标签： MATH 200

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

Posted on 2023年6月30日2023年6月30日 by statistics-lab

如果你也在怎样代写离散数学Discrete Mathematics 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。离散数学Discrete Mathematics是数学的一个分支，研究一般代数环境中的同源性。它是一门相对年轻的学科，其起源可以追溯到19世纪末的组合拓扑学（代数拓扑学的前身）和抽象代数（模块和共轭理论）的研究，主要是由亨利-庞加莱和大卫-希尔伯特提出。

离散数学Discrete Mathematics是研究同源漏斗和它们所带来的复杂的代数结构；它的发展与范畴理论的出现紧密地联系在一起。一个核心概念是链复合体，可以通过其同调和同调来研究。它在代数拓扑学中发挥了巨大的作用。它的影响逐渐扩大，目前包括换元代数、代数几何、代数理论、表示理论、数学物理学、算子矩阵、复分析和偏微分方程理论。K理论是一门独立的学科，它借鉴了同调代数的方法，正如阿兰-康尼斯的非交换几何一样。

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

Collection of well defined objects is called a set. Well defined means distinct and distinguishable. The objects are called as elements of the set. The ordering of elements in a set does not change the set. i.e. the ordering of elements can not play a vital role in the set theory. For example
$$
\mathrm{A}={a, b, c, d} \text { and } \mathrm{B}={b, a, d, c} \text { are equal sets. }
$$
The symbol $\in$ stands for ‘belongs to’. $x \in$ A means $x$ is an element of the set A. It is observed that if $\mathrm{A}$ be a set and $x$ is any object, then either $x \in \mathrm{A}$ or $x \notin \mathrm{A}$ but not both. Generally sets are denoted by capital letters A, B, C and etc.
Consider the examples of set:
$$
\mathrm{A}={2,4,6,8,10,12,14,16,18}
$$

$$
\begin{aligned}
\mathrm{B} & ={x, y, z, u, v, w} \
\mathrm{N} & ={1,2,3, \ldots} \
\mathrm{I} & ={\ldots-2,-1,0,1,2,3, \ldots}
\end{aligned}
$$
In general the set can be expressed in two ways. i.e. Tabular method (Roster method) and Set-builder method (Specification method).
Tabular Method
Expressing the elements of a set within a parenthesis where the elements are separated by commas is known as tabular method, roster method or method of extension.
Consider the example
$$
\mathrm{A}={1,3,5,7,9,11,13,15}
$$
Set Builder Method
Expressing the elements of a set by a rule or formula is known as set-builder method, specification method or method of intension. Mathematically
$$
\mathrm{S}={x \mid \mathrm{P}(x)}
$$
where $\mathrm{P}(x)$ is the property that describes the elements of the set. The symbol | stands for ‘such that’. It is not possible to write every set in tabular form. Consider an example
$$
\mathrm{S}={x \mid x \text { is an Italian }}
$$
The above set $\mathrm{S}$ can not be expressed in tabular form as it is impossible to list all Italians. Consider the examples
$$
\begin{aligned}
\mathrm{A} & ={x \mid x=2 n+1 ; 0 \leq n \leq 7 ; n \in \mathrm{I}} \
& ={1,3,5,7,9,11,13,15} \
\mathrm{B} & ={x \mid x=1, x=a, x=\operatorname{Book}, x=\mathrm{Pen}} \
& ={1, a, \text { Book, Pen }}
\end{aligned}
$$
and
From the second example given above it is clear that the elements of a set do not have any common property also.

数学代写|离散数学作业代写discrete mathematics代考|TYPES OF SETS

Here we will discuss the different types of sets.
Finite Set
A set which contains finite number of elements is known as finite set. Consider the example of finite set as
$$
\mathrm{A}={a, b, c, d, e}
$$
Infinite set
A set which contains infinite number of elements is known as infinite set. Consider the example of infinite set as
$$
\begin{aligned}
\mathrm{N} & ={1,2,3,4, \ldots} \
\mathrm{I} & ={\ldots-3,-2,-1,0,1,2,3, \ldots}
\end{aligned}
$$
Singleton Set
A set which contains only one element is known as a singleton set. Consider the example
$$
\mathrm{S}={9}
$$
Pair set
A set which contains only two elements is known as a pair set. Consider the examples
$$
\begin{aligned}
& \mathrm{S}={e, f} \
& \mathrm{S}={{a},{1,3,5}}
\end{aligned}
$$
Empty Set
A set which contains no element is known as empty set. The empty set is also known as void set or null set. Generally denoted by $\phi$. Consider the examples
(i) $\phi={x: x \neq x}$
(ii) $\phi={x: x$ is a month of the year containing 368 days $}$
2.2.6 Set of Sets
A set which contains sets is known as set of sets. Consider the example
$$
\mathrm{A}={{a, b},{1},{1,2,3,4},{u, v},{\text { Book, Pen }}}
$$
Universal Set
A set which is superset of all the sets under consideration or particular discussion is known as universal set. Generally denoted by $\mathrm{U}$ or $\mathrm{E}$ or $\Omega$.

Generally, the universal set can be chosen arbitrarily for discussion, but once chosen it is fixed for discussion. Consider the example
Let
$$
\begin{aligned}
& \mathrm{A}={a, b, c} \
& \mathrm{B}={a, e, i, o, u} \
& \mathrm{C}={p, q, r, s}
\end{aligned}
$$
So, we can take the universal set $\mathrm{U}$ as ${a, b, c, \ldots ., z}$
i.e.
$$
\mathrm{U}={a, b, c, d, e, \ldots, z}
$$

离散数学代写

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

定义良好的对象的集合称为集合。定义良好的意思是不同的和可区分的。这些对象被称为集合的元素。元素在集合中的顺序不会改变集合。也就是说，元素的排序在集合论中不能起重要作用。例如
$$
\mathrm{A}={a, b, c, d} \text { and } \mathrm{B}={b, a, d, c} \text { are equal sets. }
$$
符号$\in$代表“属于”。$x \in$ A表示$x$是集合A的一个元素。可以观察到，如果$\mathrm{A}$是一个集合，$x$是任何对象，那么要么$x \in \mathrm{A}$要么$x \notin \mathrm{A}$，但不能两者都是。集合一般用大写字母A、B、C等表示。
考虑set的例子:
$$
\mathrm{A}={2,4,6,8,10,12,14,16,18}
$$

$$
\begin{aligned}
\mathrm{B} & ={x, y, z, u, v, w} \
\mathrm{N} & ={1,2,3, \ldots} \
\mathrm{I} & ={\ldots-2,-1,0,1,2,3, \ldots}
\end{aligned}
$$
一般来说，集合可以用两种方式表示。即表列法(名册法)和集构建法(规格法)。
表格法
在用逗号分隔元素的圆括号内表示集合的元素，称为表格方法、花名册方法或扩展方法。
考虑这个例子
$$
\mathrm{A}={1,3,5,7,9,11,13,15}
$$
Set Builder方法
用规则或公式表示集合元素的方法称为集合构造法、说明法或内涵法。数学上
$$
\mathrm{S}={x \mid \mathrm{P}(x)}
$$
其中$\mathrm{P}(x)$是描述集合元素的属性。符号代表“这样”。不可能把每一组都写成表格形式。考虑一个例子
$$
\mathrm{S}={x \mid x \text { is an Italian }}
$$
由于不可能列出所有意大利人，因此不能以表格形式表示上述集合$\mathrm{S}$。考虑下面的例子
$$
\begin{aligned}
\mathrm{A} & ={x \mid x=2 n+1 ; 0 \leq n \leq 7 ; n \in \mathrm{I}} \
& ={1,3,5,7,9,11,13,15} \
\mathrm{B} & ={x \mid x=1, x=a, x=\operatorname{Book}, x=\mathrm{Pen}} \
& ={1, a, \text { Book, Pen }}
\end{aligned}
$$
和
从上面给出的第二个例子可以清楚地看出，集合的元素也不具有任何公共属性。

数学代写|离散数学作业代写discrete mathematics代考|TYPES OF SETS

这里我们将讨论不同类型的集合。
有限集
包含有限个元素的集合称为有限集合。考虑有限集的例子为
$$
\mathrm{A}={a, b, c, d, e}
$$
无限集
包含无限个元素的集合称为无限集合。考虑无穷集的例子为
$$
\begin{aligned}
\mathrm{N} & ={1,2,3,4, \ldots} \
\mathrm{I} & ={\ldots-3,-2,-1,0,1,2,3, \ldots}
\end{aligned}
$$
单例集合
只包含一个元素的集合称为单例集合。考虑这个例子
$$
\mathrm{S}={9}
$$
配对组
只包含两个元素的集合称为对集合。考虑下面的例子
$$
\begin{aligned}
& \mathrm{S}={e, f} \
& \mathrm{S}={{a},{1,3,5}}
\end{aligned}
$$
空集
不包含任何元素的集合称为空集合。空集也称为空集或空集。通常用$\phi$表示。考虑下面的例子
(i) $\phi={x: x \neq x}$
(ii) $\phi={x: x$是包含368天的一年中的一个月$}$
2.2.6集合的集合
包含集合的集合称为集合的集合。考虑这个例子
$$
\mathrm{A}={{a, b},{1},{1,2,3,4},{u, v},{\text { Book, Pen }}}
$$
通用集
一个集合是所有所考虑或特定讨论的集合的超集，称为全称集合。通常用$\mathrm{U}$或$\mathrm{E}$或$\Omega$表示。

一般情况下，可以任意选择全称集进行讨论，但一旦选择了全称集就固定了。考虑这个例子
让
$$
\begin{aligned}
& \mathrm{A}={a, b, c} \
& \mathrm{B}={a, e, i, o, u} \
& \mathrm{C}={p, q, r, s}
\end{aligned}
$$
因此，我们可以取泛集$\mathrm{U}$为${a, b, c, \ldots ., z}$
例如:
$$
\mathrm{U}={a, b, c, d, e, \ldots, z}
$$

数学代写|离散数学作业代写discrete mathematics代考请认准statistics-lab™

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。

有限元方法代写

有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。

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随机分析代写

随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。

回归分析代写

多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。

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，工具箱允许您学习和应用专业技术。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。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|离散数学作业代写discrete mathematics代考|Direct Proof

Posted on 2023年6月30日2023年6月30日 by statistics-lab

数学代写|离散数学作业代写discrete mathematics代考|Direct Proof

In this section we shall assume that you are familiar with the positive integers, or natural numbers (a detailed treatment of the natural numbers appears in Sec. 5.2). This number system ${1,2,3, \ldots}$ is denoted by the symbol $\mathbb{N}$. For now we will take the elementary arithmetic properties of $\mathbb{N}$ for granted. We shall formulate various statements about natural numbers and we shall prove them. Our methodology will emulate the discussions in earlier sections. We begin with a definition.

Definition 2.1 A natural number $n$ is said to be even if, when it is divided by 2 , there is an integer quotient and no remainder.

Definition 2.2 A natural number $n$ is said to be odd if, when it is divided by 2 , there is an integer quotient and remainder 1.

You may have never before considered, at this level of precision, what is the meaning of the terms “odd” or “even.” But your intuition should confirm these definitions. A good definition should be precise, but it should also appeal to your heuristic idea about the concept that is being defined.

Notice that, according to these definitions, any natural number is either even or odd. For if $n$ is any natural number, and if we divide it by 2 , then the remainder will be either 0 or 1 -there is no other possibility (according to the Euclidean algorithm). In the first instance, $n$ is even; in the second, $n$ is odd.

In what follows we will find it convenient to think of an even natural number as one having the form $2 m$ for some natural number $m$. We will think of an odd natural number as one having the form $2 k+1$ for some nonnegative integer $k$. Check for yourself that, in the first instance, division by 2 will result in a quotient of $m$ and a remainder of 0 ; in the second instance it will result in a quotient of $k$ and a remainder of 1 .

Now let us formulate a statement about the natural numbers and prove it. Following tradition, we refer to formal mathematical statements either as theorems or propositions or sometimes as lemmas. A theorem is supposed to be an important statement that is the culmination of some development of significant ideas. A proposition is a statement of lesser intrinsic importance. Usually a lemma is of no intrinsic interest, but is needed as a step along the way to verifying a theorem or proposition.

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

Aristotelian logic dictates that every sensible statement has a truth value: TRUE or FALSE. If we can demonstrate that a statement $\mathbf{A}$ could not possibly be false, then it must be true. On the other hand, if we can demonstrate that $\mathbf{A}$ could not be true, then it must be false. Here is a dramatic example of this principle. In order to present it, we shall assume for the moment that you are familiar with the system of rational numbers. These are numbers that may be written as the quotient of two integers (without dividing by zero, of course). We shall discuss the rational numbers in greater detail in Sec. 5.4.

Theorem 2.1 (Pythagoras) There is no rational number $x$ with the property that $x^2=2$
Proof: In symbols (refer to Chap. 1), our assertion may be written
$$
\sim\left[\exists x,\left(x \in \mathbb{Q} \wedge x^2=2\right)\right]
$$
Let us assume the statement to be false. Then what we are assuming is that
$$
\exists x,\left(x \in \mathbb{Q} \wedge x^2=2\right)
$$
Since $x$ is rational we may write $x=p / q$, where $p$ and $q$ are integers.
We may as well suppose that both $p$ and $q$ are positive and nonzero. After reducing the fraction, we may suppose that it is in lowest terms-so $p$ and $q$ have no common factors.
Now our hypothesis asserts that
$$
x^2=2
$$
or
$$
\left(\frac{p}{q}\right)^2=2
$$
We may write this out as
$$
p^2=2 q^2
$$
Observe that this equation asserts that $p^2$ is an even number. But then $p$ must be an even number ( $p$ cannot be odd, for that would imply that $p^2$ is odd by Proposition 2.2). So $p=2 r$ for some natural number $r$.

离散数学代写

数学代写|离散数学作业代写discrete mathematics代考|Direct Proof

在本节中，我们将假设您熟悉正整数或自然数(自然数的详细处理将在第5.2节中出现)。这个数字系统${1,2,3, \ldots}$用符号$\mathbb{N}$表示。现在我们将把$\mathbb{N}$的基本算术性质视为理所当然。我们将拟定关于自然数的各种表述，并加以证明。我们的方法将模仿前面章节中的讨论。我们从定义开始。

定义2.1自然数$n$被称为偶数，当它除以2时，有一个整数商而没有余数。

定义2.2一个自然数$n$被称为奇数，当它被2整除时，有一个整数商，余数为1。

在这种精确度下，您可能从未考虑过术语“奇数”或“偶数”的含义。但是你的直觉应该会证实这些定义。一个好的定义应该是精确的，但它也应该吸引你对所定义的概念的启发式想法。

注意，根据这些定义，任何自然数不是偶数就是奇数。因为如果$n$是任何自然数，如果我们把它除以2，那么余数将是0或1——没有其他可能(根据欧几里得算法)。在第一种情况下，$n$是偶数;在第二种情况下，$n$很奇怪。

下面我们会发现，对于某些自然数$m$，把偶数看作具有$2 m$形式的数是很方便的。对于非负整数$k$，我们把奇数看作是形式为$2 k+1$的整数。你自己检查一下，在第一个例子中，除以2得到的商是$m$余数是0;在第二个实例中，它将得到商$k$和余数1。

现在让我们形成一个关于自然数的表述并证明它。按照传统，我们将正式的数学陈述称为定理或命题，有时也称为引理。定理应该是一个重要的陈述，是一些重要思想发展的顶点。命题是一种内在重要性较低的陈述。引理通常没有什么内在的意义，但是在验证定理或命题的过程中需要作为一个步骤。

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

亚里士多德逻辑规定，每一个可感知的陈述都有一个真值:真或假。如果我们能证明一个陈述$\mathbf{A}$不可能是假的，那么它一定是真的。另一方面，如果我们能证明$\mathbf{A}$不可能是真的，那么它一定是假的。下面是这个原则的一个戏剧性的例子。为了介绍它，我们暂时假定你熟悉有理数系统。这些数字可以写成两个整数的商(当然，不用除以零)。我们将在第5.4节更详细地讨论有理数。

定理2.1(毕达哥拉斯)没有有理数$x$具有$x^2=2$
证明:在符号中(参看第一章)，我们的断言可以写出来
$$
\sim\left[\exists x,\left(x \in \mathbb{Q} \wedge x^2=2\right)\right]
$$
让我们假设这个陈述是假的。那么我们假设
$$
\exists x,\left(x \in \mathbb{Q} \wedge x^2=2\right)
$$
因为$x$是有理数，我们可以写成$x=p / q$，其中$p$和$q$是整数。
我们不妨假设$p$和$q$都是正数且非零。在对分数进行化简之后，我们可以假设它是最小的——所以$p$和$q$没有公因数。
我们的假设断言
$$
x^2=2
$$
或
$$
\left(\frac{p}{q}\right)^2=2
$$
我们可以把它写成
$$
p^2=2 q^2
$$
注意，这个方程断言$p^2$是一个偶数。但是$p$必须是偶数($p$不可能是奇数，因为这就意味着根据命题2.2 $p^2$是奇数)。对于某个自然数$r$是$p=2 r$。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|离散数学作业代写discrete mathematics代考|Sequential Rationality

Posted on 2023年5月9日2023年5月9日 by statistics-lab

数学代写|离散数学作业代写discrete mathematics代考|Sequential Rationality

As these examples indicate, more should be assumed about players than that they select best responses ex ante (from the point of view of the beginning of the game). Players ought to demonstrate rationality whenever they are called on to make decisions. This is called sequential rationality.

Sequential rationality: An optimal strategy for a player should maximize his or her expected payoff, conditional on every information set at which this player has the move. That is, player $i$ ‘s strategy should specify an optimal action from each of player $i$ ‘s information sets, even those that player $i$ does not believe (ex ante) will be reached in the game.
If sequential rationality is common knowledge between the players (at every information set), then each player will “look ahead” to consider what players will do in the future in response to her move at a particular information set.
To operationalize the notion of sequential rationality, one has to deal with some intricacies regarding limitations on the beliefs players have about each other at different information sets. In addition, there are versions that build on rationalizability (iterated dominance) and others that build on Nash equilibrium. Over the past few decades, researchers have developed a number of refinements based on different assumptions about beliefs and the scope of best-response behavior. In this chapter, I’ll introduce you to three of the most basic concepts. The key one, called subgame perfect Nash equilibrium, is the most straightforward and is widely applied in following chapters.

数学代写|离散数学作业代写discrete mathematics代考|Backward Induction

Perhaps the simplest way of representing sequential rationality is through a procedure that identifies an optimal action for each information set by working backward in the game tree. Consider, for instance, a game of perfect information, which you’ll recall has only singleton information sets (there are no dashed lines in the extensive-form diagram). One can start by looking at the decision nodes that are immediate predecessors of only terminal nodes. At such a decision node, the game ends after the relevant player makes her choice and so, to determine the optimal action, there is no need to think about the behavior of other players. Essentially, the player on the move has a choice among some terminal nodes, and we assume that she will select the payoff-maximizing action. Let us call the other actions “dominated.”

The procedure then moves to evaluate decision nodes whose immediate successors are either terminal nodes or the nodes we already evaluated. From each node in this second class, the payoff consequences of each action are clear because the player on the move can anticipate how other players will behave later. The process continues all the way back to the initial node.
Backward induction procedure: This is the process of analyzing a game from the end to the beginning. At each decision node, one strikes from consideration any actions that are dominated, given the terminal nodes that can be reached through the play of the actions identified at successor nodes.

For a demonstration of backward induction, examine the game in Figure 15.2. There are two nodes at which player 1 makes a decision. At the second node, player 1 decides between $\mathrm{E}$ and $\mathrm{F}$. On reaching this node, player 1’s only rational choice is $\mathrm{E}$. We can therefore cross out $\mathrm{F}$ as a possibility. Player 2 knows this and, therefore, in her lower decision node she ought to select $C$ (which she knows will eventually yield the payoff 3 for her). We thus cross out action D for player 2. Furthermore, $\mathrm{A}$ is the best choice at player 2’s upper decision node, so we cross out $B$. Finally, we can evaluate the initial node, where player 1 has the choice between $U$ and D. He knows that if he chooses $U$, then player 2 will select $\mathrm{A}$ and he will get a payoff of 1 . If he chooses $\mathrm{D}$, then player 2 will select $\mathrm{C}$, after which he will select $\mathrm{E}$, yielding a payoff of 3 . Player 1’s optimal action at the initial node is therefore $\mathrm{D}$; action U should be crossed out.

离散数学代写

数学代写|离散数学作业代写discrete mathematics代考|Sequential Rationality

正如这些例子所表明的，比起玩家事先选择最佳对策(从游戏开始的角度来看)，我们应该更多地假设玩家。当玩家被要求做出决定时，他们应该表现出理性。这被称为顺序理性。

顺序理性:玩家的最佳策略应该最大化他或她的预期收益，这取决于玩家所处的每个信息集。也就是说，玩家i的策略应该从每个玩家i的信息集中指定一个最佳行动，即使是那些玩家i不相信(事先)会在游戏中达到的行动。
如果顺序理性是参与者之间的共同知识(在每个信息集)，那么每个参与者都将“向前看”，考虑玩家在特定信息集下的行动在未来会做出什么反应。
为了操作顺序理性的概念，我们必须处理一些复杂的问题，比如玩家在不同的信息集上对彼此的信念的限制。此外，还有一些版本建立在合理化(迭代优势)和纳什均衡的基础上。在过去的几十年里，研究人员基于对信念和最佳反应行为范围的不同假设，开发了许多改进。在本章中，我将向您介绍三个最基本的概念。关键的一种，称为子博弈完美纳什均衡，是最直接的，并在以下章节中得到广泛应用。

数学代写|离散数学作业代写discrete mathematics代考|Backward Induction

也许表示顺序合理性的最简单方法是通过一个程序，通过在游戏树中向后工作来确定每个信息集的最佳行动。例如，考虑一个具有完全信息的博弈，您可能还记得它只有单个信息集(在扩展形式图中没有虚线)。可以从仅作为终端节点的直接前身的决策节点开始。在这样的决策节点上，游戏在相关玩家做出选择后结束，因此，为了确定最佳行动，不需要考虑其他玩家的行为。从本质上讲，移动中的玩家可以在一些终端节点中做出选择，我们假设他会选择收益最大化的行动。让我们称其他行为为“受支配的”。

然后，该过程移动到评估决策节点，其直接后继节点要么是终端节点，要么是我们已经评估的节点。从第二类中的每个节点来看，每个行动的回报结果都是明确的，因为移动中的玩家可以预测其他玩家随后的行为。这个过程一直延续到初始节点。
逆向归纳过程:这是从头到尾分析游戏的过程。在每个决策节点上，考虑到可以通过在后继节点上确定的动作的发挥达到的终端节点，人们可以从考虑的任何占主导地位的动作中取出。

为了演示逆向归纳，请查看图15.2中的游戏。参与人1在两个节点上做出决定。在第二个节点，参与人1在$\mathrm{E}$和$\mathrm{F}$之间做出选择。在到达这个节点时，参与人1唯一的理性选择是$\ mathm {E}$。因此，我们可以划掉$\ mathm {F}$作为一种可能性。参与人2知道这一点，因此，在她的较低决策节点，她应该选择C(她知道这最终会给她带来收益3)。因此，我们划掉了玩家2的行动D。此外，$\mathrm{A}$是玩家2上决策节点的最佳选择，所以我们划掉$B$。最后，我们可以评估初始节点，参与人1可以在$U$和d之间做出选择，他知道如果他选择$U$，那么参与人2将选择$\ mathm {A}$，他将获得1的收益。如果他选择$\ mathm {D}$，那么参与人2将选择$\ mathm {C}$，之后他将选择$\ mathm {E}$，收益为3。因此，参与人1在初始节点的最优行为是$\ mathm {D}$;action U应该被划掉。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|离散数学作业代写discrete mathematics代考|The Median Voter Theorem

Posted on 2023年5月9日2023年5月9日 by statistics-lab

数学代写|离散数学作业代写discrete mathematics代考|The Median Voter Theorem

Consider an expanded version of the location-choice model from Chapter 8 , where two political candidates (players 1 and 2) decide where to locate on the political spectrum. Suppose the policy space is given by the interval $[0,1]$, with the location 0 denoting extreme liberal and 1 denoting extreme conservative. Citizens (the voters) are distributed across the political spectrum, not necessarily uniformly as was the case in the basic location game described in Chapter 8 .

Each citizen has an ideal point (a favorite policy) on the interval [0, 1]. Let function $F$ describe the distribution of the citizens. Specifically, for any location $x \in[0,1], F(x)$ is the fraction of the citizens whose ideal point is less than or equal to $x$. Assume that $F$ is a continuous function, with $F(0)=0$ and $F(1)=1$. Technically, this means that there is a “continuum” of citizens-an infinite number, smoothly distributed across the political spectrum.

Voting takes place after the candidates simultaneously and independently select their policy locations. Each citizen is assumed to vote for the candidate who is located closest to the citizen’s ideal point. A candidate wins by obtaining more votes than the other candidate does. Assume that if the candidates select exactly the same policy position, then the voters split evenly between them and they tie in the election, with the eventual winner determined by an unmodeled Supreme Court decision. Each candidate wants to win the election, so let us assume that a candidate obtains a payoff of 1 if he wins, 0 if he loses, and $1 / 2$ if there is a tie. ${ }^7$

数学代写|离散数学作业代写discrete mathematics代考|Strategic Voting

An underlying assumption of the analysis of candidates’ policy choices in the previous section is that each citizen votes for the candidate whose policy is closest to the citizen’s ideal point. Is it, however, safe to say that voters behave in this way? Would a sophisticated citizen always vote for her favorite candidate, even if this candidate were sure to lose the election? A story and numerical example will shed light on this issue.

On October 7, 2003, in a historic recall election, Gray Davis was removed from office as the governor of California. Popular confidence in Davis had been eroded in the preceding years by revelations of a severe imbalance in the state’s budget, a crisis in the electricity market, declining trust in government officials in general, and a growing sense that Davis lacked the gumption to lead. In fact, Davis’s approval ratings had been falling well before his reelection in November 2002 , but his shrewd and disciplined political machine was able to orchestrate his reelection anyway.

The recall election was highly unusual; many people considered its rules to be ill suited to modern political decision making. On the same ballot, voters were asked whether to remove Davis from office and then, conditional on the recall passing, they were asked to select among the-get this-135 registered replacement candidates. Among the field of replacement candidates were dozens of people who entered the race for publicity or just on a whim. Candidates included a star of pornographic movies, a former child star from a television series, and many others with colorful histories. The election was won by Arnold Schwarzenegger, a movie star and former bodybuilder, who had little political experience but excellent name recognition and was himself a savvy enough politician to later get reelected. ${ }^9$

离散数学代写

数学代写|离散数学作业代写discrete mathematics代考|The Median Voter Theorem

考虑第8章中位置选择模型的扩展版本，其中两个政治候选人(玩家1和玩家2)决定自己在政治光谱中的位置。假设策略空间由区间$[0,1]$给出，其中位置0表示极端自由，1表示极端保守。公民(投票人)分布在不同的政治范围内，并不一定像第8章所描述的基本区位博弈那样是均匀的。

每个公民在区间[0,1]上有一个理想点(最喜欢的政策)。设函数F描述公民的分布。具体来说，对于任意位置$x \in[0,1]， F(x)$是理想点小于或等于$x$的公民的百分比。假设$F$是一个连续函数，其中$F(0)=0$， $F(1)=1$。从技术上讲，这意味着有一个“连续体”的公民——一个无限的数字，平稳地分布在政治光谱中。

投票是在候选人同时独立选择他们的政策所在地之后进行的。假设每个公民都投票给最接近其理想点的候选人。一个候选人赢得比另一个候选人更多的选票。假设如果候选人选择完全相同的政策立场，那么选民就会平均分配给他们，他们在选举中打成平手，最终的赢家将由最高法院的一项未模拟的裁决决定。每个候选人都想赢得选举，所以我们假设候选人获胜的收益为1，失败的收益为0，平局时收益为1 / 2。${} ^ 7美元

数学代写|离散数学作业代写discrete mathematics代考|Strategic Voting

上一节对候选人政策选择分析的一个基本假设是，每个公民投票给政策最接近其理想点的候选人。然而，可以肯定地说，选民的行为是这样的吗?一个老练的公民会永远投票给她最喜欢的候选人，即使这个候选人肯定会输掉选举吗?一个故事和数值例子将阐明这个问题。

2003年10月7日，在一次历史性的罢免选举中，格雷·戴维斯被免去了加州州长的职务。在过去的几年里，由于国家预算严重失衡、电力市场危机、对政府官员的总体信任度下降，以及越来越多的人认为戴维斯缺乏领导的进取心，公众对戴维斯的信心受到了侵蚀。事实上，戴维斯的支持率在2002年11月再次当选之前就一直在下降，但他精明而有纪律的政治机器无论如何都能够精心策划他的再次当选。

罢免选举极不寻常;许多人认为它的规则不适合现代政治决策。在同一份投票中，选民被问及是否将戴维斯免职，然后，如果罢免通过，他们被要求从135名登记的替代候选人中选择。在替补候选人中，有几十人参加竞选是为了宣传，或者只是一时兴起。候选人包括一位色情电影明星，一位曾经出演过电视剧的童星，以及其他许多有着丰富历史的人。阿诺德·施瓦辛格赢得了选举，他是一位电影明星和前健美运动员，他几乎没有政治经验，但有很高的知名度，他本人也是一位足够精明的政治家，后来再次当选。${} ^ 9美元

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|离散数学作业代写discrete mathematics代考|Cournot Duopoly Model

Posted on 2023年5月9日2023年5月9日 by statistics-lab

数学代写|离散数学作业代写discrete mathematics代考|Cournot Duopoly Model

In the early 1800 s, Augustin Cournot constructed a model of the interaction between two firms, whereby the firms compete by choosing how much to produce. ${ }^1$ Here is a version of his model. Suppose firms 1 and 2 produce exactly the same good-that is, there is no product differentiation in the market, so consumers do not care from which firm they purchase the good. To be concrete, suppose the product is red brick. Simultaneously and independently, the firms select the number of bricks to produce. Let $q_1$ denote firm 1’s quantity and $q_2$ denote firm 2’s quantity, in thousands. Assume that $q_1, q_2 \geq 0$. The total output in the industry is then $q_1+q_2$. Assume that all of the brick is sold, but the price that consumers are willing to pay depends on the number of bricks produced. ${ }^2$ The demand for bricks is given by an inverse relation between quantity and price-an inverse relation is the norm in most markets (when price drops, consumers buy more). Suppose the price is given by the simple function $p=1000-q_1-q_2$. Also suppose each firm must pay a production cost of $\$ 100$ per thousand bricks. Firms wish to maximize their profits.

To compute the equilibrium of this market game, we start by specifying the normal form. Because each firm selects a quantity, we have $S_1=[0, \infty)$ and $S_2=[0, \infty)$. Each firm’s payoff is its profit, which is revenue (price times quantity) minus cost. Thus, firm 1’s payoff is
$$
u_1\left(q_1, q_2\right)=\left(1000-q_1-q_2\right) q_1-100 q_1
$$
and firm 2’s payoff is
$$
u_2\left(q_1, q_2\right)=\left(1000-q_1-q_2\right) q_2-100 q_2 .
$$

数学代写|离散数学作业代写discrete mathematics代考|Bertrand Duopoly Model

The Cournot model may seem a bit unreasonable because it has firms selecting quantities rather than prices. In reality, firms select both prices and quantities. But consumer demand implies a definite relation between these two variables, so firms can be thought of selecting one first (quantity or price) and then setting the other to whatever the market will bear.

The technology of production typically dictates whether the firm effectively commits to quantity or price. For example, in industries with long production cycles (such as automobiles, pharmaceuticals, and agricultural products), firms have to establish production targets well in advance of sales; later, prices adjust as the quantity supplied meets demand. Thus, the Cournot model is well suited to industries with long production cycles and/or short consumption cycles.
At the other extreme are industries in which firms can produce on short notice. For instance, once a piece of software is designed, it can be freely and instantaneously copied and transmitted to consumers in any quantity. In these markets, firms set prices and then produce to accommodate demand. We can analyze price-based competition using a variant of the Cournot model presented in the previous section. Suppose that the two firms simultaneously and independently set prices and then are forced to produce exactly the number of bricks demanded by customers at these prices. As before, industry demand is given by $p=1000-q_1-q_2$, which can be written as $Q=1000-p$, where $Q=q_1+q_2$. That is, facing price $p$, consumers demand $1000-p$ thousand units of brick. Let us assume that consumers purchase brick from the firm that charges the lower price. If the firms set equal prices, then suppose the demand is split evenly – that is, each firm sells $(1000-p) / 2$ thousand units. Assume the cost of production is 100 per thousand bricks, as before. A model like this one was analyzed by Joseph Bertrand in the late $1800 \mathrm{~s} .^4$

离散数学代写

数学代写|离散数学作业代写discrete mathematics代考|Cournot Duopoly Model

19世纪初，奥古斯汀·古诺(Augustin Cournot)构建了一个两家公司之间相互作用的模型，在这个模型中，公司通过选择生产多少来竞争。${ }^1$这是他的模型的一个版本。假设公司1和公司2生产完全相同的商品，也就是说，市场上不存在产品差异化，因此消费者不关心他们从哪家公司购买商品。具体来说，假设产品是红砖。同时独立地，公司选择生产砖块的数量。设$q_1$表示公司1的数量$q_2$表示公司2的数量，单位为千。假设$q_1, q_2 \geq 0$。该行业的总产出为$q_1+q_2$。假设所有的砖都卖出去了，但消费者愿意支付的价格取决于生产的砖的数量。${ }^2$对砖的需求是由数量和价格之间的反比关系给出的——反比关系是大多数市场的常态(当价格下降时，消费者购买更多)。假设价格由简单函数$p=1000-q_1-q_2$给出。同时假设每个企业必须支付每千块砖$\$ 100$的生产成本。公司希望利润最大化。

为了计算这个市场博弈的均衡，我们首先指定范式。因为每个公司选择一个数量，我们有$S_1=[0, \infty)$和$S_2=[0, \infty)$。每家公司的支付是它的利润，也就是收入(价格乘以数量)减去成本。因此，公司1的收益是
$$
u_1\left(q_1, q_2\right)=\left(1000-q_1-q_2\right) q_1-100 q_1
$$
公司2的收益是
$$
u_2\left(q_1, q_2\right)=\left(1000-q_1-q_2\right) q_2-100 q_2 .
$$

数学代写|离散数学作业代写discrete mathematics代考|Bertrand Duopoly Model

古诺模型可能看起来有点不合理，因为它让企业选择数量而不是价格。在现实中，企业既选择价格也选择数量。但是消费者需求暗示了这两个变量之间的明确关系，因此企业可以考虑先选择一个(数量或价格)，然后将另一个设定为市场所能承受的。

生产技术通常决定了企业是否有效地承诺数量或价格。例如，在生产周期较长的行业(如汽车、制药和农产品)，企业必须在销售之前制定生产目标;随后，价格随着供给量满足需求而调整。因此，古诺模型非常适合生产周期长和/或消费周期短的行业。
另一个极端是企业可以在短时间内生产的行业。例如，一旦一款软件被设计出来，它就可以被自由地、即时地复制并以任意数量传播给消费者。在这些市场中，企业设定价格，然后根据需求进行生产。我们可以使用前一节中介绍的古诺模型的一个变体来分析基于价格的竞争。假设这两家公司同时独立地设定了价格，然后被迫以这些价格生产顾客所需的砖的数量。如前所述，行业需求由$p=1000-q_1-q_2$给出，可以写成$Q=1000-p$，其中$Q=q_1+q_2$。也就是说，面对价格$p$，消费者需要$1000-p$千块砖。让我们假设消费者从价格较低的公司购买砖。如果两家公司定价相同，那么假设需求被平均分配——即每家公司销售$(1000-p) / 2$千件。如前所述，假设生产成本为每千块砖100元。约瑟夫·伯特兰在20世纪90年代后期分析了这样一个模型 $1800 \mathrm{~s} .^4$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|离散数学作业代写discrete mathematics代考|MPCS50103

Posted on 2023年4月3日2023年4月3日 by statistics-lab

如果你也在怎样代写离散数学discrete mathematics这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

离散数学是研究可以被认为是 “离散”（类似于离散变量，与自然数集有偏射）而不是 “连续”（类似于连续函数）的数学结构。离散数学研究的对象包括整数、图形和逻辑中的语句。相比之下，离散数学不包括 “连续数学 “中的课题，如实数、微积分或欧几里得几何。离散对象通常可以用整数来列举；更正式地说，离散数学被定性为处理可数集的数学分支（有限集或与自然数具有相同心数的集）。然而，”离散数学 “这一术语并没有确切的定义。

我们提供的离散数学discrete mathematics及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(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|离散数学作业代写discrete mathematics代考|One-to-One and Onto Functions

A function $f: X \rightarrow Y$ is said to be one-to-one or injection if and only if $f\left(x_1\right)=f\left(x_2\right)$ implies that $x_1=x_2$ for all elements in $X$. In other words, if at least two different elements in the domain of a function can be found that have the same element in the codomain, then the function is not one-to-one. Using quantifiers, a function $f$ is one-to-one if $\forall x_1 \forall x_2\left(f\left(x_1\right)=f\left(x_2\right) \rightarrow x_1=x_2\right)$ or equivalently, $\forall x_1 \forall x_2\left(f\left(x_1\right) \neq\right.$ $\left.f\left(x_2\right) \rightarrow x_1 \neq x_2\right)$, where $x_1$ are $x_2$ are in the domain of the function and $f\left(x_1\right)$ and $f\left(x_2\right)$ are in the codomain of the function.

A function $f: X \rightarrow Y$ is said to be onto or surjection if and only if, for every element $\gamma \in Y$, there is at least one element $x \in X$ with $f(x)=\gamma$. In other words, if the range and codomain are not the same, then the function is not onto. Using quantifiers, a function $f$ is onto if $\forall \gamma \exists x(f(x)=\gamma)$, where $x$ and $y$ are in the domain and codomain of the function, respectively.

A function $f: X \rightarrow Y$ is said to be one-to-one correspondence or bijection if and only if it is both one-to-one and onto. When a function is a one-to-one correspondence, the elements of its domain and codomain match up perfectly.
Example 7.7
Consider the four arrow diagrams in Fig. 7.3, where each represents a function. Identify the functions that are one-to-one and those that are onto.
Solution
Only the arrow diagrams in (a) and (c) are one-to-one functions, because in each of them, different elements of the domain have distinct images (i.e., no two values in the domain are assigned to the same function value). Only the arrow diagrams in (b) and (c) are onto functions, because in each of them, all elements in the codomain are images of elements in the domain. In summary, the function in (a) is oneto-one, but not onto, the function in (b) is onto, but not one-to-one, the function in (c) is both one-to-one and onto, i.e., one-to-one correspondence, and the function in (d) is neither one-to-one nor onto.

Geometrical characterization of one-to-one and onto functions can bring about meaningful insights. Consider functions of the form $f: \boldsymbol{R} \rightarrow \boldsymbol{R}$. The graphs of such functions can be plotted in the Cartesian plane using the set of ordered pairs $((a, b) \mid a \in \boldsymbol{R}$ and $f(a)=b$ ), where the graph of a function $f$ is an aid in understanding the behavior of the function. The concepts of being one-to-one, onto, and one-to-one correspondence have some geometrical meaning, which are as follows:

If the function $f: \boldsymbol{R} \rightarrow \boldsymbol{R}$ is one-to-one, then each horizontal line intersects the graph of the function $f$ in at most one point (i.e., the number of intersection points $\leq 1$ ).
If the function $f: \boldsymbol{R} \rightarrow \boldsymbol{R}$ is onto, then each horizontal line intersects the graph of the function $f$ in at one or more points (i.e., the number of intersection points $\geq 1$ ).
If the function $f: \boldsymbol{R} \rightarrow \boldsymbol{R}$ is a one-to-one correspondence, then each horizontal line intersects the graph of the function $f$ in at exactly one point (i.e., the number of intersection points $=1$ )

数学代写|离散数学作业代写discrete mathematics代考|Compositions of Functions

In addition to simple operations on functions, such as addition and multiplication, there is a fundamentally different way, called composition, to combine two functions so as to construct a new function.

Consider the function $f: X \rightarrow Y$ and the function $g: Y \rightarrow Z$. The composition of the functions $f$ and $g$, denoted by $g \circ f$ and read as ” $g$ circle $f$,” is a function from $X$ to $Z$, defined as follows:
$$
(g \circ f)(x)=g(f(x))
$$
In order to find $(g \circ f)(x)$, we first apply the function $f$ to $x$ to obtain $f(x)$, and then we apply the function $g$ to $f(x)$ to obtain $(g \circ f)(x)=g(f(x))$. Fig. 7.5 shows the composition of functions.

In general, the domain of the function $g$ need not be the same as the codomain of the function $f$. The composition of $g \circ f$ cannot be defined unless the range of the function $f$ is a subset of the domain of the function $g$. For instance, suppose the domain of the function $g$ is the set of positive real numbers if the range of the function $f$ is the set of positive integers, then $g \circ f$ can be defined; however, if the range of the function $f$ is the set of all integers, then $g \circ f$ cannot be defined.

At a system level, the composite function can be viewed as a system with two subsystems in series, where the output of the first subsystem forms the input of the second subsystem, and the composite function represents the system output $(g \circ f)(x)$ for the system input $x$.

Note that the order of the functions matters in a composition. Even if both $g \circ f$ and $f \circ g$ are defined, though, in general, we have $g \circ f \neq f \circ g$. In other words, the commutative law does not hold for the composition of functions.

离散数学代写

数学代写|离散数学作业代写discrete mathematics代考|One-to-One and Onto Functions

一个功能 $f: X \rightarrow Y$ 据说是一对一或注射当且仅当 $f\left(x_1\right)=f\left(x_2\right)$ 暗示 $x_1=x_2$ 对于所有元素 $X$. 换句话说，如果在函数域中至少可以找到两个不同的元素在共域中具有相同的元素，则该函数不是一对一的。使用量词，函数 $f$ 是一对一的如果 $\forall x_1 \forall x_2\left(f\left(x_1\right)=f\left(x_2\right) \rightarrow x_1=x_2\right)$ 或者等价地， $\forall x_1 \forall x_2\left(f\left(x_1\right) \neq f\left(x_2\right) \rightarrow x_1 \neq x_2\right)$ ，在哪里 $x_1$ 是 $x_2$ 是在功能域和 $f\left(x_1\right)$ 和 $f\left(x_2\right)$ 在函数的 codomain中。
一个功能 $f: X \rightarrow Y$ 对于每个元素，当且仅当当且仅当 $\gamma \in Y$ ，至少有一个元素 $x \in X$ 和 $f(x)=\gamma$. 换句话说，如果 range 和 codomain 不一样，那么这个函数就不会被调用。使用量词，函数 $f$ 如果 $\forall \gamma \exists x(f(x)=\gamma)$ ，在哪里 $x$ 和 $y$ 分别在函数的域和辅域中。
一个功能 $f: X \rightarrow Y$ 被称为一对一对应或双射当且仅当它既是一对一又是一一对应。当一个函数是对应时，其域和辅域的元素完美匹配。
示例 7.7
考虑图 7.3 中的四个箭头图，其中每个箭头代表一个函数。确定一对一的功能和相关的功能。
解决方案
只有 (a) 和 (c) 中的箭头图是一对一的函数，因为在它们各自中，域的不同元素具有不同的图像 (即域中没有两个值被分配给相同的函数值). 只有 (b) 和 (c) 中的箭头图指向函数，因为在它们的每一个中， codomain 中的所有元素都是 domain 中元素的图像。综上所述，(a)中的函数是一对一的，但不是一对应的，(b)中的函数是一一对应的，但不是一一对应的，(c)中的函数既是一对一的，又是一一对应的，即一对应，(d)中的函数既不是一一对应，也不是一一对应。
一对一和函数上的几何特征可以带来有意义的见解。考虑表单的功能 $f: \boldsymbol{R} \rightarrow \boldsymbol{R}$. 可以使用有序对集在笛卡尔平面上绘制此类函数的图形 $((a, b) \mid a \in \boldsymbol{R}$ 和 $f(a)=b)$ ，其中函数图 $f$ 有助于理解函数的行为。 being one-to-one、 onto、 one-to-one correspondence 等概念具有一定的几何意义，具体如下:

如果函数 $f: \boldsymbol{R} \rightarrow \boldsymbol{R}$ 是一对一的，那么每条水平线都与函数的图形相交 $f$ 在最多一个点（即交叉点的数量 $\leq 1$ ).
如果函数 $f: \boldsymbol{R} \rightarrow \boldsymbol{R}$ 在上，那么每条水平线都与函数的图形相交 $f$ 在一个或多个点（即交叉点的数量 $\geq 1)$.
如果函数 $f: \boldsymbol{R} \rightarrow \boldsymbol{R}$ 是一一对应的，那么每条水平线都与函数的图形相交 $f$ 在恰好一个点（即交叉点的数量 $=1$ )

数学代写|离散数学作业代写discrete mathematics代考|Compositions of Functions

除了对函数的简单操作 (例如加法和乘法) 之外，还有一种根本不同的方法，称为组合，将两个函数组合起来构造一个新函数。
考虑函数 $f: X \rightarrow Y$ 和功能 $g: Y \rightarrow Z$. 功能的组成 $f$ 和 $g$, 表示为 $g \circ f$ 并读作 ” $g$ 圆圈 $f$,” 是一个函数 $X$ 到 $Z$ ，定义如下:
$$
(g \circ f)(x)=g(f(x))
$$
为了找到 $(g \circ f)(x)$ ，我们首先应用函数 $f$ 到 $x$ 获得 $f(x)$ ，然后我们应用函数 $g$ 到 $f(x)$ 获得 $(g \circ f)(x)=g(f(x))$. 图 7.5 显示了函数的组成。
一般来说，函数的域 $g$ 不必与函数的密码域相同 $f$. 的组成 $g \circ f$ 不能定义，除非函数的范围 $f$ 是函数域的子集 $g$. 例如，假设函数的域 $g$ 是正实数集，如果函数的范围 $f$ 是正整数的集合，那么 $g \circ f$ 可以定义；但是，如果函数的范围 $f$ 是所有整数的集合，那么 $g \circ f$ 无法定义。
在系统层面，复合函数可以看作是一个有两个子系统串联的系统，其中第一个子系统的输出形成第二个子系统的输入，复合函数代表系统的输出 $(g \circ f)(x)$ 对于系统输入 $x$.
请注意，功能的顺序在组合中很重要。即使两者 $g \circ f$ 和 $f \circ g$ 被定义，虽然，一般来说，我们有 $g \circ f \neq f \circ g$. 换句话说，交换律不适用于函数的组合。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年4月3日2023年4月3日 by statistics-lab

如果你也在怎样代写离散数学discrete mathematics这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的离散数学discrete mathematics及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(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|离散数学作业代写discrete mathematics代考|Basic Definitions

A function associates each member in one set, say the set of all people living in the world, with a member in another set, say the set of all positive real numbers representing their heights. Let $X$ and $Y$ be two nonempty sets of real numbers. A function from $X$ to $Y$, denoted by $f: X \rightarrow Y$, is a relation from $X$ to $Y$, a subset of $X \times Y$, that must satisfy both of the following two requirements:

Every element in $X$ is related to some element in $Y$.
No element in $X$ is related to more than one element in $Y$.
A relation from $X$ to $Y$ that contains only one ordered pair $(x, y)$ for every element $x \in X$ defines a function $f$ from $X$ to $Y$ where $y \in Y$, as shown in Fig. 7.1. For instance, assuming $X$ and $Y$ are the set of real numbers, $y=x^2$ is a function, as it meets both requirements. However, $y^2=x$ is not a function because there is an element $x>0$ in $X$ that is related to the two elements $-\sqrt{x}$ and $\sqrt{x}$ in $Y$. Note that $f(x)$ does not mean $f$ times $x$; it simply means $f$ is a function of $x$, and $f(x)$, read as ” $f$ of $x$,” is just the value of $f$ at $x$. Although it is incorrect, we may call $f(x)$ the function. $f(x)$ is the short form for a function of $x$. Note that functions may also be called maps, mappings, or transformations.

With $f$ as a function from $X$ to $Y$, the set $X$ is the domain of the function $f$, and $Y$ is the codomain of $f$. Moreover, $y$ is the image of $x$, and $x$ is a preimage or an inverse image of $\gamma$. The range of $f$ is the set of all images of elements of $X$. Note that the codomain of a function is the set of all possible values (i.e., all elements of $Y$ ), and the range is the set of all values of $f(x)$ for $x \in X$; therefore the range is a subset of the codomain.

For a function of $y=f(x)$, the variable $x$ is called the independent variable, as it can have any value from its domain, and the variable $\gamma$ is called the dependent variable because its value solely depends on the value of $x$. If $f$ represents a system, then $\gamma$ is the output corresponding to the input $x$.

As a function is defined by its domain, codomain, and the mapping of elements of the domain to elements in the codomain, two functions are equal when they have the same domain, the same codomain, and the same mapping of elements in the domain. A function is called real valued if its codomain is the set of real numbers $\boldsymbol{R}$, and it is called integer

数学代写|离散数学作业代写discrete mathematics代考|Special Functions

Let the domain and codomain of the function $f$ be subsets of the set of real numbers. $f$ is called a nondecreasing function if $f\left(x_1\right) \leq f\left(x_2\right)$ and an increasing function if $f\left(x_1\right)f\left(x_2\right)$, whenever $x_1<x_2$ and both $x_1$ and $x_2$ are in the domain of $f$. Using quantifiers and assuming $x_1$ and $x_2$ are in the domain of the function, and $f\left(x_1\right)$ and $f\left(x_2\right)$ are in the codomain of the function, we have the following:

Nondecreasing function: $\forall x_1 \forall x_2\left(x_1<x_2 \rightarrow f\left(x_1\right) \leq f\left(x_2\right)\right)$.
Increasing function: $\forall x_1 \forall x_2\left(x_1<x_2 \rightarrow f\left(x_1\right)<f\left(x_2\right)\right)$.
Nonincreasing function: $\forall x_1 \forall x_2\left(x_1<x_2 \rightarrow f\left(x_1\right) \geq f\left(x_2\right)\right)$.
Decreasing function: $\forall x_1 \forall x_2\left(x_1f\left(x_2\right)\right)$.
Example 7.4
Consider the following four functions of the form $f: \boldsymbol{R} \rightarrow \boldsymbol{R}$. Determine the intervals for which they are nondecreasing, decreasing, nonincreasing, or increasing functions.
(a) $f(x)=x^2$
(b) $g(x)=e^{-x}$.
(c) $h(x)=x^3-x$.
(d) $k(x)=x^3$
Solution
The following categorizes each function over some intervals:
(a) It is decreasing over the interval $(-\infty, 0)$ and increasing over the interval $(0, \infty)$
(b) It is decreasing over the interval $(-\infty, \infty)$.
(c) It is increasing over the intervals $\left(-\infty,-\frac{1}{\sqrt{3}}\right)$ and $\left(\frac{1}{\sqrt{3}}, \infty\right)$ and decreasing over the interval $\left(-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)$.
(d) It is increasing over the interval $(-\infty, \infty)$.
The definition of a function in mathematics generally consists of just one formula. However, many practical functions reflecting real-world applications may consist of more than one formula, depending on the values of $x$. Such a function is called a piecewise defined function. Functions that are defined piecewise are written as if-then-else statements in most programming languages.

离散数学代写

数学代写|离散数学作业代写discrete mathematics代考|Basic Definitions

一个函数将一个集合中的每个成员（比如生活在世界上的所有人的集合) 与另一集合中的一个成员相关联，比如代表他们身高的所有正实数的集合。让 $X$ 和 $Y$ 是两个非空实数集。一个函数来自 $X$ 到 $Y$ ，表示为 $f: X \rightarrow Y$ ，是来自 $X$ 到 $Y$ ，的一个子集 $X \times Y$ ，必须同时满足以下两个要求:

中的每一个元素 $X$ 与某些元素有关 $Y$.
没有元素 $X$ 与一个以上的元素有关 $Y$.
来自的关系 $X$ 到 $Y$ 仅包含一个有序对 $(x, y)$ 对于每个元素 $x \in X$ 定义一个函数 $f$ 从 $X$ 到 $Y$ 在哪里 $y \in Y$ ，如图 7.1 所示。例如，假设 $X$ 和 $Y$ 是实数集， $y=x^2$ 是一个函数，因为它满足这两个要求。然而， $y^2=x$ 不是函数，因为有一个元素 $x>0$ 在 $X$ 这与两个元素有关 $-\sqrt{x}$ 和 $\sqrt{x}$ 在 $Y$. 注意 $f(x)$ 并不意味着 $f$ 次 $x$ ；它只是意味着 $f$ 是一个函数 $x$ ，和 $f(x)$ ，读作” $f$ 的 $x$ ，”只是价值 $f$ 在 $x$. 虽然不正确，但我们可以调用 $f(x)$ 功能。 $f(x)$ 是函数的缩写形式 $x$. 请注意，函数也可以称为映射、映射或转换。
和 $f$ 作为一个函数 $X$ 到 $Y$ ，集合 $X$ 是函数的域 $f$ ，和 $Y$ 是的密码域 $f$. 而且， $y$ 是图像 $x$ ，和 $x$ 是原像或反像 $\gamma$. 的范围 $f$ 是元素的所有图像的集合 $X$. 请注意，函数的陪域是所有可能值的集合（即，函数的所有元素 $Y)$ ，范围是所有值的集合 $f(x)$ 为了 $x \in X$ ；因此范围是密码域的一个子集。
对于一个函数 $y=f(x)$ ，变量 $x$ 被称为自变量，因为它可以具有来自其域的任何值，而变量 $\gamma$ 被称为因变量，因为它的值完全取决于 $x$. 如果 $f$ 代表一个系统，那么 $\gamma$ 是输入对应的输出 $x$.
由于一个函数是由它的域、辅域以及域的元素到辅域中的元素的映射来定义的，所以当两个函数具有相同的域、相同的辅域以及相同的域中的元素映射时，它们是相等的。如果一个函数的余域是实数集，则该函数称为实值函数 $\boldsymbol{R}$, 它被称为整数

数学代写|离散数学作业代写discrete mathematics代考|Special Functions

让函数的域和辅域 $f$ 是实数集的子集。 $f$ 称为非递减函数，如果 $f\left(x_1\right) \leq f\left(x_2\right)$ 和一个增加的功能，如果 $f\left(x_1\right) f\left(x_2\right)$ ，每当 $x_1<x_2$ 和两者 $x_1$ 和 $x_2$ 属于 $f$. 使用量词并假设 $x_1$ 和 $x_2$ 在函数域中，并且 $f\left(x_1\right)$ 和 $f\left(x_2\right)$ 在函数的codomain中，我们有以下内容:

非递减函数: $\forall x_1 \forall x_2\left(x_1<x_2 \rightarrow f\left(x_1\right) \leq f\left(x_2\right)\right)$.
递增函数: $\forall x_1 \forall x_2\left(x_1<x_2 \rightarrow f\left(x_1\right)<f\left(x_2\right)\right)$.
非增函数: $\forall x_1 \forall x_2\left(x_1<x_2 \rightarrow f\left(x_1\right) \geq f\left(x_2\right)\right)$.
递减函数: $\forall x_1 \forall x_2\left(x_1 f\left(x_2\right)\right)$.
示例 7.4
考虑表单的以下四个函数 $f: \boldsymbol{R} \rightarrow \boldsymbol{R}$. 确定它们是非递减、递减、非递增或递增函数的区间。
(A) $f(x)=x^2$
(二) $g(x)=e^{-x}$.
(C) $h(x)=x^3-x$.
(四) $k(x)=x^3$
解决方案
下面对某些区间内的每个函数进行分类：
(a) 它在区间内递减 $(-\infty, 0)$ 并在间隔内增加 $(0, \infty)$
(b) 在区间内递减 $(-\infty, \infty)$.
(c) 它在时间间隔内增加 $\left(-\infty,-\frac{1}{\sqrt{3}}\right)$ 和 $\left(\frac{1}{\sqrt{3}}, \infty\right)$ 并在间隔内减少 $\left(-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)$.
(d) 在区间内增加 $(-\infty, \infty)$.
数学中函数的定义通常只包含一个公式。然而，许多反映实际应用的实用函数可能包含多个公式，具体取决于 $x$. 这样的函数称为分段定义函数。在大多数编程语言中，分段定义的函数被编写为 ifthen-else 语句。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|离散数学作业代写discrete mathematics代考|MPCS50103

Posted on 2023年3月31日2023年3月31日 by statistics-lab

如果你也在怎样代写离散数学discrete mathematics这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的离散数学discrete mathematics及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(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|离散数学作业代写discrete mathematics代考|Propositional Equivalences

It is sometimes important to replace a logical statement with an equivalent statement in a mathematical argument. One method to determine whether two compound propositions are equivalent is to use well-known logical identities to establish new logical identities. This method is quite effective, especially when there are a large number of propositional variables involved. Table 1.12 presents some important logical equivalences involving the negation, conjunction, and disjunction operators. Of all logical equivalences, De Morgan’s laws are of great importance, as they have wide applications in logic. De Morgan’s laws state that (1) the negation of an “and” statement is logically equivalent to the “or” statement in which each component is negated, and (2) the negation of an “or” statement is logically equivalent to the “and” statement in which each component is negated.

Table 1.13 presents some important logical equivalences involving conditional and biconditional statements.To verify logical equivalences or simplify logical statements, truth tables can also be used, where in each row of the truth table, the truth value of one statement is the same as the truth value of the other statement.

A compound proposition that is always true, regardless of the truth values of the propositional variables (i.e., the compound proposition contains only $T$ in the last column of its truth table), is called a tautology. In other words, a tautology is an always-true proposition regardless of the truth values of the propositional variables. A statement whose form is a tautology is a tautological statement. Note that the compound propositions $p$ and $q$ are logically equivalent if $p \leftrightarrow q$ is a tautology. Some simple examples of tautology in English are “Parents are older than their children,” “You don’t give what you don’t have,” and “Dead people do not breathe.” A simple example of tautology in logic is $p \vee \bar{p}$.

A compound proposition that is always false, regardless of the truth values of the propositional variables (i.e., the compound proposition contains only $F$ in the last column of its truth table), is called a contradiction. In other words, a contradiction is an alwaysfalse proposition regardless of the truth values of the propositional variables. A statement whose form is a contradiction is a contradictory statement. Note that the negation of a tautology is a contradiction, and the negation of a contradiction is a tautology. Some simple examples of contradiction in English are “Some are more equal than others,” “Rich people need a tax cut because they do not have enough money,” and “Texting while driving reduces chances of having a car accident.” A simple example of contradiction in logic is $p \wedge \bar{p}$.

Note that a compound proposition that is neither a tautology nor a contradiction is called a contingency. In most practical applications and statements in logic, the proposition happens to be contingency. A statement whose form is a contingency is a contingent statement. Some simple examples of contingency in English are “Politicians are dishonest” and “People in this country are not racist.” A simple example of contingency in logic is $p \rightarrow \bar{p}$.

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

To understand predicate logic, we first need to understand the concept of a predicate. A predicate refers to the part of a sentence that attributes a property to the subject. For instance, in the sentence “The United States of America is a powerful country,” “The United States of America” is the subject, and the part of the sentence from which the subject has been removed (i.e., “is a powerful country”) is the predicate. Another example is the sentence ” $x$ represents the world population”, in which the variable ” $x$ ” is the subject and “represents the world population” is the predicate.

A predicate contains a finite number of variables and becomes a propositional statement when specific values are substituted for the variables. The domain, also known as the universe of discourse or the domain of discourse, is the set of all values of a variable that can replace it.

A predicate that involves just one variable may be denoted by $P(x)$. The statement $P(x)$ is said to be the value of the propositional function $P$ at $x$. A propositional function $P$, by itself, is neither true nor false. However, once a value from the domain has been assigned to the variable $x, P(x)$ becomes a propositional statement and thus has a truth value.

A predicate that involves $n$ variables is called an $\boldsymbol{n}$-ary predicate and denoted by $P\left(x_1, x_2, \ldots, x_n\right)$. Once a set of values has been assigned to the variables $x_1, x_2, \ldots, x_n, P\left(x_1, x_2, \ldots, x_n\right)$ has a truth value, as it is then a propositional statement. Note that a predicate with two variables is called a binary predicate.

离散数学代写

数学代写|离散数学作业代写discrete mathematics代考|Propositional Equivalences

在数学论证中用等价陈述代替逻辑陈述有时很重要。确定两个复合命题是否等价的一种方法是使用众所周知的逻辑恒等式来建立新的逻辑恒等式。这种方法非常有效，尤其是涉及到大量的命题变量时。表 1.12 给出了一些重要的逻辑等价，涉及否定、合取和析取运算符。在所有的逻辑等价中，德摩根定律非常重要，因为它们在逻辑上有广泛的应用。德摩根定律指出 (1) “和”陈述的否定在逻辑上等同于“或”陈述，其中每个成分都被否定，

表 1.13 给出了一些涉及条件语句和双条件语句的重要逻辑等价。为了验证逻辑等价或简化逻辑语句，也可以使用真值表，在真值表的每一行中，一个语句的真值与真值相同其他语句的值。

一个永远为真的复合命题，不管命题变量的真值如何（即复合命题只包含吨在其真值表的最后一列中）称为重言式。换句话说，重言式是一个永远真实的命题，无论命题变量的真值如何。形式为重言式的陈述是重言式陈述。注意复合命题p和q在逻辑上是等价的如果p↔q是重言式。英语同义反复的一些简单例子是“父母比他们的孩子年长”、“你不给你没有的东西”和“死人不呼吸”。逻辑重言式的一个简单例子是p∨p¯.

无论命题变量的真值如何，始终为假的复合命题（即复合命题仅包含F在其真值表的最后一列中）称为矛盾。换句话说，无论命题变量的真值如何，矛盾总是错误的命题。形式为矛盾的陈述是自相矛盾的陈述。请注意，重言式的否定是矛盾，矛盾的否定是重言式。英语中一些简单的矛盾示例是“有些人比其他人更平等”、“富人需要减税，因为他们没有足够的钱”和“开车时发短信减少发生车祸的几率”。逻辑矛盾的一个简单例子是p∧p¯.

请注意，既不是重言式也不是矛盾的复合命题称为偶然性。在大多数实际应用和逻辑陈述中，命题恰好是偶然的。形式为偶然性的陈述是偶然性陈述。英语中偶然性的一些简单例子是“政治家是不诚实的”和“这个国家的人不是种族主义者”。逻辑中偶然性的一个简单例子是p→p¯.

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

要理解谓词逻辑，我们首先需要理解谓词的概念。谓语是指句子中将属性赋予主语的部分。例如，在”The United States of America is a power country”这句话中， “The United States of America”是主语，句子中去掉主语的部分（即“is a power country” “) 是谓词。另一个例子是句子” $x$ 代表世界人口”，其中变量 “ $x$ “是主语，“代表世界人口”是谓语。
谓词包含有限数量的变量，并在用特定值代替变量时成为命题陈述。域，也称为论域或论域，是可以替代它的变量的所有值的集合。
只涉及一个变量的谓词可以表示为 $P(x)$. 该声明 $P(x)$ 被称为命题函数的值 $P$ 在 $x$. 命题函数 $P$ ，本身既不是真的也不是假的。但是，一旦将域中的值分配给变量 $x, P(x)$ 成为命题陈述，因此具有真值。
涉及的谓词 $n$ 变量称为 $n$-ary 谓词并表示为 $P\left(x_1, x_2, \ldots, x_n\right)$. 一旦将一组值分配给变量 $x_1, x_2, \ldots, x_n, P\left(x_1, x_2, \ldots, x_n\right)$ 具有真值，因为它是一个命题陈述。请注意，具有两个变量的谓词称为二元谓词。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年3月31日2023年3月31日 by statistics-lab

如果你也在怎样代写离散数学discrete mathematics这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的离散数学discrete mathematics及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(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|离散数学作业代写discrete mathematics代考|Basic Logical Operators

Logical operators, also known as logical connectives, are used to combine two or more simple propositions to form a compound proposition. A statement form or a propositional form is an expression consisting of propositional variables and logical operators.

The truth table for a given propositional form presents the truth values that correspond to all possible combinations of truth values for the propositional variables. Two compound propositions are called logically equivalent or simply equivalent if they have identical truth tables (i.e., they have the same truth values regardless of the truth values of its propositional variables). The notation ” $\equiv “$ denotes logical equivalence.

The negation of the proposition $p$, denoted by $\bar{p}$, is the statement “It is not the case that $p$.” The simple proposition $\bar{p}$, which is read as “not $p$,” has the truth value that is the opposite of the truth value of $p$. Table 1.1 presents the truth table for the negation of a proposition $p$, where it has two rows corresponding to the two possible truth values of $p$.

For instance, if $p$ denotes hope is a good thing, then $\bar{p}$ denotes it is false (or not true) that hope is a good thing or hope is not a good thing.

The conjunction of the two propositions $p$ and $q$, denoted by $p \wedge q$ and read as ” $p$ and $q$,” is a compound proposition that is true when both $p$ and $q$ are true and is false otherwise. For instance, the compound proposition “The sun is hot and water is a liquid” is true because both its simple propositions are true, and the compound proposition “2+ $2=4$ and the United States of America is a country with a very long history” is false because not both of its simple propositions are true. Note that the word “but” sometimes is used instead of the word “and” to show conjunction. As an example, in the propositional logic, the two statements “The United States of America is the most advanced country in the world but it was built on indigenous land” and “The United States of America is the most advanced country in the world and it was built on indigenous land” are equivalent. Table 1.2 presents the truth table for the conjunction of two propositions.

The disjunction of the two propositions $p$ and $q$, denoted by $p \vee q$ and read as ” $p$ or $q$,” is a compound proposition that is false when both $p$ and $q$ are false and is true otherwise. Note that the word “or” in the propositional logic is an inclusive or, meaning a disjunction is true when at least one of the two propositions is true. In other words, $p \vee q$ implies ” $p$ or $q$ or both”; that is, it is an inclusive disjunction. For instance, the proposition “It is August or it is sunny” is true in the month of August or when it is sunny. It is false if it is not August and also it is not sunny. Table 1.3 presents the truth table for the disjunction of two propositions.

数学代写|离散数学作业代写discrete mathematics代考|Conditional Statements

Let $p$ and $q$ be propositions. The conditional statement $p \rightarrow q$, read as “if $p$, then $q$ ” or ” $p$ implies $q, “$ is a compound proposition that is false when $p$ is true and $q$ is false and is true otherwise. Table 1.7 presents the truth table for the conditional statement. There are also other ways to express this conditional statement, such as ” $p$ is sufficient for $q$,” “a sufficient condition for $q$ is $p$,” ” $q$ is necessary for $p$,” or “a necessary condition for $p$ is $q . “$ In the implication $p \rightarrow q$, $p$ is called the hypothesis, the premise, or the antecedent, and $q$ is called the conclusion or the consequence. In an implication, the hypothesis and its conclusion are not required to have related subject matters.

If the implication is true, we do not automatically know that either the hypothesis or the conclusion is true. For instance, consider the conditional statement “If you obey the law, you never go to prison.” In this implication, if you obey the law, then you do not expect to go to prison. If you do not obey the law, you may or may not go to prison depending on other factors. However, if you do obey the law but you go to prison, you feel outraged. This last scenario corresponds to the case when $p$ is true, but $q$ is false, and thus the truth value of the conditional statement $p \rightarrow q$ is false.

From an implication $p \rightarrow q$, the following well-known conditional statements, whose truth tables are presented in Table 1.8 , can be made:

The converse of $p \rightarrow q$ is $q \rightarrow p$.
The inverse of $p \rightarrow q$ is $\bar{p} \rightarrow \bar{q}$.
The contrapositive of $p \rightarrow q$ is $\bar{q} \rightarrow \bar{p}$.
Noting that logically equivalent propositions have the same truth values regardless of the truth values of its propositional variables, the implication (original conditional statement) and its contrapositive are equivalent, and the converse and the inverse of a conditional statement are also equivalent. Some people mistakenly think that an implication and its converse mean the same thing as they usually say one to mean another. In fact, their truth tables are not identical.

离散数学代写

数学代写|离散数学作业代写discrete mathematics代考|Basic Logical Operators

逻辑运算符，也称为逻辑连接词，用于将两个或多个简单命题组合成一个复合命题。语句形式或命题形式是由命题变量和逻辑运算符组成的表达式。

给定命题形式的真值表显示了对应于命题变量真值的所有可能组合的真值。如果两个复合命题具有相同的真值表（即，无论其命题变量的真值如何，它们都具有相同的真值），则它们被称为逻辑等价或简单等价。符号”≡“表示逻辑等价。

命题的否定p, 表示为p¯, 是陈述“事实并非如此p” 简单的命题p¯，读作“不p, 的真值与的真值相反p. 表 1.1 给出了命题否定的真值表p，其中有两行对应于两个可能的真值p.

例如，如果p表示希望是好事，那么p¯表示希望是好事或希望不是好事是错误的（或不正确的）。

两个命题的结合p和q, 表示为p∧q并读作“p和q, 是一个复合命题，当两者都为真时p和q为真，否则为假。例如，复合命题“太阳是热的，水是液体”是真的，因为它的两个简单命题都是真的，而复合命题“2+2=4美利坚合众国是一个历史悠久的国家”是错误的，因为它的两个简单命题并非都是正确的。请注意，有时使用“but”一词代替“and”一词来表示连词。例如，在命题逻辑中，“美利坚合众国是世界上最先进的国家，但它是建立在土著土地上”和“美利坚合众国是世界上最先进的国家”这两个陈述它建在土著土地上”是等同的。表 1.2 给出了两个命题的合取真值表。

两个命题的分离p和q, 表示为p∨q并读作“p或者q, 是一个复合命题，当两者都为假时p和q是假的，否则是真的。请注意，命题逻辑中的“或”一词是包含性或，意思是当两个命题中至少有一个为真时，析取为真。换句话说，p∨q暗示 ”p或者q或两者”; 也就是说，它是一个包容性析取。例如，命题“It is August or it is sunny”在八月或晴天时为真。如果不是八月，也不是晴天，那是假的。表 1.3 给出了两个命题析取的真值表。

数学代写|离散数学作业代写discrete mathematics代考|Conditional Statements

让p和q成为命题。条件语句p→q, 读作“如果p，然后q“ 或者 ”p暗示q,“是一个复合命题，当p是真实的并且q为假，否则为真。表 1.7 给出了条件语句的真值表。这个条件语句还有其他的表达方式，比如”p足以q”，“的充分条件q是p,” ” q是必要的p”或“一个必要条件p是q.“言外之意p→q, p被称为假设、前提或前提，并且q称为结论或后果。言外之意，假设及其结论不需要有相关的主题。

如果蕴涵为真，我们不会自动知道假设或结论为真。例如，考虑条件语句“如果你遵守法律，你永远不会进监狱。” 言外之意，如果你遵守法律，那么你就不会坐牢。如果您不遵守法律，您可能会或可能不会入狱，具体取决于其他因素。但是，如果你遵守法律却进了监狱，你会感到愤怒。最后一个场景对应于以下情况p是真的，但是q是假的，因此条件语句的真值p→q是假的。

从言外之意p→q，可以做出以下众所周知的条件语句，其真值表如表 1.8 所示：

相反的p→q是q→p.
的倒数p→q是p¯→q¯.
的对立面p→q是q¯→p¯.
注意到逻辑上等价的命题无论其命题变项的真值如何，都具有相同的真值，因此蕴涵（原条件语句）与其对立命题是等价的，条件语句的逆命题和逆命题也是等价的。有些人错误地认为一个蕴涵和它的逆蕴涵义是同一个意思，因为他们通常说一个蕴涵另一个意思。事实上，它们的真值表并不相同。

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金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|离散数学作业代写discrete mathematics代考|MPCS50103

Posted on 2023年3月27日2023年3月27日 by statistics-lab

如果你也在怎样代写离散数学discrete mathematics这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的离散数学discrete mathematics及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(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|离散数学作业代写discrete mathematics代考|MPCS50103

数学代写|离散数学作业代写discrete mathematics代考|Finite-State Machines with No Output

A finite-state machine with no output, referred to as an FSA, is an abstract model of a machine that accepts input values but does not produce output values, yet it has a set of final states. In a state diagram for an FSA, the final states are shown by using double circles.

There are two types of finite-state automata: one is the deterministic FSA, where for each pair of state and input values there is a unique next state given by the transition function; the other is the nondeterministic FSA, in which there is a list of possible next states for each pair of input value and state. The adjective deterministic is often used to emphasize that an FSA is not nondeterministic.

A deterministic FSA $M=\left(S, I, f, s_0, F\right)$ is a model that consists of the following five characterizing parts:

A finite set $S$ of states.
A finite set $I$ of input alphabet.
A transition function $f: S \times I \rightarrow S$, which maps state-input pairs to states.
An initial state (or start state) $s_0$.
A subset $F$ of $S$ consisting of final states (or accepting states).
The transition function specifies the actions of the system. For instance, if a system with half a dozen states is in state $s_3$, and the system receives the input $i$ where $f\left(s_3, i\right)=s_4$, then the system will change to $s_4$. Note that state changes occur as a result of a sequence of input alphabets. When the input string consisting of single elements causes an FSA to land in a final state, the string is said to be recognized or accepted; otherwise, it is rejected by the automaton.

Two finite-state automata are called equivalent if they recognize the same language. However, two equivalent finite-state automata may have different numbers of states. As the memory space required to store an FSA with $n$ states is approximately proportional to $n^2$, it is thus important to construct an FSA with the fewest possible states among all finite-state automata equivalent to a given FSA. In addition, simplifying an FSA to have fewer states make it easier to write a computer algorithm based on it. In general, simplification of an FSA involves identifying equivalent states that can be combined while not affecting the action of the FSA on input strings. In summary, an equivalence relation on the set of states of the automaton is defined, and a new automaton whose states are the equivalence classes of the relation is formed.

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

As stated earlier, by assigning a value to the variable $x$, the propositional function $P(x)$ becomes a propositional statement with a truth value. Another way to obtain a proposition from a propositional function is to add quantifiers. For instance, the propositions “Few people are very compassionate,” “Some people are racist,” “All people are mortal,” “None of them are good,” “One even prime number exists,” and “Every day the sun rises” each contains a word indicating a quantity, such as “few,” “some,” “all,” “none,” “one,” and “every.” These words are called quantifiers, as each word reveals for how many elements a given predicate is true. In other words, quantification is a way to express the extent to which a predicate is true over a range of elements. There are two widely known quantifications in predicate logic, namely, universal quantification and existential quantification.

Universal quantification indicates that a predicate is true for every element under consideration. In other words, universal quantification asserts that a predicate is true for all values of a variable in a given domain. Because the domain specifies the possible values of a variable, by changing the domain, the meaning of the universal quantification of a predicate may change. For instance, if the domain consists of all real numbers greater than 1 , then the assertion that every number, say 2 , is greater than its inverse (i.e., $\frac{1}{2}$ ) is true, as we have $\frac{1}{2}<2$. However, if the domain changes and includes all positive real numbers, then the assertion that every number, say $\frac{1}{3}$, is greater than its inverse (i.e., 3) is false, as we have $\frac{1}{3}<3$.

The universal quantification of $P(x)$, which is the statement $P(x)$ for all values of $x$ in the domain, is denoted by $\forall x P(x)$. The symbol $\forall$ is called the universal quantifier and read as “for all” or “for every.” Note that if a domain is not specified when a universal quantifier is used, then the universal quantification of a statement is not defined. The statement $\forall x P(x)$ is defined to be true if and only if $P(x)$ is true for every $x$ in the domain, and it is defined to be false for at least one $x$ in the domain. A value of $x$ for which $P(x)$ is false is called a counterexample to the universal statement $\forall x P(x)$. Moreover, if the domain is empty, then $\forall x P(x)$ is true for any $P(x)$, as there exists no element $x$ in the domain for which $P(x)$ is false.

离散数学代写

数学代写|离散数学作业代写discrete mathematics代考|Finite-State Machines with No Output

没有输出的有限状态机，称为 FSA，是一种接受输入值但不产生输出值但具有一组最终状态的机器的抽象模型。在 FSA 的状态图中，最终状态使用双圆圈表示。
有两种类型的有限状态自动机：一种是确定性 FSA，其中对于每对状态和输入值，都有一个由转换函数给出的唯一下一个状态；另一种是非确定性 FSA，其中每对输入值和状态都有一个可能的下一个状态列表。形容词 deterministic 通常用于强调 FSA 不是非确定性的。
确定性 $\mathrm{FSA} M=\left(S, I, f, s_0, F\right)$ 是由以下五个特征部分组成的模型:

有限集 $S$ 状态。
有限集 $I$ 输入字母表。
一个过渡函数 $f: S \times I \rightarrow S$ ，它将状态输入对映射到状态。
初始状态 (或开始状态) $s_0$.
一个子集 $F$ 的 $S$ 由最终状态 (或接受状态) 组成。
转换函数指定系统的动作。例如，如果具有六个状态的系统处于状态 $s_3$ ，系统接收输入 $i$ 在哪里 $f\left(s_3, i\right)=s_4$ ，那么系统将变为 $s_4$. 请注意，状态变化是输入字母序列的结果。当由单个元素组成的输入字符串导致 FSA 进入最终状态时，该字符串被称为被识别或接受; 否则，它会被自动机拒绝。
如果两个有限状态自动机识别相同的语言，则它们被称为等价的。然而，两个等价的有限状态自动机可能有不同数量的状态。作为存储 FSA 所需的内存空间 $n$ 状态大约与 $n^2$ ，因此在等价于给定 FSA 的所有有限状态自动机中构建具有最少可能状态的 FSA 非常重要。此外，将 FSA 简化为具有更少的状态，可以更轻松地基于它编写计算机算法。通常，FSA 的简化涉及识别可以组合的等效状态，同时不影响 FSA 对输入字符串的操作。综上所述，在自动机的状态集上定义了一个等价关系，并形成了一个新的自动机，其状态是该关系的等价类。

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

如前所述，通过为变量赋值 $x$ ，命题函数 $P(x)$ 成为具有真值的命题陈述。从命题函数得到命题的另一种方法是添加量词。例如，“很少有人非常富有同情心”、“有些人是种族主义者”、“所有人都会死”、没有人是好人”、“存在一个偶数”、“每天太阳升起”等命题” each 包含一个表示数量的词，例如“few”、”some”、
“all”、”none”、”one”和”every”。这些词称为量词，因为每个词都揭示了给定谓词有多少元素为真。换句话说，量化是一种表达谓词在一定范围内为真的程度的方法。谓词逻辑中有两种广为人知的量化，即全称量化和存在量化。
通用量化表明谓词对于所考虑的每个元素都是正确的。换句话说，通用量化断言谓词对于给定域中变量的所有值都为真。因为定义域指定了一个变量的可能取值，通过改变定义域，谓词的全称量化的意义可能会改变。例如，如果域由大于 1 的所有实数组成，则断言每个数字 (例如 2 ) 都大于其倒数 (即， $\frac{1}{2}$ ) 是真的，因为我们有 $\frac{1}{2}<2$. 但是，如果域发生变化并包含所有正实数，则断言每个数字，比如说 $\frac{1}{3}$ ，大于它的倒数（即 3）是假的，因为我们有 $\frac{1}{3}<3$.
的通用量化 $P(x)$ ，这是声明 $P(x)$ 对于所有值 $x$ 在域中，表示为 $\forall x P(x)$. 符号 $\forall$ 被称为全称量词，读作 “for all”或“for every”。请注意，如果在使用全称量词时末指定域，则不会定义语句的全称量词。该声明 $\forall x P(x)$ 被定义为真当且仅当 $P(x)$ 对每一个都是真的 $x$ 在域中，并且它被定义为至少有一个是假的 $x$ 在域中。的价值 $x$ 为了哪个 $P(x)$ 是假的被称为全称命题的反例 $\forall x P(x)$. 此外，如果域为空，则 $\forall x P(x)$ 对任何情况都适用 $P(x)$ ，因为不存在元素 $x$ 在哪个领域 $P(x)$ 是假的。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写