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

2023年4月3日

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

A relation $R$ may not have a desired property, such as reflexivity, symmetry, or transitivity. If there is a relation containing $R$ and having the desired property, then the smallest such relation is the closure of the relation $R$ with respect to the property. Assuming $R$ is a relation on a set $A$ with $n$ elements, the reflexive closure of $R$, the symmetric closure of $R$, and the transitive closure of $R$ exist. Moreover, these closures are also unique, in that there cannot be even two distinct reflexive closures, symmetric closures, or transitive closures of some relation.

The reflexive closure of the relation $R$ is the smallest relation $R_r$, such that $R \subset R_r$ and $R_r$ is reflexive on the set $A$. The relation $R_r$ is obtained by $\operatorname{simply~adding~to~} R$ all pairs of the form $(a, a)$ with $a \in A$ that do not already belong to $R$. In other words, the reflexive closure of $R$ is $R \cup \Delta_A$, where $\Delta_A={(a, a) \mid a \in A}$ is known as the diagonal relation on $A$.

The symmetric closure of the relation $R$ is the smallest relation $R_s$, such that $R \subset R_s$ and $R_s$ is symmetric on the set $A$. The relation $R_s$ is obtained by simply adding to $R$ all pairs in the form $(b, a)$ whenever $(a, b)$ belongs to $R$. In other words, the symmetric closure of $R$ is $R \cup R^{-1}$, where $R^{-1}={(b, a) \mid(a, b) \in R}$.

The transitive closure of the relation $R$ is the smallest relation $R_t$, such that $R \subset R_t$ and $R_t$ is transitive on the set $A$ with $n$ elements. Every possible matched pair of the form $(a, b) \leftrightarrow(b, c)$ is examined, and then make sure that the ordered pair $(a, c)$ is either in the relation or is added to the relation. Obviously, obtaining the transitive closure is more complicated than obtaining either the reflexive closure or the symmetric closure. The relation $R_t$ is obtained by simply including all pairs that belong to the relations $R, R^2=R \circ R, \ldots$, and $R^n=R^{n-1} \circ R$. In other words, the transitive closure of $R$ is $R \cup R^2 \cup \ldots \cup R^n$.

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

Having relations to order some or all of the elements of sets are often very much needed. A relation on the set of tasks required to build a house and a relation on the words listed in a dictionary are some examples of partial orderings. Let $R$ be a relation defined on a set $A$. $R$ is a partial order relation or partial ordering relation if and only if $R$ is reflexive, antisymmetric, and transitive, as these properties characterize relations that can be employed to order the elements of sets. For instance, the less-than-or-equal-to $(\leq)$ relation on a set of real numbers, the subset $(\subseteq)$ relation on the power set of sets, and the divisibility $($ I) relation on a set of positive integers are all partial orderings.

A set $A$ together with a partial ordering $R$ is called a partially ordered set, or poset, and is denoted by the pair $(A, R)$. Members of the set $A$ are called elements of the poset. Note that if $R$ is a partial order on a set $A$, the notation $a \leqslant b$ is sometimes used to indicate that $(a, b) \in R$ in an arbitrary poset $(A, R)$. Suppose that $R$ is a partial order relation on a set $A$. If $a, b \in A$ and either $a \leqslant b$ or $b \leqslant a$, then the elements $a$ and $b$ are called comparable; otherwise, they are incomparable. In other words, if $a, b \in A$ and neither $a \leqslant b$ nor $b \leqslant a$, then the elements $a$ and $b$ are called incomparable or noncomparable.

If $R$ is a partial order relation on a set $A$, and every pair of elements in $A$ is comparable, then $R$ is called a total order relation or linear order relation on $A$, and $A$ is called totally ordered set or linearly ordered set. For instance, the less-than-or-equal-to relation $R$ on the positive integers $A$ is a total order, for if $a$ and $b$ are integers, either $a \leq b$ or $b \leq a$, whereas the divisibility relation $R$ on the set of positive integers $A$ is a partial order, as it has both comparable elements, such as 4 and 8 , and incomparable elements, such as 5 and 9.

The directed graph for a finite poset can be simplified quite significantly. For instance, because a partial order is reflexive, each vertex has a loop, which can be deleted. In addition, all edges whose existence is implied by transitivity can be dropped. Moreover, if the remaining edges are drawn upward and all arrows are removed, the resulting diagram is called the Hasse diagram of a poset.To recover the directed graph of a relation from the Hasse diagram, these steps are required: first, reinsert the direction markers on the arrows making all arrows point upward, next add loops at each vertex, and finally for each sequence of arrows from one node to a second and that second node to a third, add an arrow from the first node to the third.

# 离散数学代写

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

simply adding to $R$ 所有形式的对 $(a, a)$ 和 $a \in A$ 还 不属于 $R$. 换句话说，自反闭包 $R$ 是 $R \cup \Delta_A$ ，在哪里 $\Delta_A=(a, a) \mid a \in A$ 被称为对角线关系 $A$.

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。