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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• 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|Set Theory

A set is a fundamental building block in mathematics, and it is defined as a collection of well-defined objects. The elements in a set are of the same kind, and they are distinct with no repetition of the same element in the set. ${ }^2$ Most sets encountered in computer science are finite, as computers can only deal with finite entities. Venn diagrams ${ }^3$ are often employed to give a pictorial representation of a set, and they may be used to illustrate various set operations such as set union, intersection and set difference.

There are many well-known examples of sets including the set of natural numbers denoted by $\mathbb{N}$, the set of integers denoted by $\mathbb{Z}$, the set of rational numbers denoted by $\mathbb{Q}$, the set of real numbers denoted by $\mathbb{R}$ and the set of complex numbers denoted by $\mathbb{C}$.

A finite set may be defined by listing all of its elements. For example, the set $A={2,4,6,8,10}$ is the set of all even natural numbers less than or equal to 10 . The order in which the elements are listed is not relevant: i.e. the set ${2,4,6,8,10}$ is the same as the set ${8,4,2,10,6}$.
$$\left(\begin{array}{l} a \ b \end{array}\right)^A$$
Sets may be defined by using a predicate to constrain set membership. For example, the set $S={n: \mathbb{N}: n \leq 10 \wedge n \bmod 2=0}$ also represents the set ${2,4,6$, $8,10}$. That is, the use of a predicate allows a new set to be created from an existing set by using the predicate to restrict membership of the set. The set of even natural numbers may be defined by a predicate over the set of natural numbers that restricts membership to the even numbers. It is defined by
$$\text { Evens }={x \mid x \in \mathbb{N} \wedge \operatorname{even}(x)} .$$
In this example, even $(x)$ is a predicate that is true if $x$ is even and false otherwise. In general, $A={x \in E \mid P(x)}$ denotes a set $A$ formed from a set $E$ using the predicate $P$ to restrict membership of $A$ to those elements of $E$ for which the predicate is true.

The elements of a finite set $S$ are denoted by $\left{x_1, x_2, \ldots x_n\right}$. The expression $x \in S$ denotes that the element $x$ is a member of the set $S$, whereas the expression $x \notin S$ indicates that $x$ is not a member of the set $S$.

## 数学代写|离散数学作业代写discrete mathematics代考|Set Theoretical Operations

Several set theoretical operations are considered in this section. These include the Cartesian product operation; the power set of a set; the set union operation; the set intersection operation; the set difference operation; and the symmetric difference operation.

The Cartesian product allows a new set to be created from existing sets. The Cartesian ${ }^5$ product of two sets $S$ and $T$ (denoted by $S \times T$ ) is the set of ordered pairs ${(s, t) \mid s \in S, t \in T}$. Clearly, $S \times T \neq T \times S$ and so the Cartesian product of two sets is not commutative. Two ordered pairs $\left(s_1, t_1\right)$ and $\left(s_2, t_2\right)$ are considered equal if and only if $s_1=s_2$ and $t_1=t_2$.

The Cartesian product may be extended to that of $n$ sets $S_1, S_2, \ldots, S_n$. The Cartesian product $S_1 \times S_2 \times \ldots \times S_n$ is the set of ordered tuples $\left{\left(s_1, s_2, . ., s_n\right) \mid s_1 \in S_1, s_2 \in S_2, . ., s_n \in S_{\mathrm{n}}\right}$. Two ordered $n$-tuples $\left(s_1, s_2, \ldots, s_n\right)$ and $\left(s_1{ }^{\prime}, s_2{ }^{\prime}, \ldots, s_n{ }^{\prime}\right)$ are considered equal if and only if $s_1=s_1{ }^{\prime}, s_2=s_2{ }^{\prime}, \ldots$, $s_n=s_n{ }^{\prime}$.

The Cartesian product may also be applied to a single set $S$ to create ordered $n$ tuples of $S:$ i.e. $S^n=S \times S \times \ldots \times S$ ( $n$ times).
Power Set
The power set of a set $A$ (denoted by $\mathbb{P A}$ ) denotes the set of subsets of $A$. For example, the power set of the set $\mathrm{A}={1,2,3}$ has 8 elements and is given by
$$A={\emptyset,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}} .$$

There are $2^3=8$ elements in the power set of $A={1,2,3}$ and the cardinality of $\mathrm{A}$ is 3 . In general, there are $2^{|\mathrm{A}|}$ elements in the power set of $A$.

# 离散数学代写

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

$(a b)^4$

Evens $=x \mid x \in \mathbb{N} \wedge \operatorname{even}(x)$

## 数学代写|离散数学作业代写discrete mathematics代考|Set Theoretical Operations

$$A=\emptyset, 1,2,3,1,2,1,3,2,3,1,2,3 .$$

## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师