n\left(a_k \cdot b_k\right)$

## 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|Multisets

As defined earlier, a set is an unordered collection of objects, where the multiplicity of objects is ignored, and the membership of an object has a binary status, that is, either an element belongs to the set or it does not. We now deviate from this general definition of a set to briefly introduce multisets, where the multiplicity of an object is explicitly significant, and later present fuzzy sets, where membership of an object is not binary but a continuum of values.

A multiset (short form for multiple-membership set), also known as a bag, is an unordered collection of objects where an object can occur as a member of a set more than once, that is, repeated occurrences of objects are allowed. For instance, multisets ${7,8,9}$ and ${9,8,7}$ are the same, but multisets ${7,8,9}$ and ${7,8,7,9}$ are different. The number of occurrences, given for each element, is called the multiplicity of the element in the multiset. A multiset corresponds to an ordinary set if the multiplicity of every element is one.

Example of multisets may include the multiset of prime factors of an integer, such as the integer 360 that has the prime factorization $360=2^3 \times 3^2 \times 5^1$, which gives the multiset ${2,2,2,3,3,5}$. The sets of distinct letters forming the words “are,” “era,” “ear,” and “rear” are the same, which is ${r, a, e}$; however, their multisets of letters forming these words are different, as the multiset of the words “are,” “era,” and “ear” is ${r, a, e}$, whereas that for the word “rear” is ${r, r, a, e}$.

## 数学代写|离散数学代写Discrete Mathematics代考|Fuzzy Sets

In a world of many shades of gray, a black-white dichotomy is an unnecessary artificial imposition. The concept of fuzzy sets is an important and practical generalization of the notion of classical sets. For instance, if the universe of discourse consists of knowledgeable people, then in fuzzy set theory, members of a set can have varying degrees of knowledge. Fuzzy sets, introduced by Lotfi Zadeh, where each member of the set is defined by the degree of fuzziness, have an array of applications in modeling, control systems, linguistics, information retrieval, decision-making, and of course artificial intelligence, where information is incomplete or imprecise.

In classical set theory, a set $A$ is defined in terms of its characteristic function $\mu_A(x)$, a mapping from the universal set $U$ to the binary set ${0,1}$, where $x$ belongs to $A$ if and only if $\mu_A(x)=1$ and $x$ does not belong to $A$ if and only if $\mu_A(x)=0$. In fuzzy set theory, a set $A$ is defined in terms of its membership function $\mu_A(x)$, a mapping from the universal set $U$ to the unit interval $[0,1]$, where $x$ in the fuzzy set $A$ has a certain degree of membership. Therefore the fuzzy set $A$ is denoted by listing the elements with their degrees of membership.

## 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|Proofs by Contraposition and Proofs by Contradiction

When we cannot easily employ a direct proof, we make use of an indirect proof. Indirect proofs do not start with the premises and end with the conclusion. There are two general types of indirect proofs, namely, proofs by contraposition and proofs by contradiction. A proof by contraposition is based on the law of contrapositive, that is, the conditional statement $p \rightarrow q$ is equivalent to its contrapositive $\bar{q} \rightarrow \bar{p}$. In other words, in a proof by contraposition of $p \rightarrow q$, we take $\bar{q}$ as a premise, and we show that $\bar{p}$ must follow.
Example $4.7$
Prove the following statements using a proof by contraposition.
(a) If a real number is irrational, then its square root is irrational.
(b) If $r=m n$, where $m$ and $n$ are positive integers, then $m \leq \sqrt{r}$ or $n \leq \sqrt{r}$.
Solution
(a) By letting $x$ be an arbitrary real number, we need to prove that if $x$ is irrational, then $\sqrt{x}$ is irrational. Using a proof by contraposition, we want to prove that if $\sqrt{x}$ is not irrational, then $x$ is not irrational, or equivalently if $\sqrt{x}$ is rational, then $x$ is rational. If $\sqrt{x}$ is rational, then $\sqrt{x}=\frac{m}{n}$ for some integers $m$ and $n \neq 0$. As a result, we have $x=\frac{m^2}{n^2}$, which is the quotient of integers. Hence $x$ is rational. We just showed the negation of the hypothesis of the original conditional statement is true.

(b) Using a proof by contraposition, we want to prove that if $m \leq \sqrt{r}$ or $n \leq \sqrt{r}$ is false, then $r=m n$ is false, or equivalently if both $m>\sqrt{r}$ and $n>\sqrt{r}$ are true, then $r=m n$ is false. However, if $m>\sqrt{r}$ and $n>\sqrt{r}$, then $m n>r$. This shows that $m n \neq r$, which contradicts the premise $m n=r$. We just showed the negation of the hypothesis of the original conditional statement is true.

## 数学代写|离散数学代写Discrete Mathematics代考|Proof by Cases and Proofs by Exhaustion

Sometimes we need to partition the proof into several disjoint parts whose union is the complete theorem and then prove each part individually. Suppose we must prove $p \rightarrow q$ and that $p$ is equivalent to $p_1 \vee p_2 \vee \ldots \vee p_n$ (where $p_1, p_2, \ldots, p_n$ are the cases). To prove a conditional statement of the form $\left(p_1 \vee p_2 \vee \ldots \vee p_n\right) \rightarrow q$, we prove $\left(p_1 \rightarrow q\right) \wedge\left(p_2 \rightarrow q\right) \wedge \ldots \wedge\left(p_n \rightarrow q\right)$, as the two statements are equivalent. Such a proof is called a proof by cases, as we have
$$(p \rightarrow q) \leftrightarrow\left(\left(p_1 \vee p_2 \vee \ldots \vee p_n\right) \rightarrow q\right) \leftrightarrow\left(\left(p_1 \rightarrow q\right) \wedge\left(p_2 \rightarrow q\right) \wedge \ldots \wedge\left(p_n \rightarrow q\right)\right)$$
Example $4.9$
Assuming $k$ is a positive integer, show that $m=k^3-k$ is an even integer.
Solution
Using a proof by cases, we consider two mutually exclusive cases for $k$, that is, $k$ is even, and $k$ is odd, as every positive integer falls into one of these two mutually exclusive cases. Assuming $k$ is even, then for some integer $n$, we have $k=2 n \rightarrow m=k^3-k=k(k+1)(k-1)=2 n(2 n+1)(2 n-1)$, that is, $m$ is even, as it is a multiple of 2 . Assuming $k$ is odd, then for some integer $n$, we have $k=2 n+1 \rightarrow m=k^3-k=(k-1) k(k+1)=2 n(2 n+1)(2 n+2)$, that is, $m$ is even, as it is a multiple of 2 .

## 数学代写|离散数学代与Discrete Mathematics代考|Proofs by Contraposition and Proofs by Contradiction

(a) 如果一个实数是无理数，那么它的平方根也是无理数。
(b) 如果 $r=m n$ ，在哪里 $m$ 和 $n$ 是正整数，那么 $m \leq \sqrt{r}$ 或者 $n \leq \sqrt{r}$. 解决方案
(a) 通过让 $x$ 是任意实数，我们需要证明如果 $x$ 是无理数，那么 $\sqrt{x}$ 是不合理的。使用反证法，我们 想证明如果 $\sqrt{x}$ 不是无理数，那么 $x$ 不是不合理的，或者等效地如果 $\sqrt{x}$ 是有理数，那么 $x$ 是理性的。 如果 $\sqrt{x}$ 是有理数，那么 $\sqrt{x}=\frac{m}{n}$ 对于一些整数 $m$ 和 $n \neq 0$. 结果，我们有 $x=\frac{m^2}{n^2}$ ，这是整数的 商。因此 $x$ 是理性的。我们只是证明了原始条件语句假设的否定为真。
（b）使用反证法，我们想证明如果 $m \leq \sqrt{r}$ 或者 $n \leq \sqrt{r}$ 是假的，那么 $r=m n$ 是假的，或者如果 两者都是 $m>\sqrt{r}$ 和 $n>\sqrt{r}$ 是真的，那么 $r=m n$ 是假的。然而，如果 $m>\sqrt{r}$ 和 $n>\sqrt{r}$ ，然 后 $m n>r$. 这表明 $m n \neq r$ ， 这与前提相矛盾 $m n=r$. 我们只是证明了原始条件语句假设的否定为 真。

## 数学代写|离散数学代写Discrete Mathematics代考|Proof by Cases and Proofs by Exhaustion

$$(p \rightarrow q) \leftrightarrow\left(\left(p_1 \vee p_2 \vee \ldots \vee p_n\right) \rightarrow q\right) \leftrightarrow\left(\left(p_1 \rightarrow q\right) \wedge\left(p_2 \rightarrow q\right) \wedge \ldots \wedge\left(p_n \rightarrow q\right)\right)$$

$k=2 n \rightarrow m=k^3-k=k(k+1)(k-1)=2 n(2 n+1)(2 n-1)$ ， 那是， $m$ 是偶数，因 为它是 2 的倍数。假设 $k$ 是奇数，那么对于某个整数 $n$ ，我们有
$k=2 n+1 \rightarrow m=k^3-k=(k-1) k(k+1)=2 n(2 n+1)(2 n+2)$ ，那是， $m$ 是偶 数，因为它是 2 的倍数。

## 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|Lots of Identities

There are many interesting relations valid for the Fibonacci numbers. For example, what is the sum of the first $n$ Fibonacci numbers? We have
\begin{aligned} 0 &=0, \ 0+1 &=1, \ 0+1+1 &=2, \ 0+1+1+2 &=4, \ 0+1+1+2+3 &=7, \ 0+1+1+2+3+5 &=12, \ 0+1+1+2+3+5+8 &=20, \ 0+1+1+2+3+5+8+13 &=33 . \end{aligned}
Staring at these numbers for a while, it is not hard to recognize that by adding 1 to the right-hand sides we get Fibonacci numbers; in fact, we get Fibonacci numbers two steps after the last summand. As a formula, we have
$$F_0+F_1+F_2+\cdots+F_n=F_{n+2}-1 .$$
Of course, at this point this is only a conjecture, an unproven mathematical statement we believe to be true. To prove it, we use induction on $n$ (since the Fibonacci numbers are defined by recurrence, induction is the natural and often only proof method at hand).

## 数学代写|离散数学代写Discrete Mathematics代考|A Formula for the Fibonacci Numbers

How large are the Fibonacci numbers? Is there a simple formula that expresses $F_n$ as a function of $n$ ?

An easy way out, at least for the author of a book, is to state the answer right away:
Theorem 4.3.1 The Fibonacci numbers are given by the formula
$$F_n=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right) .$$

Proof. It is straightforward to check that this formula gives the right value for $n=0,1$, and then one can prove its validity for all $n$ by induction.

## 数学代写|离散数学代写Discrete Mathematics代考|Lots of Identities

$0=0,0+1=1,0+1+1=2,0+1+1+2=4,0+1+1+2+3=7,0+1+1+2+3+5=12,0+1+2+3+8=20$

-个式，我们有
$$F_0+F_1+F_2+\cdots+F_n=F_{n+2}-1$$

## 数学代写|离散数学代写Discrete Mathematics代考|A Formula for the FibonacciNumbers

$$F_n=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right) .$$

## 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。